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In this lesson, several popular

3 D

figures like cylinders, cones, and spheres will be explored. To better understand their characteristics, their volume and surface area formulas will be introduced.

Catch-Up and Review

Here are a few recommended readings before getting started with this lesson.

Challenge

Relating the Volumes of Cones and Cylinders

The year is about to end and New Year's Eve parties are right around the corner. Tiffaniqua and her grandmother decide to bake chewy chocolate chip cookies for their gathering. As a part of their baking steps, Tiffaniqua fills a cone-shaped cup with flour and pours it into a cylindrical cup.

Animation to fill up a cylinder using a cone

The cylindrical cup and cone-shaped cup have the same radius of

8

centimeters. The height of the cylindrical cup is

12

centimeters. What is the height of the cone-shaped cup if it takes three of them, completely filled with flour, to fill one cylindrical cup?

Discussion

Cylinder

The first $3 D$ figure of this lesson to breakdown is the cylinder. Here, the method of how to calculate the volume and surface area of a cylinder will be understood.

Rule

Volume of a Cylinder

The volume of a cylinder is calculated by multiplying the area of its base by its height.

If the radius of the base of a cylinder is $r,$ the volume can be calculated by the following formula.

$V = π r^{2} h$

Note that the surface area of a cylinder consists of the two equal circular areas and one rectangular lateral face.

Rule

Surface Area of a Cylinder

Consider a cylinder of height $h$ and radius $r .$

Cylinder with a radius of r and height of h

The surface area of this cylinder is given by the following formula.

$S = 2 π r h + 2 π r^{2}$

Example

Finding the Volume and Surface Area of a Cylinder

Tiffaniqua loves fireworks. Her and her grandmother team up to buy a box full of them to celebrate the new year.

One of the types of fireworks that came in the box Tiffaniqua bought is in the shape of a cylinder. The firework's height is

18

centimeters and its radius is

4

centimeters.

a Calculate the volume of the cylindrical-shaped firework. Round the answer to the nearest integer.

b Calculate the surface area of the cylindrical-shaped firework. Round the answer to the nearest integer.

Hint

a The volume of a cylinder is the product of

π,

the square of its radius, and its height.

b The surface area of a cylinder is the sum of the areas of the base and top circles and the lateral face.

Solution

a It is a given that the firework is in the shape of a cylinder. This means we can use the formula for the volume of a cylinder to determine its volume.

A cylinder with a base radius of 4 cm and height of 18 cm positioned vertically.

It is known that

h = 18

and

r = 4 .

Substitute these two values into the formula.

V = π r^{2} h

SubstituteII

$h = 18$ , $r = 4$

V = π (4^{2}) (18)

▼

Evaluate right-hand side

CalcPow

Calculate power

V = π (16) (18)

Multiply

V = 288 π

UseCalc

Use a calculator

V = 904.778684 \dots

RoundInt

Round to nearest integer

V \approx 905

The volume of a firework is about

905

cubic centimeters.

b It is now time to calculate the surface area of a cylindrical firework. Notice that the surface area of a cylinder consists of the areas of the base and top circles and the lateral face.

Recall the surface area formula as the sum of these three areas.

S A = 2 π r h + 2 π r^{2}

Once again, substitute

h = 18

and

r = 4

into the surface area formula.

S A = 2 π r h + 2 π r^{2}

SubstituteII

$h = 18$ , $r = 4$

S A = 2 π (4) (18) + 2 π (4)^{2}

▼

Evaluate right-hand side

CalcPow

Calculate power

S A = 2 π (4) (18) + 2 π (16)

Multiply

S A = 144 π + 32 π

AddTerms

Add terms

S A = 176 π

UseCalc

Use a calculator

S A = 552.920307 \dots

RoundInt

Round to nearest integer

S A \approx 553

The surface area of a cylindrical firework is about

553

square centimeters.

Pop Quiz

Practice Finding the Surface Area and Volume of a Cylinder

Find the volume or surface area of the cylinder. The radius and height are given in centimeters.

Discussion

Cone

The second $3 D$ figure of this lesson to understand is the cone. One by one, the formulas of the volume and surface area of a cone will be considered.

Rule

Volume of a Cone

The volume of a cone is one third the product of its base area and its height.

The base area $B$ is the area of the circle and the height $h$ is measured perpendicular to the base.

$V = \frac{1}{3} B h$

Since the base is a circle, its area depends on its radius. Therefore, the base area can also be expressed in terms of the radius $r .$

$V = \frac{1}{3} π r^{2} h$

Note that the surface area of a right cone consists of a circular base and a curved lateral face.

Rule

Surface Area of a Cone

Consider a right cone with radius $r$ and slant height $ℓ .$

The surface area of a right cone is the sum of the base area and the lateral area. The area of the base is given by $π r^{2}$ and the lateral area is $π r ℓ .$

$S A = π r^{2} + π r ℓ$

Example

Finding the Volume and Surface Area of a Cone

During her winter vacation, Tiffaniqua wants to design a paper Christmas tree that spins! The tree will be made in a conical shape. She plans to use

136 -

centimeter sticks to make the framework of the lateral face. In that case, the radius of the base will be

64

centimeters.

She needs to make some calculations to finish the tree's design.

a Help her calculate the volume of the conical Christmas tree. Round the answer to the nearest cubic centimeter.

b Help her calculate the surface area of the conical Christmas tree. Round the answer to the nearest square centimeter.

Hint

a The volume of a cone is one-third of the product of the area of its base and height. Use the Pythagorean Theorem to find the height of the cone.

b The surface area of a cone is the sum of the area of its base and the area of its lateral face.

Solution

a The volume of a cone is one-third of the product of the area of its base and height. The base is already known, therefore, start by calculating the height of the conical Christmas tree. Recall that the distance from the vertex of the cone to its base is the height of the cone. The slant height of the cone is

136

centimeters and a radius is

64

centimeters.

The height, radius, and slant height of the cone form a right triangle,

△ A B C .

Use the Pythagorean Theorem to find the height of the cone,

A B .

A C^{2} = A B^{2} + B C^{2}

SubstituteII

$A C = 136$ , $B C = 64$

136^{2} = A B^{2} + 64^{2}

▼

Solve for

A B

CalcPow

Calculate power

18496 = A B^{2} + 4096

SubEqn

$LHS - 4096 = RHS - 4096$

14400 = A B^{2}

RearrangeEqn

Rearrange equation

A B^{2} = 14400

SqrtEqn

$LHS = RHS$

A B^{2} = 14400

SqrtPowToNumber

$a^{2} = a$

A B = 120

Since a negative root does not make sense for a side length, consider only the positive root. The height of the cone is

120

centimeters. Now, recall the volume formula of the cone.

V = \frac{1}{3} π r^{2} h

Here,

r

is the radius and

h

is the height of the cone. Substitute

r = 64

and

h = 120

into the volume formula.

V = \frac{1}{3} π r^{2} h

SubstituteII

$r = 64$ , $h = 120$

V = \frac{1}{3} π (64)^{2} (120)

▼

Evaluate right-hand side

CalcPow

Calculate power

V = \frac{1}{3} π (4096) (120)

Multiply

V = \frac{1}{3} \cdot π \cdot 491520

MoveRightFacToNumOne

$\frac{1}{b} \cdot a = \frac{a}{b}$

V = \frac{4 9 1 5 2 0 π}{3}

CalcQuot

Calculate quotient

V = 163840 π

UseCalc

Use a calculator

V = 514718.540364 \dots

RoundInt

Round to nearest integer

V \approx 514719

The volume of the cone is about

514719

cubic centimeters.

b This time, the surface area of the cone will be calculated. Recall that the surface area of a cone is the sum of the base area and lateral face.

S A = π r^{2} + π r ℓ

Note that the radius of the base

r

64

centimeters and the slant height of the cone

ℓ

136

centimeters. Substitute these two values into the formula.

S A = π r^{2} + π r ℓ

SubstituteII

$r = 64$ , $ℓ = 136$

S A = π (64)^{2} + π (64) (136)

▼

Evaluate right-hand side

CalcPow

Calculate power

S A = π (4096) + π (64) (136)

Multiply

S A = π (4096) + π (8704)

AddTerms

Add terms

S A = 12800 π

UseCalc

Use a calculator

S A = 40212.385965 \dots

RoundInt

Round to nearest integer

S A \approx 40212

The conical tree will be about

40212

square centimeters. Note that this means Tiffaniqua needs to have at least

40212

square centimeters green paper to design the moving Christmas tree.

Pop Quiz

Practice Finding the Volume and Surface Area of a Cone

The applet shows right cones. The dimensions of the figure are given in decimeters. Use the given information to answer the question. If necessary, round the answer to one decimal place.

Discussion

Sphere

The next $3 D$ figure of the lesson is a perfectly rounded figure called the sphere.

Concept

Sphere

A sphere is the set of all points in the space that are equidistant from a given point called the center of the sphere.

The distance from the center to any point on the sphere is called the radius of the sphere.

Now, how to calculate the volume and surface area of a sphere will be examined.

Rule

Volume of a Sphere

The volume of a sphere with radius $r$ is four-thirds the product of pi and the radius cubed.

Rule

Surface Area of a Sphere

The surface area of a sphere with radius $r$ is four times pi multiplied by the radius squared.

Example

Finding the Volume and Radius of a Sphere

It snowed a lot before winter break. Tiffaniqua's school organized an inter-class snowperson competition on the last day before the holiday. The person who makes the biggest snowperson wins. Tiffaniqua and her class worked together. They decided to make two huge spherical snowballs. Then, they will put them on top of each other and shape the snowmperson. They plan that the head of the snowperson will have $1$ meter of radius and the body will have $32$ square meters surface area.

a Calculate the volume of a spherical snowball with a radius of

1

meter. Round the answer to the nearest cubic meter.

b Calculate the radius of a spherical snowball with a surface area of

32

square meters. Round the answer to one decimal place.

Hint

a Recall the formula for the volume of a sphere.

b Recall the formula for the surface area of a sphere.

Solution

a Start by recalling the formula for the volume of a sphere.

V = \frac{4}{3} π r^{3}

It is known that the radius of the spherical snowball will be

1

meter. Substitute

r = 1

into the formula and calculate

V .

V = \frac{4}{3} π r^{3}

Substitute

$r = 1$

V = \frac{4}{3} π (1)^{3}

BaseOne

$1^{a} = 1$

V = \frac{4}{3} π \cdot 1

IdPropMult

Identity Property of Multiplication

V = \frac{4}{3} π

UseCalc

Use a calculator

V = 4.188790 \dots

RoundInt

Round to nearest integer

V \approx 4

They need approximately

4

cubic meters of snow to make a snowball with a radius of

1

meter.

b In the second scenario, they plan to make a snowball that has

32

square meters surface area. Recall the formula for the surface area of a sphere.

S A = 4 π r^{2}

Now, substitute

S A = 32

into the formula to find

r .

S A = 4 π r^{2}

Substitute

$S A = 32$

32 = 4 π r^{2}

▼

Solve for

r

DivEqn

$LHS / 4 π = RHS / 4 π$

\frac{3 2}{4 π} = r^{2}

RearrangeEqn

Rearrange equation

r^{2} = \frac{3 2}{4 π}

SqrtEqn

$LHS = RHS$

r^{2} = \frac{3 2}{4 π}

SqrtPowToNumber

$a^{2} = a$

r = \frac{3 2}{4 π}

UseCalc

Use a calculator

r = 1.595769 \dots

RoundDec

Round to $1$ decimal place(s)

r \approx 1.6

The radius of the snowball will be

1.6

meters for this case. It looks more likely that a snowperson made from two snowballs with radii of

1

meter and

1.6

meters will be the greatest sized snowperson.

Discussion

Hemisphere

The final figure is the half of a sphere which is called hemisphere.

Concept

Hemisphere

A hemisphere is a three-dimensional object formed by half of a sphere and a flat circular base. Any plane that goes through the center of a sphere divides it into two hemispheres.

The radius

r

of a hemisphere is the segment that connects the center

O

with any point on the hemisphere. The radius of a sphere is the same as the radius of any of its hemispheres. The volume of a hemisphere with radius

r

is half the volume of a sphere with radius

r .

Volume V = \frac{2}{3} π r^{3}

The surface area of a hemisphere with radius

r

is the circular flat area plus half the surface area of a sphere with radius

r .

Surface Area 2 π r^{2} + π r^{2} = 3 π r^{2}

Example

Finding the Surface Area and Radius of a Hemisphere

Tiffaniqua's father bought a big snow globe as a gift for her Christmas present.

a Since this is a present for Tiffaniqua, her father wants to use beautiful wrapping paper to make it even more special. The radius of the hemispherical part is

11

centimeters. Help him to find the surface area of this present to determine the amount of the wrapping paper. Round the answer to the nearest square centimeter.

b Her father also has a box in the shape of a hemisphere to put the gift into it. The box has the volume of

3

cubic centimeters. Does the snow globe fit this box?

Hint

a Recall the surface area formula of a hemisphere.

b Recall the volume formula of a hemisphere.

Solution

a The snow globe has a radius of

11

centimeters. Notice that the surface area of this hemispherical globe consists of a circular base and a half of the surface area of a sphere with the same radius,

11

centimeters. Recall the sum of these formulas.

S A = 2 π r^{2} + π r^{2} = 3 π r^{2}

Now, substitute

r = 11

centimeters into the formula and calculate it.

S A = 3 π r^{2}

Substitute

$r = 11$

S A = 3 π (11)^{2}

CalcPow

Calculate power

S A = 3 π \cdot 121

Multiply

S A = 363 π

UseCalc

Use a calculator

S A = 1140.398133 \dots

RoundInt

Round to nearest integer

S A \approx 1140

Tiffaniqua's father needs approximately

1140

square centimeters of paper to wrap the present.

b This time, the volume of the snow globe will be calculated. Recall that the volume of a hemisphere is half of the volume of sphere with the same radius.

V = \frac{2}{3} π r^{3}

Now, substitute

r = 11

into the volume formula.

V = \frac{2}{3} π r^{3}

Substitute

$r = 11$

V = \frac{2}{3} π (11)^{3}

CalcPow

Calculate power

V = \frac{2}{3} π (1331)

MoveRightFacToNum

$\frac{a}{c} \cdot b = \frac{a \cdot b}{c}$

V = \frac{2 6 6 2 π}{3}

UseCalc

Use a calculator

V = 2787.639881 \dots

RoundInt

Round to nearest integer

V \approx 2788

Notice that the box is

3

cubic decimeters. Multiply it by

100

to convert cubic decimeters to cubic centimeters.

3 dm^{3} = 3000 cm^{3}

Since the volume of the box is greater than

2788

cubic centimeters, the gift fits this box. The answer is yes.

Pop Quiz

Finding the Volume or Surface Area of Different Spheres and Hemispheres

In the following applet, calculate either the volume or the surface area of the given sphere or hemisphere. The radius is given in centimeters. Round the answer to the nearest integer.

Closure

Calculating the Volumes of Cones and Cylinders

The challenge presented at the beginning of this lesson said that Tiffaniqua and her grandmother filled a cone-shaped cup with flour and poured it into a cylindrical cup.

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Catch-Up and Review

Hint

Solution

Hint

Solution

Hint

Solution

Hint

Solution