MATH STUDENT BOOK. 6th Grade Unit 8

MATH STUDENT BOOK

6th Grade | Unit 8

Unit 8 | Geometry and Measurement

MATH 608 Geometry and Measurement INTRODUCTION |3

1. PLANE FIGURES

5

PERIMETER |5 AREA OF PARALLELOGRAMS |11 AREA OF TRIANGLES |17 AREA OF COMPOSITE FIGURES |21 AREA OF CIRCLES |27 PROJECT: ESTIMATING AREA |32 SELF TEST 1: PLANE FIGURES |36

2. SOLID FIGURES

39

SOLID FIGURES |39 SURFACE AREA OF RECTANGULAR PRISMS |47 VOLUME OF RECTANGULAR PRISMS |51 FINDING MISSING DIMENSIONS |56 PROJECT: TRIANGULAR PRISMS |60 SELF TEST 2: SOLID FIGURES |64

3. REVIEW

66

GEOMETRY AND MEASUREMENT |66 PLANE FIGURES |67 GLOSSARY |73

LIFEPAC Test is located in the center of the booklet. Please remove before starting the unit. |1

Geometry and Measurement | Unit 8

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Unit 8 | Geometry and Measurement

Geometry and Measurement Introduction In this unit, you will be introduced to the topics of geometry and measurement. You will learn about plane figures and solid figures and how they are measured. You will find that length is measured in various units, while area is measured in square units and volume is measured in cubic units. For plane figures, you will measure the perimeter and area of rectangles, parallelograms, triangles, circles, and composite figures. For solid figures, you will measure surface area and volume. You will use nets, two-dimensional views, and three-dimensional views to help visualize the figures. For each of these measurements, you will arrive at a general formula that will work for any rectangular prism. These basic tools will be useful in your future explorations in geometry.

Objectives Read these objectives. The objectives tell you what you will be able to do when you have successfully completed this LIFEPAC. When you have finished this LIFEPAC, you should be able to: z Find the perimeter of a polygon. z Review finding the circumference of a circle. z Find the area of a parallelogram, a triangle, a

circle, and simple composite figures. z Classify solid figures.

z Find the surface area and volume of a

rectangular prism. z Find a missing dimension of a rectangular

prism, given the surface area or volume. z Use correct units for measurement.

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Geometry and Measurement | Unit 8

Survey the LIFEPAC. Ask yourself some questions about this study and write your questions here.

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Unit 8 | Geometry and Measurement

1. PLANE FIGURES PERIMETER Bob bought a house on a large lot. He’s decided to install a fence around the property. How many feet of fence does he need? What information does Bob need to decide how much fence he needs?

Bob needs to find the perimeter of his property to know how many feet of fence to buy. In this lesson, you will learn how to find perimeter for different figures and solve real-life problems such as Bob’s. You will also review finding the circumference of a circle.

Objectives Review these objectives. When you have completed this section, you should be able to: z Find

the perimeter of a polygon.

z Review z Find

how to find the circumference of a circle.

the area of a parallelogram.

z Understand

the relationship between the area of parallelograms and triangles.

z Find

the area of a triangle.

z Find

the area of simple composite figures.

z Find

the area of a circle.

z Estimate

the area of irregular figures.

Vocabulary area. The measurement of the space inside a plane figure. base. The length of a plane figure. circumference. The distance around the outside of a circle. composite figure. A geometric figure that is made up of two or more basic shapes. diameter. The distance across a circle through the center. height . The perpendicular width of a plane figure. perimeter. The distance around the outside of a plane figure. pi. The ratio of the circumference of a circle to its diameter; approximately 3.14. radius. The distance from the center of a circle to any point on the circle. semicircle. One half of a circle, divided by the diameter. square units. The unit of measure for area. trapezoid. A quadrilateral with one pair of parallel sides. Note: All vocabulary words in this LIFEPAC appear in boldface print the first time they are used. If you are not sure of the meaning when you are reading, study the definitions given.

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Geometry and Measurement | Unit 8

Perimeter is a measurement of length because it’s the distance around a figure. You could think of it as the length you would walk if you could walk around a figure.

4 feet

4 feet

feet 44 feet

22feet feet

2 feet

2 feet

2 feet 2 feet

To find perimeter, we need to know the length of each side of the figure. Then, we can add the side lengths. In the rectangle above, we can see that the lengths of the sides are 2 feet, 4 feet, 2 feet, and 4 feet. So, the perimeter is 12 feet: 2 feet + 4 feet + 2 feet + 4 feet = 12 feet

44feet feet

To find the perimeter of Bob’s property, we just need to know the lengths of each side of the lot.

Did you know? Perimeter comes from the Greek perimetros: from peri-, meaning “around,” and metron, meaning “measure.” 100 feet

Example: Bob wants to build a fence around his property. How many feet of fencing does he need?

110 feet

Solution:

150 feet

To find the perimeter, we will add the side lengths. 100 feet + 150 + feet + 105 feet + 60 feet + 110 feet = 525 feet

60 feet 105 feet

So, Bob will need to buy 525 feet of fencing. Key point!

Always label the units for any measurement. Doing this will make it clear what type of measurement it is (length, area, volume), as well as what units were used.

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Unit 8 | Geometry and Measurement

60 feet

60 feet

any other regular polygon, we could shorten the process of finding the perimeter. We could add the side lengths to find the perimeter.

60 feet

60 feet

60 feet

Did you notice that the shape of Bob’s property is a pentagon? If it were a regular pentagon, or

60 ft + 60 ft + 60 ft + 60 ft + 60 ft = 300 ft However, since we know that a regular pentagon has five congruent sides, we can multiply one side length by five. 5(60 ft) = 300 ft Since any regular polygon has congruent sides, we can just multiply the number of sides by the side length (s).

Example: Find the perimeter of each regular polygon below.

Solution: To find the perimeter, we will multiply the number of sides by s, the side length. Regular Octagon A regular octagon has eight congruent sides, so we will use the formula 8s to find the perimeter. 8s = perimeter 8(2 m) = 16 m Regular Hexagon

2m

A regular hexagon has six congruent sides, so we will use the formula 6s to find the perimeter. 6s = perimeter 6(4 feet) = 24 feet Square

4 ft

A square has four congruent sides, so we will use the formula 4s to find the perimeter. 4s = perimeter

3 in

4(3 inches) = 12 inches = 1 foot

Section 1 |7

Geometry and Measurement | Unit 8

A square is a type of rectangle. Earlier, we found the perimeter of a rectangle, but we can shorten that process also.

We can multiply the length plus the width by two: (4 feet + 2 feet) + (4 feet + 2 feet) =

Remember that the opposite sides of a rectangle are congruent. So, instead of adding the length and the width two times:

2(4 feet + 2 feet) = 2(6 feet) = 12 feet For any rectangle, if l is the length and w is the width, the perimeter (p) is found using the formula p = 2(l + w).

4 feet + 2 feet + 4 feet + 2 feet = 12 feet

4 feet 2 feet

2 feet 4 feet

Example: Bob plans to add a rectangular swimming pool to his property. It will be 25 feet wide and 50 feet long. He wants to include a border of dark blue tile around the edge of the pool. How many feet of tile will he need?

50 feet 25 feet

Solution: We need to find the distance around the pool: the perimeter. Since the pool is rectangular, we will use the formula p = 2(l + w) to find the perimeter. p = 2(l + w) p = 2(50 feet + 25 feet) Substitute the length and width. p = 2(75 feet)

Add.

p = 150 feet

Multiply.

So, Bob will need 150 feet of tile.

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Unit 8 | Geometry and Measurement

CIRCUMFERENCE Aside from polygons, we can also find the perimeter of a circle, called the circumference. We c d

know that pi (π) is the ratio of the circumference to the diameter (π__). So, the circumference is pi

times the diameter: C = πd

Example:

6 feet

Bob has decided to include a circular whirlpool next to the pool, with the same dark blue tile around the edge. If the pool is 6 feet across, how many feet of tile will he need?

Solution: We will use the formula C = πd to find the circumference. We know that the diameter of the pool is 6 feet, and we will use 3.14 to approximate pi. C = πd

S-T-R-E-T-C-H

C = 3.14(6 feet)

Now that Bob knows the length of tile he needs, he can calculate the cost. If each tile was 4 inches long and cost $1.79 each, can you find out how much the tiles for the pool and whirlpool will cost?

C = 18.84 feet So, Bob will need about 19 feet of tile for the whirlpool.

Let’s Review! Before going on to the practice problems, make sure you understand the main points of this lesson.

99Perimeter is the distance around a figure. It is found by adding the side lengths. 99If the figure is a regular polygon, we can multiply the number of sides by the side length. 99The perimeter of a circle is called the circumference. It is found using the formula C = πd.

Section 1 |9

Geometry and Measurement | Unit 8

Match the following items. 1.1 _________ the distance around the outside of a circle

a. circumference



b. perimeter

_________ the distance around the outside of a plane figure

Circle the letter of each correct answer. 1.2_ A square has a side length of 5 inches. What is the perimeter of the square? a. 5 inches b. 10 inches c. 15 inches d. 20 inches 1.3_ A rectangle is 6 meters long and 4 meters wide. What is the perimeter of the rectangle? a. 10 meters b. 20 meters c. 10 square meters d. 20 square meters 1.4_ What is the perimeter of this figure? a. 16 cm b. 46 cm c. 48 cm d. 110 cm

4 cm

5 cm

8 cm 8 cm

5 cm 16 cm

1.5_ What is the perimeter of a regular hexagon if you know that one side is 4 cm long? a. 24 cm b. 20 cm c. 32 cm d. can’t be determined 1.6_ A square has a perimeter of 36 inches. How long is each side? a. 4 inches b. 6 inches c. 9 inches

d. 12 inches

1.7_ The length and width of each rectangle is given. Which rectangle will not have the same perimeter as the others? a. l = 12, w = 12 b. l = 14, w = 9 c. l = 8, w = 16 d. l = 13, w = 11 1.8_ What is the perimeter of this figure? a. 40 feet b. 36 feet c. 32 feet d. 28 feet

10 feet

10 feet

2 feet

2 feet 16 feet

1.9_ What is the circumference of a circle with a diameter of 100 m? a. 100 m b. 157 m c. 300 m

d. 314 m

Answer true or false. 1.10 _______________ The circumference of a circle with diameter of 6 inches will be greater than the perimeter of a square with side length 6 inches.

10| Section 1

Unit 8 | Geometry and Measurement

AREA OF PARALLELOGRAMS Each of these figures is a parallelogram. We know that they have several things in common (opposite sides are parallel and congruent). There are also some differences (right angles, number of congruent sides). So, is finding the area for each parallelogram the same, or different? In this lesson, we will discuss what area is and how it is measured, and we will explore the area of a parallelogram and how to find it.

AREA OF RECTANGLES Area is the amount of space that a plane figure takes up. How do we measure, or count, the amount of space? The number of square spaces or units that are inside the figure is our measure of its area. So, area is measured in square units. Can you find the area of the rectangle? How many squares are inside?

If the rectangle’s length and width are measured in inches, then each square inside the rectangle is 1 inch long and 1 inch wide: 1 square inch. The area is 8 square inches. 1 in 1 in 1 in2

If we count the squares, we can see that there are 8. So, the rectangle’s area is 8 square units. We use the same measures for area as we do for length: feet, inches, meters, etc. However, we refer to square feet, square inches, square meters.

Section 1 |11

Geometry and Measurement | Unit 8

Can you find the area of this rectangle? 8 feet

8 squares in each row

7 feet 7 rows

We could count each of the squares inside the rectangle one at a time to find the area, but there is an easier way. Notice that there are 8 square feet in each row of the rectangle because it is 8 feet long. There are 7 rows of 8 squares because the rectangle is 7 feet wide.

We can multiply 8 by 7 to find the number of squares. 8 × 7 = 56 So, the area of the rectangle is 56 square feet. So, to find the area (A) of any rectangle, we can just multiply the length (l) by the width (w). A=l×w It is important to understand area because it comes up in everyday situations.

Key point! We could also have looked at the rectangle as 8 columns of 7 squares each: 7 × 8 = 56. Remember, multiplication is commutative; order does not matter: 7×8=8×7

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Unit 8 | Geometry and Measurement

6.5 ft.

Example: Bob wants to paint a wall in his kitchen. He needs to know the area of the wall so he can decide how much paint to buy. The wall is 8 feet high and 6.5 feet long. What is the area of the wall?

Solution:

8 ft.

To find the area of the wall, we will multiply the length by the width because the wall is a rectangle. A = lw The length of the wall is 6.5 feet. Its width is 8 feet. 6.5 feet × 8 feet = 52 ft2 So, Bob will need enough paint to cover 52 ft2.

Did you know? Units of area are often written as exponents: ft2, m2, cm2... When we find area we multiply length times width, so just as 4 × 4 = 42, or 4 squared, feet × feet = ft2, or square feet.

Key point! Always label the units for any measurement. Doing this will make it clear what type of measurement it is (length, area, volume), as well as what units were used. You can tell right away if the measurement is area if the units are square units. Even though the wall is not made up entirely of squares, the area will be 52 square feet. This is because the half-squares can be added up to form whole squares. In this case, there are 8 halfsquares, which is the equivalent of 4 whole squares.

Section 1 |13

Geometry and Measurement | Unit 8

Example: Bob has another wall he’d like to paint, but the label on the can of paint says it can cover only 120 square feet. If the wall is 8 feet high, how long of a wall could he paint?

Solution: Using the formula A = lw, we can solve for the length of the wall. A = lw 120 ft2 = l(8 ft)

Substitute known measures.

We know that 8 feet multiplied by another length of feet is 120 ft2. So, if we divide 120 ft2 by 8 ft, we’ll find the length. 120 ft2 ÷ 8 ft = l 15 ft = l Bob has enough paint as long as the wall is less than 15 feet long.

AREA OF PARALLELOGRAMS We know that a rectangle is a parallelogram, so can we find the area of a parallelogram the same way? How many square units do you think there are in this parallelogram?

So, the parallelogram has an area of 8 square units. Notice that this is the same area as the first rectangle we looked at. Also, notice that each figure is 4 units long and 2 units tall.

2 cm 4 cm

Not all of the area is whole squares, but we can combine some pieces to make whole squares. There are 6 whole squares, and 4 half squares. 1 2

1 2

1 2

2 cm

1 2

6 + __ + __ + __ + __ 1 2

1 2

1 2

1 2

Add. 6+1+1=8

Using this method, we can find the area of a parallelogram in a similar way to finding the area of a rectangle. So, the area (A) of a parallelogram is the base (b) multiplied by the height (h). A = bh

14| Section 1

A = bh = (4 cm)(2 cm) = 8 cm2

4 cm

Group halves together to make 1. 6 + (__ + __) + (__ + __)

A = bh = (4 cm)(2 cm) = 8 cm2

Unit 8 | Geometry and Measurement

Example: Sue is re-tiling her bathroom. She is considering these two tiles whose area is measured in square inches. She would like to use the larger tile so that she will need to buy fewer tiles. Which tile should she buy?

3 in 3 in

5 in 2 in

Solution: We need to find the area of each parallelogram to see which tile is larger. We will multiply the base by the height to find the area. A = bh

This might help!

The base of the first parallelogram is 3 inches. Its height is also 3 inches.

For rectangles, we use length and width to name the dimensions. For parallelograms, and other figures, we use base and height. length

3 inches × 3 inches = 9 in2 The base of the second parallelogram is 5 inches. Its height is 2 inches. 5 inches × 2 inches = 10 in2 So, Sue should buy the second tile because it has a larger area.

side length

height base

Base is the length of the bottom side, not the length of the entire figure. Height is the vertical width of the figure, not the side length.

Let’s Review! Before going on to the practice problems, make sure you understand the main points of this lesson.

99Area is the space inside a figure and is measured in square units. 99Area for a parallelogram is found by multiplying the base by the height.

Section 1 |15

Geometry and Measurement | Unit 8

Match these items. 1.11 _________ the measurement of the space inside a plane figure

a. square units



_________ the length of a plane figure

b. base



_________ the perpendicular width of a plane figure

c. area



_________ the unit of measure for area

d. height

Circle the letter of each correct answer. 1.12_ A rectangle is 6 centimeters long and 4 centimeters wide. What is the area of the rectangle? b. 20 cm2 c. 24 cm2 d. 48 cm2 a. 10 cm2 1 2

1.13_ What is the area of a sheet of binder paper? (Binder paper is 8__ inches by 11 inches.)

a. 88 inches

1 2

b. 88__ in2

1.14_ Which of the following is a unit of area? b. feet a. cm2

1 2

c. 93__ in2 c. m3

4 feet 1.15 What is the area of this parallelogram if each square is 1 square foot? 3 feet a. 12 b. 12 feet d. 12 ft3 c. 12 ft2 1.16_ What is the area of a parallelogram with a height of 4 inches and a base of 5 inches? c. 25 in a. 20 in b. 20 in2

d. 94 in2 d. inches

d. 25 in2

1.17_ A parallelogram has an area of 48 m2. If the base is 12 m long, what is the height? a. 4 m b. 8 m c. 12 m d. 36 m 1.18_ Which figure has sides that are perpendicular to each other?

a.

c.

b.



d.

1.19_Parallelogram A has a base of 4 cm and a height of 9 cm. Which figure described below has the same area as parallelogram A? a. A rectangle 6 cm long and 5 cm wide. b. A parallelogram with a base of 8 cm and a height of 4 cm. c. A rectangle 9 cm long and 4 cm wide. d. A parallelogram with a base of 5 cm and a height of 8 cm. 16| Section 1

Unit 8 | Geometry and Measurement

AREA OF TRIANGLES You know that area is measured in square units. Can you find the area of this triangle? It would be a little tricky to count the whole square and parts of squares accurately.

We can find the area of the first triangle we looked at by thinking of it as half of a parallelogram.

Luckily, there is an easier way! In this lesson we will explore finding the area of a triangle.

base: Counting the units, we can see that the base of the parallelogram is 5 units long. To find the area of a triangle, we will use our knowledge of the area of a parallelogram. We can make a parallelogram with two congruent triangles, if the triangles share one corresponding side:

height: Counting the vertical units, we can see that parallelogram has a height of 2 units. The area of the parallelogram is 10 square units, multiplying the base by the height (5 × 2 = 10). The area of the triangle is half of that: 1 __ (10) = 5 2

So, the area of the triangle is 5 square units.

Since we know how to find the area of a parallelogram, we can use that information to find the area of a triangle. Since the triangles that make the parallelogram are congruent, their areas are equal. So, 1 the area of each triangle is __ the area of the 2 parallelogram: We know that the area of a parallelogram is found by multiplying the base by the height:

4 cm

4 cm 3 cm

3 cm

4 cm

4 cm

3 cm

3 cm

Remember that the height of a figure is the vertical width. Each of these triangles has a height of 4 cm: In fact, each of these triangles has the same area because they all have the same width!

1 bh A = __ 2

Substitute the base and height.

A = bh



1 2

A = __(3 cm)(4 cm)

Multiply base times height. 1 __ 2

+

1 __ 2

=

1

1

Since the area of the triangle is __ the area of 2 the parallelogram, the formula for the area of the triangle is: 1 A = __bh 2



1 2

A = __(12 cm2)

Divide by 2. A = 6 cm2 So, each triangle has an area of 6 square centimeters.

Section 1 |17

Geometry and Measurement | Unit 8

FIND THE AREA OF TRIANGLES Example:

Think about it!

There are three slices of pizza for Chris and his two friends. If Chris wants the largest slice, which slice should he choose?

Notice that the height of a right triangle (half of a rectangle) is the same as the side length. This is because that side of the triangle is vertical.

Solution:

We need to find the area of each slice to find which is largest. We will use the formula for area of a triangle. First slice: 1 2 1 __ A = (3 in)(8 in) 2 1 __ A = (24 in2) 2

A = __bh

A = 12 in2

8 in Substitute the base and height. Multiply base times height. Divide by 2. 3 in

Second slice: 1 2 1 __ A = (4 in)(6 in) 2 1 __ A = (24 in2) 2

A = __bh

A = 12 in2

Substitute the base and height.

6 in

Multiply base times height. Divide by 2.

4 in

Third slice: 1 2 1 A = __(3.5 in)(7 in) 2 1 A = __(24.5 in2) 2

A = __bh

A = 12.25 in2

Substitute the base and height. Multiply base times height. Divide by 2.

So, the third slice is the largest, but just barely!

18| Section 1

7 in

3.5 in

Unit 8 | Geometry and Measurement

Key point! Always label the units for any measurement. Doing this will make it clear what type of measurement it is (length, area, volume), as well as what units were used. You can tell right away if area is being measured if the units are square units. If we know the area of a triangle and either the base or the height, we can solve for the other.

Example: Sarah has a crowded backyard, but she thinks she has room for a triangular garden. There is room for the triangular garden to be 4 feet wide. She would like the garden to have an area of 17 square feet. How long should the garden be?

Solution: We can use the formula for the area of a triangle and solve for the length of the garden. The garden is 4 feet wide, so that is the base. A = bh 17 ft2 = (4 ft)h Substitute the base and area. 17 ft2 = (2 ft)h Divide by 2. 2 ft times the height is 17 ft2. So, if we divide 17 ft2 by 2 ft, we’ll find the height. 17 ft2 ÷ 2 ft = 8.5 ft So, Sarah’s garden needs to be 8.5 feet long.

Let’s Review! Before going on to the practice problems, make sure you understand the main points of this lesson.

99Any parallelogram can be divided into two congruent triangles. 99The area for a triangle is found by taking one half of the base multiplied by the height: 1 __ bh 2

Match each word to its definition. 1.20 _________ the measurement of the space inside a plane figure

_________ the length of a plane figure



_________ the perpendicular width of a plane figure



_________ the unit of measure for area

a. height b. base c. area d. square units

Section 1 |19

Geometry and Measurement | Unit 8

Circle the letter for each correct answer. A

1.21_ If the area of parallelogram ABCD is 27 square feet, what is the area of triangle ABC? b. 54 ft2 a. 27 ft2 2 d. 13 ft c. 13.5 ft

B

D

C

1.22_ The base of a triangle and a parallelogram are the same length. Their heights are also the same. If the area of the parallelogram is 48 m2, what is the area of the triangle? b. 24 m2 c. 48 m2 d. 96 m2 a. 12 m2 1.23_ The base of a parallelogram and a triangle are the same length, and both figures have the same area. What is true about height of the triangle? a. It is the same as the parallelogram’s height. b. It is half of the parallelogram’s height. c. It is twice the parallelogram’s height. d. Its height is twice its base. 1.24_ A triangle is 6 centimeters long and 4 centimeters high. What is the area of the triangle? b. 20 cm2 c. 24 cm2 d. 48 cm2 a. 12 cm2 1.25_ What is the area of this triangle? b. 16 in2 a. 24 in2 2 c. 15 in d. 12 in2

5 in

5 in 4 in 6 in

1.26_ The area of a triangle is 3.6 cm2. If the triangle has a base of 6 cm, what is the height? a. 0.6 cm b. 1.2 cm c. 12 cm d. 3 cm

1.27_ Which of these triangles does not have the same base length as the others? a. A b. B c. C d. D

1.28_ What is the area of the triangle if the base is 5 centimeters and the height is 6 centimeters? b. 12 feet a. 15 cm2 d. 16 inches c. 4 m3

20| Section 1

A.

B.

C.

D.

Unit 8 | Geometry and Measurement

AREA OF COMPOSITE FIGURES

12 feet

Kelsey has a small bedroom that includes her closet, her bed, a desk, a dresser, and some shelves. Her parents are going to put carpet in the room, so they need to find the area of the floor to know how much carpet to buy.

9 feet

The room is not a rectangle, a parallelogram, or even a triangle. So, how will they find the area of the floor? In this lesson, we will answer this question and learn how to solve problems like these using what we already know about finding the area.

15 feet

6 feet

6 feet

6 feet

Can you see a way to divide the figure into rectangles? There are three ways we could find the area using rectangles. In Example 1, the room is divided into two rectangles: 6 feet

6 feet

9 feet

Before we look at some new figures, let’s take a minute to review what we already know about area.

15 feet

We know that the area of a parallelogram is found by multiplying the base by the height. The area of a triangle is found by taking half of the base multiplied by the height. We found that the area of a triangle was half the area of a parallelogram. If we need to find the area of more complicated shapes, called composite figures, we can usually divide the figure into basic shapes. Then, we can find the area of each shape and add them together. Let’s take a look at the shape of Kelsey’s bedroom without the furniture, closet, or door, and with measurements marked.

Example 1 One is 15 feet long and 6 feet wide, and the other is 9 feet long and 6 feet wide. We can find the area of each rectangle and add their areas together. A = lw + lw Substitute the lengths and widths of the rectangles. A = (15 feet)(6 feet) + (9 feet)(6 feet) Multiply, then add.

A = 90 ft2 + 54 ft2 = 144 ft2

Section 1 |21

Geometry and Measurement | Unit 8

In Example 2, the room is again divided into two rectangles:

12 feet

12 feet

9 feet

15 feet

6 feet Example 3 6 feet Example 2 One is 12 feet long and 9 feet wide, and the other is 6 feet long and 6 feet wide. We can find the area of each rectangle and add their areas together. A = lw + lw Substitute the lengths and widths of the rectangles. A = (12 feet)(9 feet) + (6 feet)(6 feet) Multiply, then add. A = 108 ft2 + 36 ft2 = 144 ft2

22| Section 1

In Example 3, we can look at the room as one rectangle, 15 feet long and 12 feet wide. Then, we can subtract the part of the rectangle that is not included in the room: a rectangle 6 feet long and 6 feet wide. A = lw – lw Substitute the lengths and widths of the rectangles. A = (15 feet)(12 feet) – (6 feet)(6 feet) Multiply, then subtract. A = 180 ft2 – 36 ft2 = 144 ft2 In each example, the area comes out to be 144 square feet. So, when we work with composite figures, there is often more than one way to break up the shape. You can choose whichever way works best for you!

Unit 8 | Geometry and Measurement

Example:

4 cm

What is the area of the figure shown?

2 cm 4 cm

Solution: To find the area, we will divide the figure into two rectangles. We can find the area of each rectangle and then add the areas together.

8 cm

4 cm

The first rectangle is 8 cm long and 4 cm wide, and the second rectangle is 4 cm long and 4 cm wide.

4 cm 2 cm

A = lw + lw

4 cm

Substitute the lengths and widths of the rectangles. 4 cm

(8 cm)(4 cm) + (4 cm)(4 cm) Multiply, then add.

4 cm

32 cm2 + 16 cm2 = 48 cm2 So, the area of the figure is 48 square centimeters.

TRAPEZOIDS A figure that is similar to a parallelogram is a trapezoid. It is a quadrilateral that has one pair of parallel sides. The parallel sides are also called bases. This trapezoid has a height of 4 cm and bases of 4 cm and 10 cm.

8 cm

4 cm

4 cm

4 cm

10 cm

Section 1 |23

Geometry and Measurement | Unit 8

We can find the area of the trapezoid by dividing it into a parallelogram and a triangle. 4 cm

congruent. Since the bottom base is 4 cm, the length of the triangle base is 6 cm (10 cm – 4 cm = 6 cm). Now we can find the area by adding the area of the parallelogram and the area of the triangle: 1 2

A = bh + __bh

4 cm

Substitute the base and height of each figure. 10 cm

By drawing a line parallel to the right side, a parallelogram is created because we already know that the bases are parallel, so each pair of opposite sides is parallel. We also know that the bottom side of the parallelogram is 4 cm because opposite sides of a parallelogram are

(4 cm)(4 cm) + (6 cm)(4 cm) Multiply, then add. 16 cm2 + 12 cm2 = 28 cm2­ So, the trapezoid has an area of 28 square centimeters.

6 ft

Example: Tom is going to paint the wall of a bedroom in his attic and he needs to know how much paint to buy. The wall is 8 feet high and includes a window.

Solution: The wall is a trapezoid, so we can divide it into a parallelogram and triangle to find the area. For now, we’ll ignore the window and subtract its area at the end.

2 ft 3 ft

8 ft

16 ft

The bottom base of the parallelogram is 6 feet, since it is the same length as its opposite side. The base of the triangle is 10 feet, subtracting the parallelogram base from the trapezoid base (16 feet – 6 feet = 10 feet). Now we can find the area by adding the area of the parallelogram and the area of the triangle: 1 2

A = bh + __bh 1 2

(6 feet)(8 feet) + __(10 feet)(8 feet)

Substitute the base and height of each figure.

48 feet2 + 40 feet2 = 88 feet2

Multiply, then add.

The wall is 88 square feet. However, the window area needs to be subtracted. The window is 2 feet long and 3 feet wide. A = lw = (2 feet)(3 feet) = 6 ft2 Now we can subtract the window area from the wall area. 88 ft2 – 6 ft2 = 82 ft2 So, the trapezoid has an area of 82 square centimeters.

24| Section 1

Unit 8 | Geometry and Measurement

Let’s Review! Before going on to the practice problems, make sure you understand the main points of this lesson.

99The area of composite figures can be found by dividing the figure into rectangles, parallelograms, and triangles.

99The area of a trapezoid can be found by dividing it into a parallelogram and a triangle. Match each word to its definition. 1.29 _________ a geometric figure that is made up of two or more basic shapes

_________ a quadrilateral with one pair ofparallel sides

a. trapezoid b. composite figure

Place a check mark next to each correct answer (you may select more than one answer).

1.30_ How could this figure be divided to find its area, assuming side lengths are known?

……

……

……

……

Section 1 |25

Geometry and Measurement | Unit 8

Circle each correct answer. 1.31_ What is the area of this figure? b. 118 in2 a. 128 in2 c. 108 in2 d. 48 in2

10 in 8 in 6 in 16 in

1.32_ A wall is 12 feet long and 8 feet tall. There is a square window 4 feet long in the wall. What is the area of the wall surface? b. 92 ft2 c. 80 ft2 d. 32 ft2 a. 96 ft2 1.33_ What is the area of this figure? b. 24 m2 a. 20 m2 d. can’t be determined c. 32 m2

4m

4m

2m 8m

1.34_ If this picture frame can display a picture 6 inches long and 4 inches wide, what is the area of the frame itself? b. 30 in2 a. 24 in2 2 d. 78 in2 c. 54 in

1.35_ The corners of a square are cut off two centimeters from each corner to form an octagon. If the octagon is 10 centimeters wide, what is its area? b. 92 cm2 a. 84 cm2 c. 100 cm2 d. can’t be determined

9 in 6 in 6 in

4 in

2 cm 2 cm 10 cm

6 ft

1.36_ What is the area of this trapezoid? b. 80 ft2 a. 60 ft2 c. 64 ft2 d. 24 ft2

8 ft

10 ft

1.37_ What is the area of the figure? b. 84 ft2 a. 40 ft2 c. 96 ft2 d. can’t be determined

10 feet

10 feet

2 feet

2 feet 16 feet

26| Section 1

Unit 8 | Geometry and Measurement

AREA OF CIRCLES Steve has installed a sprinkler in the middle of his lawn. He is thinking about adding sprinklers in the corners and possibly the sides of the lawn. However, he would like to know the area that the sprinkler covers. How can Steve find the area of this circular space? It’s definitely not a parallelogram or a triangle, and it’s not a composite figure. In this lesson we will explore how to find the area of a circle. We can think of a circle as a regular polygon with many, many sides. Notice that as the polygon has more sides, it looks more like a circle. the congruent triangles in the polygon and get closer and closer to the area of a circle as the number of sides increased. In a way, a circle is a composite figure!

dodecagon

20-gon

36-gon

Remember that pi, usually shown as the Greek letter p (π), is the ratio of the circumference to the diameter in a circle. c π = __ d

In fact, this is how the area of a circle was calculated long ago. A Greek named Archimedes discovered that he could find the area of

This ratio, and Archimedes’ discovery, leads us to the formula for the area of a circle.

Section 1 |27

Geometry and Measurement | Unit 8

The area of any regular polygon can be found by changing it into a parallelogram as long as the distance to the center is known. 1p __ 2

The circumference is pi multiplied by the diameter, which is twice the radius. C=π×d d = 2r Substitute the values for C and d. 1 2

A = __π × 2r × r

h

Simplify.

1 A = __ ph 2

A = π × r2

The area of the parallelogram is the base multiplied by the height. The height is the distance 1

to the center of the octagon and the base is __ 2 of the perimeter of the octagon. 1 2

A = __ph

Archimedes knew that as the polygon, or circle, was divided into more and more narrower secReminder The radius is half the length of the diameter. If the diameter is 20 m, the radius is 10 m.

1C __ 2

tions, the area was getting closer and closer to pi r squared. So, the area of a circle is found by multiplying pi by the square of the radius.

r

If the octagon were a circle, it would look like this instead. Now the height of the parallelogram is the radius of the circle, and the base of 1

the parallelogram is __ of the circumference of 2 the circle. 1 2

A = __C × r

A = πr2 Although pi is a ratio, the decimal form does not repeat. Pi has been calculated to millions 22 7

of decimal places. We will use 3.14, or ___ , as a close approximation for pi, or we can leave the answer in terms of pi.

Let’s find the area of a circle using all three forms. We’ll use the formula for the area of a circle.

r = 10 d = 20

A = π r2

A = π r2

A = π r2

= π(10cm)2

= (3.14)(10cm)2

= ___(10 cm)2

= 100π cm2

= (3.14)100π cm2 ≈ 314 cm2

28| Section 1

22 7 22 = ___100 cm2 7 2200 = _____ cm2 7

Substitute values for r and p.

≈ 314.3 cm2

Divide by 7.

Multiply 10 × 10. Multiply by the value for π.

Unit 8 | Geometry and Measurement

The second and third answers are approximations, and the accuracy we need depends on the situation. In fact, if we want a quick estimate for the area of a circle, we can use 3 for pi. This is also an easy way to check that your answer makes sense.

FIND THE AREA OF A CIRCLE AND SEMICIRCLES Although 100π is an exact answer, using the answer in terms of π is not very practical in real life situations where we need a measurement. However, it is very useful in many mathematical situations. 22 7

For real life situations, we will use 3.14, or ___.

Example: Steve’s sprinkler sprays water 7 feet. What area of the lawn does it water?

Solution: The sprinkler will cover a circular area, and the radius of the circle (the distance it sprays water) is 7 feet. We 22

will use the formula for the area of a circle, and ___ for 7 pi. A = πr2 22 7

Substitute values for r and p.

A = ___(49 ft2)

22 7

Multiply 7 × 7.

A = 22(7 ft2)

Divide 49 by 7.

A = 154 ft2

Multiply.

A = ___(7 ft)2

So, the sprinkler will water an area of about 154 square meters. Let’s double check that our answer makes sense. We’ll use the formula again, but we’ll use 3 for pi. When we squared the radius, we got 49 square feet. Multiplying by 3 for pi, we get 147 square feet (3 × 49 = 147). So, our answer makes sense because pi is a little more than 3, and 154 is a little more than 147. For this situation, perhaps an estimate would have been enough accuracy.

Sometimes we need to find the area of half of a circle, called a semicircle. To find the area of a semicircle, we just divide the area of the whole circle by two, since there are two halves. The diameter is the edge of the semicircle.

A=

πr2 ___ 2

semicircle

Section 1 |29

Geometry and Measurement | Unit 8

Example: Sally is painting the area above her door with expensive gold leaf paint. The door is 3 feet wide. She needs to know the area so she knows how much paint to buy.

Solution: An estimate will not be useful here. Sally does not want to spend any more money on the paint than she needs to, so she wants an accurate answer. We will use the formula for the area of a circle, and then divide that result by two since the area is a semicircle. The diameter of the semicircle is 3 feet, so the radius is 1.5 feet (3 ÷ 2 = 1.5). We’ll use 3.14 for pi. A = πr2 A = (3.14)(1.5 ft)2

Substitute values for pi and r.

A = (3.14)2.25 ft2

Multiply 1.5 by 1.5.

A ≈ 7.065 ft

Be careful!

This is the area of the circle with a diameter of 3 feet. However, Sally is painting a semicircle, so we need to divide the result by 2.

Sometimes the diameter is given instead of the radius. Be sure to divide the diameter by two to find the radius and use it in the formula.

2

7.065 ft2 ÷ 2 = 3.5325 ft2 So, Sally needs to buy enough paint to cover about 3.5 square feet.

S-T-R-E-T-C-H If a circle has an area of 144π square units, what is the radius of the circle?

Let’s Review! Before going on to the practice problems, make sure you understand the main points of this lesson.

99The area of a circle is found using the formula A = πr . πr 99The area of a semicircle is found by dividing the area of the circle by 2: A = ____ 2 2

2

30| Section 1

Unit 8 | Geometry and Measurement

Match each word to its definition. 1.38 _________ the distance across a circle through the center

a. diameter



_________ the ratio of the circumference of a circle to its diameter; approximately 3.14

b. pi



_________ the distance from the center of a circle to any point on the circle



_________ One half of a circle, divided by the diameter

c. semicircle d. radius

Circle each correct answer. 1.39_ Which measure is not the area of a circle with radius 20 mm?

a. 400π mm2

b. 1256 mm2

8800 7

_____ m2 c. 628.57 mm2 d.

1.40_ A circle has an area of π cm2. What is its radius? a. 1 cm b. 2 cm 1 c. __ cm 2

d. can’t be determined

1.41_ What is the area of this circle? b. 50.24 m2 a. 64π m2 176 7

____ m2 c. 100.48 m2 d.

d=8m

1.42_ The logo for Chris’s Calculator Company is 3 semicircles. The logo will be placed on the company building and will be 4 feet tall. What is the area of the logo? b. 37.68 ft2 a. 18.84 ft2 2 c. 50.24 ft d. 75.36 ft2

4 feet

1.43_ A dog is tied to a 14 foot long leash in the middle of the yard. How much area does the dog 22 7

have to run around? (use ___ for pi)



a. 88 ft2

b. 616 ft2

c. 44 ft2

1.44_ A circle has a diameter of 6 cm. What is its area? b. 37.68 cm2 c. 28.26 cm2 a. 18.84 cm2

d. 308 ft2 d. 9.42 cm2

Section 1 |31

Geometry and Measurement | Unit 8

1.45_ A circle has a radius of 5 inches. A semicircle has a radius of 10 inches. How do the areas compare? a. The areas are equal. b. The semicircle has twice the area of the circle. c. The circle has twice the area of the semicircle. d. The semicircle has four times the area of the circle.

1.46_ The area behind the free throw line on a basketball court is a semicircle with a 6 foot radius. What is the area of the semicircle? 132 7



____ft2 a. 56.52 ft2 b.



c. 113.04 ft2

264 7

d. ____ft2

PROJECT: ESTIMATING AREA In this section, you have learned about area and found the area of different plane figures: triangles, rectangles, parallelograms, circles, and composite figures. However, how do we find the area of a figure that isn’t any of the shapes we’ve explored? In this project, you will explore how to find the area of figures like this.

Materials Pencil Grid Paper

32| Section 1

6 feet

Unit 8 | Geometry and Measurement

ESTIMATING AREA The area of an irregular figure can be found accurately using a type of mathematics called Calculus. The method involves filling the space with narrower and narrower rectangles. Then, the areas of the rectangles are added. As the rectangles get narrower, less space is unaccounted for. If you continue your math studies, you will learn this method later in high school.

We can use a few different strategies to estimate the area. First, notice that the figure is within a 4 × 4 square, so we know the area is less than 16 cm2, (4 cm × 4 cm = 16 cm2). This gives us a rough estimate. We could count the number of empty squares and subtract that area from 16. There are about 2 whole squares and 5 half-squares that are empty. = 1 cm2 1

1 2

3

For now, we will use estimation to find the area of irregular figures. By placing a grid over the figure and using squares the size of the unit we need to measure, we can get a reasonable estimate.

4

5

2

Start by multiplying. = 1 cm

2



1 2

2 + 5(__) =

5 2 5 Change __ to a mixed number and add. 2 1 1 2 + 2__ = 4__ 2 2



2 + __ =

The figure is within an area of 16 cm2, and about 4 of the squares are empty. So, the area 1 2

of the figure is about 11__ cm2.

1 2

1 2

16 – 4__ = 11__

Section 1 |33

Geometry and Measurement | Unit 8

We could also estimate the area by counting the squares inside the figure. It looks like there are about 8 whole squares and 6 half-squares that are filled.

= 1 cm2

Multiply, then add.

1 __ 8 + 6( 2 ) =



8 + 3 = 11

1

1 3 5 5

Using this method, our result was 11 cm . So, we arrived at a slightly different answer (our 2

2

3

6

7

8

2 4 4

6

1 __

earlier estimate was 11 2 cm2), but both results are an estimate, so either answer is valid.

FIND THE AREA OF IRREGULAR FIGURES 1. On a piece of grid paper, draw three different irregular figures and estimate the area of each one.

2. Draw two different irregular figures with about 12 square units.

Answer the following questions, corresponding to the numbered steps above. 1.47_ List three figures that you have found the area of in this unit. _________________________________________________________________________________________________ _________________________________________________________________________________________________ _________________________________________________________________________________________________ 1.48_ What strategy did you use to draw each figure? ________________________________________________ _________________________________________________________________________________________________ _________________________________________________________________________________________________ _________________________________________________________________________________________________ _________________________________________________________________________________________________ _________________________________________________________________________________________________ _________________________________________________________________________________________________ _________________________________________________________________________________________________

34| Section 1

Unit 8 | Geometry and Measurement

1.49_ What method(s) did you use to estimate the area of the figures you drew? ____________________ _________________________________________________________________________________________________ _________________________________________________________________________________________________ _________________________________________________________________________________________________ _________________________________________________________________________________________________ _________________________________________________________________________________________________ _________________________________________________________________________________________________ _________________________________________________________________________________________________ 1.50_ How accurate do you think your estimates were? ______________________________________________ _________________________________________________________________________________________________ _________________________________________________________________________________________________ _________________________________________________________________________________________________ _________________________________________________________________________________________________ _________________________________________________________________________________________________ _________________________________________________________________________________________________ _________________________________________________________________________________________________ 1.51_ Can you think of any real life situations where we would need to find the area of irregular figures? ________________________________________________________________________________________ _________________________________________________________________________________________________ _________________________________________________________________________________________________ _________________________________________________________________________________________________ _________________________________________________________________________________________________ _________________________________________________________________________________________________ _________________________________________________________________________________________________ _________________________________________________________________________________________________

TEACHER CHECK

initials

date

Review the material in this section in preparation for the Self Test. The Self Test will check your mastery of this particular section. The items missed on this Self Test will indicate specific areas where restudy is needed for mastery. Section 1 |35

Geometry and Measurement | Unit 8

SELF TEST 1: PLANE FIGURES Circle each correct answer (each answer, 4 points). Use parallelogram ABCD to answer questions 1.01 – 1.04. 1.01_

What is the perimeter? a. 14 cm b. 13 cm c. 12 cm d. 11 cm

1.02_

What is the area? a. 7 cm2 c. 12 cm2

A

b. 9 cm2 d. 12.25 cm2

3 cm

3 cm

3.5 cm

1.03_

What is the area of triangle ABC? b. 3.5 cm2 a. 7 cm2 2 c. 6 cm d. 4.5 cm2

1.04_

What is the perimeter of ACD? a. 9.5 cm b. 7.5 cm2 c. 8.5 cm2 d. 5.5 cm

1.05_

What is the circumference of a circle with a diameter of 5 meters? (Use 3.14 for pi.) a. 78.5 m b. 31.4 m c. 15.7 m d. 8.14 m

3.5 cm

D

C

3 cm

3m

1.06_

What is the perimeter of the figure? a. 28 m b. 22 m c. 20 m d. 14 m

5m 4m

1m 7m

1.07_

A regular pentagon has a perimeter of 60 feet. How long is each side? a. 5 feet b. 6 feet c. 10 feet d. 12 feet

1.08_

If the area of the parallelogram is 15 cm2, what is the area of the green triangle? b. 15 cm2 a. 30 cm2 c. 7.5 cm2 d. 8 cm2

1.09_

36| Section 1

What is the height of the triangle? a. 2 units b. 3 square units c. 3 units d. can’t be determined

B

Unit 8 | Geometry and Measurement

1.010_ The area of a triangle is 18 square feet. If the base is 3 feet, what is the height of the triangle? a. 6 feet b. 3 feet c. 12 feet d. 9 feet 1.011_ The area of a rectangle is 51 square inches. If the width of the rectangle is 6 inches, what is the length? a. 19.5 inches b. 13 inches c. 9 inches d. 8.5 inches 1.012_ Steve is adding wallpaper to a living room wall and he needs to know how much wallpaper to buy. If the wall is 8.5 feet tall and 12.5 wide, how much wallpaper should he buy? b. 96.25 ft2 c. 42 ft d. 108 ft2 a. 106.25 ft2 1.013_ A square has a perimeter of 12 cm. What is its area? b. 18 cm2 c. 36 cm2 a. 9 cm2 1.014_ What is the area of the triangle? b. 14 cm2 a. 15 cm2 c. 12 cm2 d. 24 cm2

d. 144 cm2 6 cm

8 cm

1.015_ What is the area of a circle with a diameter of 12 m? b. 37.68 m2 c. 75.36 m2 a. 18.84 m2

d. 113.04 m2

6 in

1.016_ What is the area of the trapezoid? b. 128 in2 a. 88 in2 2 c. 96 in d. 48 in2

8 in

16 in

16 cm

1.017_ A square picture frame has a round circle cut out to show the picture. What is the area of the picture frame? b. 193.2 cm2 a. 177.5 cm2 d. 256 cm2 c. 334.5 cm2

1.018_ An arched entrance to a stadium is made by combining a square and a semicircle. What is the area of the opening? b. 150 ft2 a. 178.5 ft2 2 c. 139.25 ft d. 100 ft2 1.019_ What is the area of the composite figure? b. 72 m2 a. 84 m2 2 d. 96 m2 c. 108 m

10 cm

10 ft

6m 8m

4m 12 m Section 1 |37

Geometry and Measurement | Unit 8

1.020_ What is the area of the hexagon? b. 80 m2 a. 60 m2 2 c. 100 m d. 120 m2

6m 5m

10 m

5m 6m Answer true or false (each answer, 5 points). 1.021 _____________ To find the perimeter and area of a square, use the same formula.

68

85

38| Section 1

SCORE

TEACHER

initials

date

MAT0608 – Apr ‘15 Printing 804 N. 2nd Ave. E. Rock Rapids, IA 51246-1759 800-622-3070 www.aop.com

ISBN 978-0-7403-3472-6

9 780740 334726