CIRCUMFERENCE Lesson 8-1. Vocabulary Start-Up A circle is the set of all points in a plane that are the same distance from a point, called the center.

CIRCUMFERENCE Lesson 8-1

Vocabulary Start-UpA circle is the set of all points in a plane that are the same distance from a point, called the center. The circumference is the distance around a circle. The diameter is the distance across a circle through its center. The radius is the distance from the center to any point on the circle.

Fill in the box with one of the vocabulary terms.

Real-World LinkThe table shows the approximate measurements of two sizes of hula hoops.

a. Describe the relationship between the diameter and radius of each hula hoop.

The diameter is twice the radiusb. Describe the relationship between the circumference and diameter of each hula hoop.

The circumference is about three times the diameter.

Radius and DiameterWords: The diameter d of a circle is twice its radius r. The radius r of a circle is half of its diameter d.

Symbols: d = 2r r =

Example 1 & 2a. The diameter of a circle is 14 inches. Find the radius.

r = = = 7

The radius is 7 inches.

b. The radius of a circle is 8 feet. Find the diameter.

d = 2r = 2(8) = 16

The diameter is 16 inches.

Got it? 1 & 2Find the radius or diameter for each circle with the given dimensions.

a. d = 23 cm b. r = 3 inches

c. d = 16 yards d. r = 5.2 meters

11.5 cm 6 inches

8 yards 10.4 meters

CircumferenceWords: The circumference of a circle is equal to times its diameter or times twice its radius.

Symbols: C = d or C = 2r Model:

The value of pi () is 3.1415926… or .

Example 3Find the circumference of a circle with a radius of 21 inches.

Since 21 is a multiple of 7, use for . C = 2r

C = 2()(21)C = 22(3)C = 132

The circumference is about 132 inches.

Got it? 3Find the circumference of each circle. Use for .

a. b.

220 inches

5 feet

Example 4Big Ben is a famous clock tower in London, England. The diameter of the clock face is 23 feet. Find the circumference of the clock face. Round to the nearest tenth.

C = dC 3.14(23)

C 72.2

So the distance around the clock is about 72.2 feet.

Got it? 4A circular fence is being placed to surround a tree. The diameter of the fence is 4 feet. How much fencing is used? Use 3.14 for . Round to the nearest tenth if necessary.

about 12.6 feet

AREA OF CIRCLESLesson 8-2

Real-World Link1. Adrianne wants to find the distance the dog runs when it runs one circle with the leash fully extended. Should she calculate the circumference or area? Explain.Circumference; the circumference is the distance

around the circle

2. Suppose she wants to find the amount of running room the dog has with the leash fully extended. Should she calculate the circumference or area? Explain.Area; the area is the interior region of an enclosed

figure

Find the Area of a CircleWords: The area A of a circle equals the product of and the square of its radius r.

Symbols: A = r2 Model:

Cross out the formula that is not used in finding area of a circle.

A = r2 A = r2 A = r2 A =

Example 1Find the area of the circle. Use 3.14 for pi.

A = r2

A • 22 A 3.14 • 4A 12.56

The area of the circle is approximately 12.56 square inches.

Example 2Find the area of the circle with a radius of 14 centimeters. Use for pi.

A = r2

A • 142 A • 196A 616

The area of the circle is approximately 616 square centimeters.

Got it? 1 & 2Find the area of a circle with a radius of 3.2 centimeters. Round to the nearest tenth.

32.2 cm2

Example 3Find the area of the face of the Virginia quarter with a diameter of 24 millimeters. Use 3.14 for pi. Round to the nearest tenth if necessary.

The radius is (24) of 12 millimeters.

A = r2

A • 122 A 3.14 • 144

A 452.16

The area is approximately 452.2 square millimeters.

Got it? 3The bottom of a circular swimming pool with a diameter of 30 feet is painted blue. How many square feet are blue? Round to the nearest tenth.

706.5 ft2

Example 4 – Area of SemicirclesFind the area of the semicircle. Use 3.14 for pi. Round to the nearest tenth.

A = r2

A • 82 A 3.14 • 64

A 100.5

Got it? 4Find the approximate area of a semicircle with a radius of 6 centimeters. Round to the nearest tenth.

56.5 cm2

Example 5On a basketball count, there is a semicircle above the free-throw line that has a radius of 6 feet. Find the area of the semicircle. Use 3.14 for pi. Round to the nearest tenth.

A = r2

A • 62 A 3.14 • 36

A 56.5

So, the area of the semicircle is approximately 56.5 square feet.

BRAIN POP VIDEO1. Search “Circles”.2. Watch Video.3. Take quiz.4. Write the score of your quiz by

your name on your Lesson 3 Independent Practice.

AREA OF COMPOSITE FIGURESLesson 8-3

Find the Area of a Composite FigureA composite figure is made up of two or more shapes. To find the area of a composite figure, decompose the figure into shapes with areas you know. Then find the sum of these areas.

Shape FormulaParallelogram A = (base)(height)

Triangle A = (base)(height)

Trapezoid A = (height)(base 1 + base 2)

Circle A = r2

Example 1Find the area of the composite figure.The figure can be separated into a semicircle and a triangle.

The area of the figure is about 14.1 + 33 or 47.1 square meters.

Area of a semicircle

A = r2

A 32

A 14.1

Area of a triangleA = A =

A = 33

Got it? 1Find the area of the figure. Round to the nearest tenth if necessary.

482.5 in2

Example 2A miniature gold hole is composed of a trapezoid and a parallelogram. How many square feet of turf does the hole cover?

So, 7.5 + 15 or 22.5 square feet of turf will be needed.

Got it? 2Pedro’s father is building a shed. How many square feet of wood are needed to build the back of the shed shown?

210 ft2

Example 3 – Find Area of Shaded Region

Find the area of the rectangle and subtract the area of the four triangles.

The area of the shaded region is 60 – 2 or 58 square inches.

Area of a triangle

A = 4 A = 4 A = 2

Area of a rectangle

A = A =

A = 60

Example 4 – Find Area of Shaded Region

The blueprint for a hotel swimming area is represented by the figure shown. The shaded area represents the pool. Find the area of the pool.

The area of the shaded region is 1,050 – 440 or 610 square meters.

Area of the entire rectangle

A = A =

A = 1,050

Area not shaded

A = A =

A = 440

Got it? 3 & 4A diagram for a park is shown. The shaded area represented the picnic sections. Find the area of the picnic sections.

2,250 yd2

BRAIN POP VIDEO1. Search “Area of Polygons”.2. Watch Video. 3. Take quiz. You will need formulas for a

triangle, parallelogram, and trapezoid.4. Write the score of your quiz by your

name on your Lesson 4 Independent Practice.

VOLUME OF PRISMSLesson 8-4

Volume of a Rectangular PrismWords: The volume V of a rectangle prism is the product of the length , the width w, and the height h. It is also the area of the base B times the height h.

Symbols: V = wh or V = Bh

Model:

Volume of a Rectangular PrismThe volume of a three-dimensional figure is the measure of space it occupies. It is measured in cubic units such as centimeters (cm3) or cubic inches (in3).

It takes 2 layers of 36 cubes to fill the box. So, the volume of the box is 72 cubic centimeters.

Example 1Find the volume of the rectangular prism.

V = whV = 5 • 4 • 3

V = 60

The volume is 60 cubic centimeters or 60 cm3.

Got it? 1Find the volume of the rectangular prism shown below.

142.5 m3

Volume of a Triangular PrismWords: The volume V of a triangular prism is the product of the base B times the height h.

Symbols: V = Bh, where B is the area of the base

Model:

Volume of a Triangular PrismThe diagram below shows that the volume of a triangular prism is also the product of the area of the base B and the height h of the prism.

Example 2Find the volume of the triangular prism shown. The area of the triangle is • 6 • 8 so, replace B with • 6 • 8.

V = BhV = ( • 6 • 8)(9)V = 216

Got it? 2Find the volume of the triangular prism shown below.

70 in3

Example 3 Which lunch box holds more food?Find the volume of each lunch box. Then compare.

Since 285 in3> 281.25 in3, Lunch Box B holds more food.

VOLUME OF PYRAMIDSLesson 8-5

Real – World LinkDion is helping his mother build a sand sculpture at the beach in the shape of a pyramid. The square pyramid has a base with length and width of 12 inches an d the height of 14 inches. 1. Label the dimensions of the sand sculpture on the square pyramid below.

2. What is the area of the base? 144 in2

3. What is the volume of a square prims with the same dimensions?

2,016 in3

Volume of a PyramidWords: The volume V of a pyramid is one third the area of the base B times the height h of the pyramid.

Symbols: V = Bh, where B is the area of the base

Model:

VOCABULARY:In a polyhedron, any face that is not the base is called a lateral face. The lateral faces meet at a common point or vertex.

Example 1Find the volume of the pyramid. Round to the nearest tenth.

V = BhV = (3.2 x 1.4)2.8

V 4.2

The volume is about 4.2 cubic inches.

Example 2Find the volume of the pyramid. Round to the nearest tenth.

(What is the shape of the base?)V = Bh

V = ( ∙ 8.1 ∙ 6.4)11V = 95.04

The volume is about 95.04 cubic meters.

Got it? 1 & 2Find the volume of a pyramid that has a height of 9 centimeters and a rectangular base with a length of 7 centimeters and a width of 3 centimeters.

63 cm3

Example 3The rectangular pyramid shown has a volume of 90 cubic inches. Find the height of the pyramid.

V = Bh90 = (5 ∙ 9)h

90 = 15h6 = h

The height is 6 inches.

Example 4A triangular pyramid has a volume of 44 cubic meters. It has an 8-meter back and a 3-meter height. Find the height of the pyramid.

V = Bh44 = ( ∙ 8 ∙ 4)h

44 = 4h11 = h

The height is 11 meters.

Got it? 3 & 4a. A triangular pyramid has a volume of 840 cubic inches. It has a base of 20 inches and a height of 21 inches. Find the height of the pyramid.

12 inches

b. A rectangular pyramid has a volume of 525 cubic feet. It has a base of 25 feet by 18 feet. Find the height of the pyramid.

3.5 feet

Example 5Kamilah is making a model for the Food Guide Pyramid for a class project. Find the volume of the square pyramid.

V = BhV = (12 x 12)12

V = 576

The volume is about 576 cubic inches.

SURFACE AREA OF PRISMSLesson 8-6

Real-World LinkMembers of a local recreation center are permitted to post messages on 8.5-inch by 11-inch paper on the board. Assume the signs are posted vertically and do not overlap. 1. Suppose 6 messages fit across the board widthwise. What is the width of the board in inches? _______ inches

2. Suppose 3 messages fit down the board lengthwise. What is the length of the board in inches? _______ inches

3. What is the area in square inches of the message board?

1,683 in2

4. What is the total area of the front and back of the board?

3,366 in2

51

33

Surface Area of a Rectangular PrismWords: The surface area S.A. of a rectangular prism with base , width w, and height h is the sum of the areas of its faces. .

Symbols: Model: S.A. = 2h + 2w + 2hw

The surface area is the area of all the faces added together. When you find surface area, the units are square units, not cubic units.

Example 1Find the surface area of the rectangular prism. Replace with with 9, w with 7, and h with 13.

Surface area = 2h + 2w + 2hw = 2 13 + 2 7 + 2 13 7

= 234 + 126 + 182= 542

The surface area of the prism is 542 square inches.

Got it? 1Find the surface area of each prism. a. b.

216 m2 726 mm2

Example 2Domingo built a toy box 60 inches long, 24 inches wide, and 36 inches high. He has 1 quart of paint that covers about 87 square feet of surface. Does he have enough to pain the toy box? Justify your answer (show your work).

Surface area = 2h + 2w + 2hw = 2 36 + 2 24 + 2 36 24

= 8,928 in2

Example 2Domingo built a toy box 60 inches long, 24 inches wide, and 36 inches high. He has 1 quart of paint that covers about 87 square feet of surface. Does he have enough to pain the toy box? Justify your answer.

Find the number of square inches the paint will cover. 1 ft2 = 1 ft x 1 ft

= 12 in x 12 in = 144 in2

So, 87 square feet is equal to 87 x 144 or 12,528 square inches.

Since 12,528 > 8,928, Domingo has enough paint.

Got it? 2The largest corrugated cardboard box ever constructed measured about 23 feet long, 9 feet high, and 8 feet wide. Would 950 square feet of paper be enough to cover the box? Justify your answer.

Yes, the surface area of the box is 926 ft2 and 950 ft2 > 926 ft2.

Surface Area of Triangular PrismTo find the surface area of a triangular prism, it is more efficient to find the area of each face and calculate the sum of all of the faces rather than using a formula.

Example 3

Marty is mailing his aunt the package shown. How much cardboard is used to create the shipping container?

Find the area of each face and add.

The area of each triangle is 4 3 or 6.

The area of two of the rectangles is 14 3.6 or 50.5. The area of the third rectangle is 14 4 or 56.

The sum of the areas of the faces is 6 + 6 + 50.4 + 50.4 + 56 or 168.8 squared inches.

Got it? 3Find the surface area of the triangular prism.

38 cm2

SURFACE AREA OF PYRAMIDSLesson 8-7

Vocabulary Start-UpA right square pyramid has a square base and four isosceles triangles that make up the lateral faces. The lateral surface area is sum of the areas of its lateral faces. The height of each lateral face is called the slant height.

1. Fill in the blanks.

2. Draw a net of a square pyramid.

Surface Area of a PyramidLateral Surface Area

Words: The lateral surface area L.A. of a regular pyramid is half the perimeter P of the base times the slant height .

Symbols: L.A. = P

Model:

Surface Area of a PyramidTotal Surface Area

Words: The total surface area S.A. of a regular pyramid is the lateral area L.A. plus the area of the base B.

Symbols: S.A. = B + L.A. or S.A. = B + P

Model:

Surface Area of a Regular Pyramid

Example 1Find the total surface area of the pyramid. Round to the nearest tenth.

S.A = B + PS.A = 16 + ((9)

S.A = 88

The surface area is 88 square inches.

Example 2Find the total surface area of the pyramid. Round to the nearest tenth.

S.A = B + PS.A = 111 + ((20)

S.A = 591

The surface area is 591 square meters.

Example 3Find the total surface area of the pyramid. Round to the nearest tenth.

S.A = B + PS.A = 43.5 + ((12)

S.A = 223.5

The surface area is 223.5 square feet.

Got it? 1 – 3 a. Find the surface area of a square pyramid that has a slant height of 8 centimeters and a base length of 5 centimeters.

105 cm2

b. Find the total surface area of the pyramid.

332.4 m2

Example 4Sal is wrapping gift boxes that are square pyramids for party favors. They have a slant height of 3 inches and base edges 2.5 inches long. How many square inches of card stock are used to make one gift box?

S.A = B + PS.A = 6.25 + ((3)

S.A = 21.25

So, 21.25 square inches of card stock are used to make one gift box.

Got it? 4Amado purchased a bottle of perfume that is in the shape of a square pyramid. The slant height of the bottle is 4.5 inches and the base is 2 inches. Find the surface area.

22 in2

VOLUME AND SURFACE AREA OF COMPOSITE FIGURESLesson 8-8

Example 1Find the volume of the composite figure.Find the volume of each prism.

The volume is 768 + 384 or 1,152 cubic inches.

V = whV = 8 ∙ 6 ∙ 16 or

768

V = whV = 8 ∙ 6 ∙ 8 or 384

Example 2Find the volume of the composite figure.Find the volume of each prism.

The volume is 512 + 106.7 or 618.7 cubic feet.

V = whV = 8 ∙ 8 ∙ 8 or 512

V = BhV = (8 ∙ 8)5 or

106.7

Got it? 1 & 2Find the volume of the composite figure.

228 cm3

Example 3Find the surface area of the composite figure.Find the surface area of each prism.

Total Surface Area is 2(192) + 2(96) + 4(48) or 768 in2.

A = w + wA = (8 ∙ 16) +

(8 ∙ 8)A = 128 + 64

or 192 in2

A = wA = 6 ∙ 16A = 96 in2

A = wA = 6 ∙ 8A = 48 in2

Example 4Find the surface area of the composite figure.Find the surface area of each prism.

The surface area is 5(64) + 4(25.6) or 422.4 square feet.

A = wA = 8 ∙ 8 or 64

A = BhA = (8 ∙ 6.4) or

25.6

Got it? 3 & 4Find the surface area of this composite figure.

30 ft2