Chapter 8 Volume and Surface Area
Lesson 1: Volume of Cylinders
Volume of a Cylinder:
Words: The volume V of a cylinder with radius r is the area of the base B times the height h.
Symbols: V = Bh, where B = πr2 or V = πr2h
Model:

– the measure of space occupied by a solid

– a three-dimensional figure with two parallel congruent circular
bases.
What is the formula do you use to find the area of the base of the cylinder?
Example 1: Find the volume of the cylinder. Round to the nearest tenth.
V =πr2h
V = π(5)2(8.3)
Use a calculator.
V = 651.8804756
The volume is about 651.9 cubic centimeters.
Example 2: Find the volume of the cylinder with a diameter of 16 inches and a height of 20
inches. Round to the nearest tenth.
If the diameter is 16, the radius is 8.
V =πr2h
V = π(8)2(20)
Use a calculator.
V = 4,021.2
The volume is about 4,021.2 cubic inches.
Got it? 1 & 2
a.
b. Diameter: 12 mm, Height: 5 mm
Example 3: A metal paperweight is in the shape of a cylinder. The paperweight has a
height of 1.5 inches and a diameter of 2 inches. How much does the paperweight weigh if 1
square inch weighs 1.8 ounces? Round to the nearest tenth.
V =πr2h
Multiply the volume by 1.8.
2
V = π(1) (1.5)
4.7(1.8) = 8.46
Use a calculator.
The paperweight weighs about 8.46 ounces.
V = 4.7
Got it? 3
Example The Roberts family uses a container shaped like a cylinder to recycle aluminum
cans. It has a height of 4 feet and a diameter of 1.5 feet. The container is full. How much
do the contents weigh if the average weight of the aluminum cans is 37 ounces per square
foot? Round to the nearest tenth.
Example 4: Tonya uses cube-shaped beads to make jewelry. Each
bead has a circular hole through the middle. Find the volume of the bead.
Rectangular Prism
Cylinder
V =lwh
V =πr2h
V = (12)(12)(12)
V = π(1)2(12)
V = 1,728
V = 37.7
The volume of the bead is 1,728 – 37.7 or 1,690.3 cubic millimeters.
Got it? 4
The Service Club is building models of storage chests, like the
one shown, to donate to charity. Find the volume of the chest to
the nearest tenth.
Guided Practice:
Find the volume of each cylinder. Round to the nearest tenth.
1.
2. diameter 8 inches; height 8 inches
3. A platform like the one shown was built to hold a sculpture for an art exhibit. What is the
volume of the figure?
4. A scented candle is in the shape of a cylinder. The radius is 4 centimeter and the height is 12
centimeters. Find the mass of the wax needed to make the candle if 1 cubic centimeter of wax
has a mass of 3.5 grams. Round to the nearest tenth.
Journal: How is the formula for the volume of a cylinder similar to the formula for the volume of
a rectangular prism?
Lesson 2: Volume of Cones
Volume of a Cone: A cone is a three-dimensional figure with one circular base.
Words: the volume V of a cone with radius r is one third the area of the base B times the height h.
1
1
Symbol: 𝑉 = 3 𝐵ℎ 𝑜𝑟 𝑉 = 3 𝜋𝑟2ℎ
Model:
Example 1: Find the volume of the cone. Round to the nearest tenth.
1
V =3πr2h
1
V = 3π(3)2(6)
V ≈ 56.5
The volume of the cone is about 56.5 cubic inches.
Got it? 1
Find the volume of each cone. Round to the nearest tenth.
a.
b.
Example 2: A cone-shaped paper cup is filled with water. The height of the cup is 10
centimeters and the diameter is 8 centimeters. What is the volume of the paper cup?
Round to the nearest tenth.
1
The volume of the paper cup is about 167.6
V =3πr2h
cubic centimeters.
1
V = 3π(4)2(10)
V ≈ 167.6
Got it? 2
April is filling six identical cones for her piñata. Each cone has a radius of 1.5 inches and a
height of 9 inches. What is the total volume of the cones? Round the nearest tenth.
Example 3: Find the volume of the solid. Round to the nearest tenth.
1 2
Volume of Cylinder:
V
=
πr h
3
V =πr2h
1
V = π(4)2(5)
V = π(4)2(4)
3
V ≈ 83.8
V ≈ 201.1
Volume of Cone:
The total volume is 201.1 + 83.8 or 284.9 cubic feet.
Got it? 3
Find the volume of the solid. Round to the nearest tenth.
Lesson 3: Volume of Spheres
Words: the volume V of a sphere is four thirds the product of π and the cube of the radius r.
4
Symbol: 𝑉 = 𝜋𝑟3
Model:
3
Example 1: Find the volume of this sphere.
4
V =3πr3
4
V =3π(6)3
V ≈ 904.8
The volume is about 904.8 cubic millimeters.
Got it? 1
Find the volume of each sphere.
a.
b.
Example 2: A spherical stone in the courtyard of the National Museum of Costa Rica has a
diameter of about 8 feet. Find the volume of the spherical stone. Round to the nearest
tenth.
4
V =3πr3
4
V =3π(4)3
V ≈ 268.1 feet3
The stone is about 268.1 cubic feet.
Got it? 2
A dish contains a spherical scoop of vanilla ice cream with a radius of 1.2 inches. What is
the volume of the ice cream? Round to the nearest tenth.
Example 3: A volleyball has a diameter of 10 inches. A pump can inflate the ball at a rate
of 325 cubic inches per minute. How long will it take to inflate the ball? Round to the
nearest tenth.
4
V =3πr3
4
V =3π(5)3
V ≈ 523.6 inches3
523.6
= 1.6
325
It would take 1.6 minutes to inflate the ball.
Got it? 3
A snowball has a diameter of 6 centimeters. How long would it take the snowball to melt if
it melts at a rate of 1.8 cubic centimeters per minute? Round to the nearest tenth.
Example 4: Find the volume of the hemisphere. Round to the nearest tenth.
1 4
V =(2) 3πr3
1 4
V =(2) 3π(5)3
V ≈ 261.8 cm3
The volume is about 261.8 cubic centimeters.
Got it? 4
a.
Guided Practice:
1.
b.
2.
3. Sarah is blowing up spherical balloons for her brother’s birthday party. One of the balloons
has a radius of 3 inches. Round to the nearest tenth.
a. What it the volume of the balloon?
b. Suppose Sarah can inflate the balloon at a rate of 200 cubic inches per minute. How long will
it take her to inflate the balloon?
Find the volume of each hemisphere. Round to the nearest tenth.
4.
5.
Journal: True or False? The volume of a sphere is two-thirds the volume of a cylinder with the
same radius r and height of 2r. Explain your reasoning.
Lesson 4: Surface Area of Cylinders
Lateral Surface Area
Words: The lateral area (L.A.) of a cylinder with height h and radius r is the circumference of
the base times the height.
Symbols: L.A. = 2πrh
Total Surface Area
Words: The surface area (S.A) is the lateral surface area plus the area of the two circular bases.
Symbols: S.A. = L.A. + 2πr2 or S.A. = 2πrh + 2πr2
Finding the surface area with a net:
What’s the difference between lateral and total surface area?
Example 1: Find the surface area of the cylinder. Round to the nearest tenth.
S.A. = 2πrh + 2πr2
S.A. = 2π(2)(7) + 2π(2)2
S.A. ≈ 113.1
The surface area is about 113.1 square meters.
Got it? 1
Find the surface area of each cylinder. Round to the nearest tenth.
a.
b.
Example 2: A circular fence that is 2 feet high is to be built around the outside of a carousel. The
distance from the center of the carousel to the edge of the fence will be 35 feet. What is the area of
the fencing material that is needed to make the fence around the carousel?
You will need to find the lateral surface area. Radius is 35 and height is 2.
L.A. = 2πrh
L.A. = 2π(35)(2)
L.A. ≈ 439.8
About 440 square feet of material is needed to make the fence.
Got it? 2
a. Find the area of the label of a can of tuna with a radius of 5.1 centimeters and a height of
2.9 centimeters. Round to the nearest tenth.
b. Find the total surface areas of a cylindrical candle with a diameter of 4 inches and a
height of 2.5 inches. Round to the nearest tenth.
Lesson 5: Surface Area of Cones
Words: The lateral surface area (L.A) of a cone is π times the radius times the slant height ℓ.
Symbols: L.A. = πrℓ
Model:
Example 1: Find the lateral area of the cone. Round to the nearest tenth.
L.A. = πrℓ
L.A. = π(5)(13)
L.A. ≈ 204.2
The lateral surface area is about 204.2 square millimeters.
Got it? 1
a. Find the lateral area of a cone with a radius of 4 inches and a slant height of 9.5 inches.
Round to the nearest tenth.
b. Find the lateral area of a cone with a diameter of 16 centimeters and a slant height of 10
centimeters. Round to the nearest tenth.
Words: The surface area (S.A) of a cone is the lateral surface area plus the area of the base.
Symbols: S.A. = πrℓ + πr2
Example 2: Find the surface area of the cone. Round to the nearest tenth.
S.A. = πrℓ + πr2
S.A. = πrℓ + πr2
S.A. = π(6)(6.2) + π(6)2
S.A. ≈ 230.0 in2
The surface area is about 230.0 square inches.
Got it? 2
Find the surface area of the cone. Round to the nearest tenth.
S.A. = πrℓ + πr2
S.A. = πrℓ + πr2
S.A. = π(4.5)(20) + π(4.5)2
S.A. ≈ 346.4 mm2
The surface area is about 346.4 square millimeters.
Example 3: A tepee has a radius of 5 feet and a slant height of 12 feet.
What formula would you use to find the area of the fabric?
Lateral Surface Area
Find the lateral surface area.
L.A. = πrℓ
L.A. = π(5)(12)
L.A. ≈ 118.5 ft2
The amount of fabric used is about 118.5 square feet.
Got it? 3
Rachel bought a party hats that were in the shape of a cone. Each hat has a diameter of 8
inches and a slant height of 11 inches.
What formula would you use to find the area of the hat?
Find the lateral surface area.
Lesson 6: Change in Dimensions
Surface Area of Similar Solids
Solids are similar if that have the same shape, and their measurements are proportional.
To find the surface area of the big cube, multiply the S.A. of the small cube by 4 or 22.
“2” because the scale factor is 2. “Squared” because it’s area.
Example 1: The surface area of a rectangular prism is 78 square centimeters. What is the
surface area of a similar prims that is 3 times as large?
S.A. = 78 x 32
S.A. = 78 x 9
S.A. = 702 cm2
The surface area of the larger rectangular prism is 702 squared centimeters.
Got it? 1
a. The surface area of a triangular prism is 34 square inches. What is the surface area of a
similar prims that is twice as large?
b. The world’s largest box of raisins has a surface area of 352 square feet. If a similar box
𝟏
is smaller than the largest box by a scale factor of 𝟒𝟖 , what is its surface area?
Volume of Similar Solids
Solids are similar if that have the same shape, and their measurements are proportional.
To find the volume of the big cube, multiply the volume. of the small cube by 8 or 23.
“2” because the scale factor is 2. “Cubed” because it’s volume.
Example 2: A triangular prims has a volume of 432 cubic yards. If the prism is reduced to
one third its original size, what is the volume of the new prism?
1
V = 432 x (3)3
1
V = 432 x 27
V = 16 yards3
The new prism will have a volume of 16 cubic yards.
Got it? 2
a. A square pyramid has a volume of 512 cubic centimeters. What is the volume of the
square pyramid with dimensions one-forth of the original?
b. A cylinder has a volume of 432 cubic meters. What is the volume of a cylinder with
dimensions one-third its size?
Example 3: The measurements for a standard hockey puck is shown. The giant hockey
pock is 40 times the size of a standard puck. Find the volume and surface area of the giant
puck.
Surface Area and Volume of Original:
Surface Area and Volume of Original:
The volume of the giant hockey puck is 452,160 cubic inches, and the surface area is about
37,680 square inches.
Guided Practice:
1. The surface area of a rectangular prism is 35 square inches. What is the surface area of a
similar solid that has been enlarged by a scale factor of 7?
2. The volume of a cylinder is about 425 cubic centimeters. What is the volume, to the nearest
1
tenth, of a similar solid that is smaller by a scale factor of 3?
3. A sink with a sliding lid is Josh’s art studio measures 16 inches by 15 inches by 6 inches. A
1
second sink used just for paintbrushes has a similar shape and is smaller by a scale factor of 2.
Find the volume and surface area of the second sink.
Journal: How is volume of a prism affected when its dimensions is tripled?