Download Volume and Surface Area Notes

Volume of a Cylinder
Lesson 1
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:
 Volume – the measure of space occupied by a solid
 Cylinder – 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?
A = πr2
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
Find the volume of each cylinder. Round to the nearest tenth.
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
V = π(1)2(1.5)
Use a calculator.
V = 4.7
Multiply the volume by 1.8.
4.7(1.8) = 8.46
The paperweight weighs about 8.46 ounces.
Got it? 3
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.
Composite Shapes – objects made up of
more than one type of solid.
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
V =lwh
V = (12)(12)(12)
V = 1,728
Cylinder
V =πr2h
V = π(1)2(12)
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.
Rectangular Prism
V =lwh
Half Cylinder
1
V = πr2h
2
Volume of Cones
Lesson 2
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.
Symbol: 𝑉 =
Model:
1
𝐵ℎ
3
𝑜𝑟 𝑉 =
1
𝜋𝑟2ℎ
3
Example 1
Find the volume of the cone. Round to the nearest tenth.
V
𝟏 2
= πr h
𝟑
𝟏
π(3)2(6)
𝟑
V=
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 2
V = πr h
𝟏
𝟑
V = π(4)2(10)
𝟑
V ≈ 167.6
The volume of the paper cup is about 167.6 cubic
centimeters.
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.
Volume of Cylinder:
V =πr2h
V = π(4)2(4)
V ≈ 201.1
Volume of Cone:
𝟏 2
V = πr h
𝟏
𝟑
V = π(4)2(5)
𝟑
V ≈ 83.8
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.
Volume of 1st Cone:
𝟏 2
V = πr h
𝟑
Volume of 2nd Cone:
𝟏 2
V = πr h
𝟑
Volume of Spheres
Lesson 3
Real-World Link
Brittani purchased a necklace that contained a round pearl
with a diameter of 7.5 millimeters. What is the circumference
of the largest circle around the outside of the pearl? Round to
the nearest tenth.
𝐶 = 2𝜋𝑟
𝐶 = ____________
𝐶 = ______
Volume of a Sphere
Words: the volume V of a sphere is four thirds the product of π
and the cube of the radius r.
Symbol: 𝑉 =
Model:
4
π𝑟3
3
Example 1
Find the volume of this sphere.
Volume of Sphere:
𝟒 3
V = πr
𝟑
𝟒
= π(6)3
𝟑
V
V ≈ 𝟗𝟎𝟒. 𝟖
The volume is about 904.8 cubic
millimeters.
(Three lines of work)
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
𝟑
𝟒
= π(4)3
𝟑
V = πr3
V
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
𝟑
𝟒
= π(5)3
𝟑
V = πr3
V
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.
Volume of a Hemisphere
Example 4:
Find the volume of the hemisphere.
Round to the nearest tenth.
𝟏 𝟒
𝟐 𝟑
𝟏 𝟒
=( ) π(5)3
𝟐 𝟑
V =( ) πr3
V
V ≈ 261.8 cm3
The volume is about 261.8 cubic centimeters.
Got it? 4
Find the volume of each hemisphere. Round to the nearest tenth.
a.
b.
Surface Area of Cylinders
Lesson 4
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.
Surface Area of Cones
Lesson 5
Real-World Link
Bobby is making waffle cones from scratch. Use the
Pythagorean Theorem to find the slant height ℓ of the cone if
the radius is 2 inches and the height is 6 inches.
a2 + b2 = c2
Lateral Surface Area of a Cone
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.
Total Surface Area of a Cone
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
Model:
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.
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?
Lateral Surface Area
Find the lateral surface area.
L.A. = πrℓ
L.A. = π(𝟒)(𝟏𝟏)
L.A. ≈ 138.2 in2
The amount of fabric used is about 138.2 square
inches.
Ticket Out The Door
Write the formula for the lateral surface area.
Explain what each variable means. Use complete
sentences. (2-3 sentences)
Change in Dimensions
Lesson 6
Real-World Link
Stephen is creating a model of the Washington Monument for
1
history class. The model will be
of the monument’s actual size.
100
The square pyramid that sits atop the monument’s obelisk shape
has a slant height of about 57.6 feet. Each side of the pyramid’s
base is about 34 feet.
What is the are of the of the
triangular faces of the actual
pyramid?
Real-World Link
Stephen is creating a model of the Washington Monument for
1
history class. The model will be
of the monument’s actual size.
100
The square pyramid that sits atop the monument’s obelisk shape
has a slant height of about 57.6 feet. Each side of the pyramid’s
base is about 34 feet.
What is the slant height of the pyramid of the model Stephen is
creating?
Real-World Link
Stephen is creating a model of the Washington Monument for
1
history class. The model will be
of the monument’s actual size.
100
The square pyramid that sits atop the monument’s obelisk shape
has a slant height of about 57.6 feet. Each side of the pyramid’s
base is about 34 feet.
What is the length of one base of the pyramid on the model?
Real-World Link
Stephen is creating a model of the Washington Monument for
1
history class. The model will be
of the monument’s actual size.
100
The square pyramid that sits atop the monument’s obelisk shape
has a slant height of about 57.6 feet. Each side of the pyramid’s
base is about 34 feet.
What is the area of one of the triangular faces of the model
pyramid?
Real-World Link
Write a ratio comparing the area of the triangular side of the model
to the actual monument.
𝒎𝒐𝒅𝒆𝒍 𝒂𝒓𝒆𝒂
𝒂𝒄𝒕𝒖𝒂𝒍 𝒂𝒓𝒆𝒂
=
𝟎.𝟎𝟗𝟕𝟗𝟐
𝟗𝟕𝟗𝟐
=
𝟏
𝟏𝟎,𝟎𝟎𝟎
The actual area of the monument is 10,000
times as large as the scale model.
Surface Area of Similar Solids
Solids are similar if that have the same shape, and their
measurements are proportional.
The scale factor is 2.
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 is smaller than the largest box
1
by a scale factor of
, what is its surface area?
48
Volume of Similar Solids
Solids are similar if that have the same shape, and their
measurements are proportional.
The scale factor is 2.
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 3
V = 432 x ( )
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.