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. 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 Link
The 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 radius
b. Describe the relationship between the circumference and
diameter of each hula hoop.
The circumference is about three times the diameter.
Radius and Diameter
Words: 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=
𝑑
𝑟
Example 1 & 2
a. The diameter of a circle is 14
inches. Find the radius.
r=
𝑑
2
=
14
2
=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 & 2
Find the radius or diameter for each circle with the given
dimensions.
a. d = 23 cm
11.5 cm
c. d = 16 yards
8 yards
b. r = 3 inches
6 inches
d. r = 5.2 meters
10.4 meters
Circumference
Words: The circumference of a circle is equal to 𝜋 times its
diameter or 𝜋 times twice its radius.
Symbols: C = 𝜋d or C = 2𝜋r
Model:
The value of pi (𝜋) is 3.1415926… or
22
.
7
Example 3
Find the circumference of a circle with a radius of 21 inches.
Since 21 is a multiple of 7, use
22
7
for 𝜋.
C = 2𝜋r
C=
22
2( )(21)
7
C = 22(3)
C = 132
The circumference is about 132 inches.
Got it? 3
Find the circumference of each circle. Use
a.
𝟐𝟐
𝟕
for 𝝅.
b.
220 inches
1
2
5 feet
Example 4
Big 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 = 𝜋d
C ≈3.14(23)
C ≈ 72.2
So the distance around the clock is about 72.2 feet.
Got it? 4
A 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 𝝅. Round to the nearest tenth if necessary.
about 12.6 feet
AREA OF CIRCLES
Lesson 8-2
Real-World Link
1. 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 Circle
Words: The area A of a circle equals the product of 𝜋 and the
square of its radius r.
Symbols: A = 𝜋r2
Model:
Cross out the formula that is not used in finding area of a circle.
A = 𝜋r2
A = 3.14r2
A=
22 2
r
7
1
2
A = 𝑏ℎ
Example 1
Find the area of the circle. Use 3.14 for pi.
A = 𝜋r2
A ≈ 3.14 • 22
A ≈ 3.14 • 4
A ≈ 12.56
The area of the circle is approximately 12.56 square inches.
Example 2
Find the area of the circle with a radius of 14 centimeters.
𝟐𝟐
Use for pi.
𝟕
22 2
A= r
7
22
A ≈ • 142
7
22
A ≈ • 196
7
A ≈ 616
The area of the circle is approximately 616 square
centimeters.
Got it? 1 & 2
Find the area of a circle with a radius of 3.2 centimeters.
Round to the nearest tenth.
32.2 cm2
Example 3
Find 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
1
(24)
2
of 12 millimeters.
A = 𝜋r2
A ≈ 3.14 • 122
A ≈ 3.14 • 144
A ≈ 452.16
The area is approximately 452.2 square millimeters.
Got it? 3
The 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 Semicircles
Find the area of the semicircle. Use 3.14 for pi. Round to
the nearest tenth.
A=
A≈
A≈
1
2
1
2
1
2
𝜋r2
3.14 • 82
3.14 • 64
A ≈ 100.5
Got it? 4
Find the approximate area of a semicircle with a radius of
6 centimeters. Round to the nearest tenth.
56.5 cm2
Example 5
On a basketball count, there is a semicircle above the freethrow 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=
A≈
A≈
1
2
1
2
1
2
𝜋r2
3.14 • 62
3.14 • 36
A ≈ 56.5
So, the area of the semicircle is approximately 56.5 square feet.
AREA OF COMPOSITE
FIGURES
Lesson 8-3
Find the Area of a Composite Figure
A 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
Formula
Parallelogram
A = (base)(height)
Triangle
A = (base)(height)
Trapezoid
A=
Circle
A = 𝜋r2
1
2
1
(height)(base
2
1 + base 2)
Example 1
Find the area of the composite figure.
The figure can be separated into a semicircle and a triangle.
Area of a semicircle
A=
1
2
1
2
𝜋r2
A≈
3.14
A ≈ 14.1
Area of a triangle
A=
32
A=
1
2
1
2
bh
• 11 • 6
A = 33
The area of the figure is about 14.1 + 33 or 47.1 square
meters.
Got it? 1
Find the area of the figure. Round to the nearest tenth if
necessary.
482.5 in2
Example 2
A 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? 2
Pedro’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.
Area of a triangle
A= 4
A= 4
1
2
1
2
bh
•1•1
A= 2
Area of a rectangle
A = bh
A = 12 • 5
A = 60
The area of the shaded region is 60 – 2 or 58 square inches.
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.
Area of the entire rectangle
A = 𝑙𝑤
A = 25 • 42
A = 1,050
Area not shaded
A = 𝑙𝑤
A = 20 • 22
A = 440
The area of the shaded region is 1,050 – 440 or 610 square
meters.
Got it? 3 & 4
A diagram for a park is shown. The shaded area
represented the picnic sections. Find the area of the
picnic sections.
2,250 yd2
VOLUME OF PRISMS
Lesson 8-4
Volume of a Rectangular Prism
Words: The volume V of a rectangle prism is the product
of the length ℓ, the width w, and the height h. It is also the
area of the base B times the height h.
Symbols: V = ℓwh or V = Bh
Model:
Volume of a Rectangular Prism
The 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 1
Find the volume of the rectangular prism.
V = ℓwh
V=5•4•3
V = 60
The volume is 60 cubic centimeters or 60 cm3.
Got it? 1
Find the volume of the rectangular prism shown below.
142.5 m3
Volume of a Triangular Prism
Words: 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 Prism
The 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 2
Find the volume of the triangular prism shown.
1
2
1
2
The area of the triangle is • 6 • 8 so, replace B with • 6 • 8.
V = Bh
V=
1
(
2
• 6 • 8)(9)
V = 216
Got it? 2
Find 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 PYRAMIDS
Lesson 8-5
Real – World Link
Dion 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 Pyramid
Words: The volume V of a pyramid is one third the area of
the base B times the height h of the pyramid.
1
3
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 1
Find the volume of the pyramid. Round to the nearest tenth.
V=
1
Bh
3
1
3
V = (3.2 x 1.4)2.8
V ≈ 4.2
The volume is about 4.2 cubic inches.
Example 2
Find the volume of the pyramid. Round to the nearest tenth.
(What is the shape of the base?)
V=
1
Bh
3
1 1
3 2
V = ( ∙ 8.1 ∙ 6.4)11
V = 95.04
The volume is about 95.04 cubic meters.
Got it? 1 & 2
Find 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 3
The rectangular pyramid shown has a volume of 90 cubic
inches. Find the height of the pyramid.
1
3
V = Bh
1
3
90 = (5 ∙ 9)h
90 = 15h
6=h
The height is 6 inches.
Example 4
A 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=
44 =
1 1
(
3 2
1
Bh
3
∙ 8 ∙ 4)h
44 = 4h
11 = h
The height is 11 meters.
Got it? 3 & 4
a. 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 5
Kamilah is making a model for the Food Guide Pyramid
for a class project. Find the volume of the square
pyramid.
V=
V=
1
Bh
3
1
(12
3
x 12)12
V = 576
The volume is about 576 cubic inches.
SURFACE AREA OF
PRISMS
Lesson 8-6
Real-World Link
Members 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
51 inches
is the width of the board in inches? _______
2. Suppose 3 messages fit down the board lengthwise. What
33 inches
is the length of the board in 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
Surface Area of a Rectangular Prism
Words: The surface area S.A. of a rectangular prism with
base ℓ, width w, and height h is the sum of the areas of its
faces. .
Symbols:
S.A. = 2ℓh + 2ℓw + 2hw
Model:
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 1
Find the surface area of the rectangular prism.
Replace with ℓ with 9, w with 7, and h with 13.
Surface area = 2ℓh + 2ℓw + 2hw
= 2  9  13 + 2  9  7 + 2  13  7
= 234 + 126 + 182
= 542
The surface area of the prism is 542 square inches.
Got it? 1
Find the surface area of each prism.
a.
b.
216 m2
726 mm2
Example 2
Domingo 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 = 2ℓh + 2ℓw + 2hw
= 2  60  36 + 2  60  24 + 2  36  24
= 8,928 in2
Example 2
Domingo 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? 2
The 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 Prism
To 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.
1
2
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? 3
Find the surface area of the triangular prism.
38 cm2
SURFACE AREA OF
PYRAMIDS
Lesson 8-7
Vocabulary Start-Up
A 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 Pyramid
Lateral 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 ℓ.
1
2
Symbols: L.A. = Pℓ
Model:
Surface Area of a Pyramid
Total 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.
1
2
Symbols: S.A. = B + L.A. or S.A. = B + Pℓ
Model:
Surface Area of a Regular Pyramid
Example 1
Find the total surface area of the pyramid. Round to the
nearest tenth.
1
2
S.A = B + Pℓ
1
2
S.A = 16 + (16)(9)
S.A = 88
The surface area is 88 square inches.
Example 2
Find the total surface area of the pyramid. Round to the
nearest tenth.
1
2
S.A = B + Pℓ
1
2
S.A = 111 + (48)(20)
S.A = 591
The surface area is 591 square meters.
Example 3
Find the total surface area of the pyramid. Round to the
nearest tenth.
1
2
S.A = B + Pℓ
1
2
S.A = 43.5 + (30)(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 4
Sal 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?
1
2
1
(10)(3)
2
S.A = B + Pℓ
S.A = 6.25 +
S.A = 21.25
So, 21.25 square inches of card stock are used to make one
gift box.
Got it? 4
Amado 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
FIGURES
Lesson 8-8
Example 1
Find the volume of the composite figure.
Find the volume of each prism.
V = ℓwh
V = 8 ∙ 6 ∙ 16 or 768
V = ℓwh
V = 8 ∙ 6 ∙ 8 or 384
The volume is 768 + 384 or 1,152 cubic inches.
Example 2
Find the volume of the composite figure.
Find the volume of each prism.
V = ℓwh
V = 8 ∙ 8 ∙ 8 or 512
1
3
V = Bh
1
3
V = (8 ∙ 8)5 or 106.7
The volume is 512 + 106.7 or 618.7 cubic feet.
Got it? 1 & 2
Find the volume of the composite figure.
228 cm3
Example 3
Find the surface area of the composite
figure.
Find the surface area of each prism.
A = ℓw + ℓw
A = (8 ∙ 16) + (8 ∙ 8)
A = 128 + 64 or
192 in2
A = ℓw
A = 6 ∙ 16
A = 96 in2
A = ℓw
A= 6 ∙ 8
A = 48 in2
Total Surface Area is 2(192) + 2(96) + 4(48) or 768 in2.
Example 4
Find the surface area of the composite figure.
Find the surface area of each prism.
A = ℓw
A = 8 ∙ 8 or 64
1
2
A = Bh
1
2
A = (8 ∙ 6.4) or 25.6
The surface area is 5(64) + 4(25.6) or 422.4 square feet.
Got it? 3 & 4
Find the surface area of this composite figure.
30 ft2