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This lesson will work with the circumference and area of a circle as well as the surface area and volume of a cylinder.

Catch-Up and Review

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

Circle
Area
Volume

Challenge

Investigating the Volume of Different Cylinders

Consider the three-dimensional figure given in the diagram. By dragging the point, the figure can be skewed. However, its height and the radius of the circles that make the base and the top of the cylinder remain the same. Think about the volume — space occupied — of the solid. Does the volume change when the solid is skewed?

cylinder

Discussion

Definition of a Circle

Before moving on, the definition of a circle and some of its parts will be reviewed.

A circle is the set of all the points in a plane that are equidistant from a given point. There are a few particularly notable features of a circle.

Center - The given point from which all points of the circle are equidistant. Circles are often named by their center point.
Radius - A segment that connects the center and any point on the circle. Its length is usually represented algebraically by $r .$
Diameter - A segment whose endpoints are on the circle and that passes through the center. Its length is usually represented algebraically by $d .$
Circumference - The perimeter of a circle, usually represented algebraically by $C .$

The following circle can be referred to as

⊙ O,

or circle

O,

since it is centered at

O .

Parts of a circle

In any given circle, the lengths of any radius and any diameter are constant. They are called the radius and the diameter of the circle, respectively.

Next, the formula for calculating the circumference of a circle will be discussed.

Rule

Circumference of a Circle

The circumference of a circle is calculated by multiplying its diameter by $π .$

$C = π d$

This can be visualized in the following diagram.

Animation unrolling a circle

Since the diameter is twice the radius, the circumference of a circle can also be calculated by multiplying $2 r$ by $π .$

$C = 2 π r$

Proof

Consider two circles and their respective diameters and circumferences.

By the Similar Circles Theorem, all circles are similar. Therefore, their corresponding parts are proportional.

\frac{C _{B}}{C _{A}} = \frac{d _{B}}{d _{A}}

This proportion can be rearranged.

\frac{C _{B}}{C _{A}} = \frac{d _{B}}{d _{A}}

Rearrange equation

$LHS \cdot C_{A} = RHS \cdot C_{A}$

C_{B} = \frac{d _{B}}{d _{A}} (C_{A})

$LHS / d_{B} = RHS / d_{B}$

\frac{C _{B}}{d _{B}} = \frac{1}{d _{A}} (C_{A})

MoveRightFacToNumOne

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

\frac{C _{B}}{d _{B}} = \frac{C _{A}}{d _{A}}

Therefore, for any two circles, the ratio of the circumference to the diameter is always the same. This means that this ratio is constant. This constant is defined as

π .

With this information, it can be shown that the circumference of a circle is the product between its diameter and

π .

$\frac{C}{d} = π \Rightarrow C = π d$

Finally, the formula for calculating the area of a circle will be seen.

Rule

Area of a Circle

The area of a circle is the product of $π$ and the square of its radius.

Circle

Proof

Informal Justification

A circle with radius $r$ will be divided into a number of equally sized sectors. Then, the top and bottom halves of the circle will be distinguished by filling them with different colors. Because the circumference of a circle is $2 π r,$ the arc length of each semicircle is half this value, $π r .$

Now, the above sectors will be unfolded. By placing the sectors of the upper hemisphere as teeth pointing downwards and the sectors of the bottom hemisphere as teeth pointing upwards, a parallelogram-like figure can be formed. As such, the area of the figure below should be the same as the circle's area.

It can be noted that if the circle is divided into more and smaller sectors, then the figure will begin to look more and more like a rectangle.

Here, the shorter sides become more vertical and the longer sides become more horizontal. If the circle is divided into infinitely many sectors, the figure will become a perfect rectangle with base

π r

and height

r .

Since the area of a rectangle is the product of its

height

and its

base,

the following formula can be derived.

$A = π r \cdot r \Leftrightarrow A = π r^{2}$

It has been shown that the area of a circle is the product of $π$ and the square of its radius.

Example

Finding the Circumference and Area of a Circle

Izabella loves to use geometry in her art. Her school noticed her skills and hired her to paint the school's soccer field — for a substantial payment. The diagram shows how the field should look.

soccer field

Izabella needs the field's measurements. She already knows the radius of the circle located at the center of the field to be

6.5

meters. For logistical reasons, she wants to find the circumference and area of this circle. Help Izabella calculate these values to one decimal place.

Hint

The circumference is twice the product of $π$ and the radius. The area is the product of $π$ and the square of the radius.

Solution

The circumference of a circle with radius $r$ is twice the product of $π$ and $r .$ Furthermore, its area is the product of $π$ and $r^{2} .$ Since the radius is $6.5$ meters, the circumference and area can be calculated.

	Formula	Substitute	Simplify	Approximate
Circumference	$C = 2 π r$	$C = 2 π (6.5)$	$C = 13 π$	$C \approx 40.8$
Area	$A = π r^{2}$	$A = π (6.5^{2})$	$A = 42.25 π$	$A \approx 132.7$

The circumference and area of the circle located at the center of the soccer field are about $40.8$ meters and $132.7$ square meters, respectively.

Example

Investigating the Area of a Circle

Maya bought $20$ meters of fencing with the hopes of constructing a circular dog run for her dog Opie. Because Opie is a large Saint Bernard, she wants the area of the dog run to be at least $30$ square meters.

circular playground

With a circumference of

20

meters, is the area of the dog run at least

30

square meters?

Hint

Start by finding the radius of the circle.

Solution

Because the dog run is in the shape of a circle, its area is the product of

π

and the square of the radius. Therefore, to find the area of the circle, the first step is to find the radius. Since it is already known that the circumference is

20

meters, the formula for the circumference will be used to find the radius.

C = 2 π r

Since Maya bought and used

20

meters of fencing, the circumference measures

20

meters. This value can be substituted in the above formula to find the radius

r .

C = 2 π r

$C = 20$

20 = 2 π r

Solve for

r

$LHS / 2 = RHS / 2$

10 = π r

$LHS / π = RHS / π$

\frac{1 0}{π} = r

Rearrange equation

r = \frac{1 0}{π}

Use a calculator

r = 3.183098 \dots

Round to $3$ significant digit(s)

r \approx 3.18

The radius of Opie's circular dog run is about

3.18

meters.

With this information, the area of the dog run can be calculated.

A = π r^{2}

To do so,

3.18

will be substituted for

r

in the formula for the area of a circle.

A = π r^{2}

$r = 3.18$

A = π (3.18^{2})

Evaluate right-hand side

Calculate power

A = π (10.1124)

Multiply

A = 31.769041

Round to $3$ significant digit(s)

A \approx 31.8

The area of Opie's dog run is about

31.8

square meters. This is enough for him to have happy and healthy playtime!

Pop Quiz

Practice Finding the Circumference and Area of a Circle

For the following questions, approximate the answers to one decimal place. Do not include units in the answer.

area and circumference

Discussion

Cavalieri's Principle

Now, two-dimensional figures will be left aside to move forward into solids, which are three-dimensional shapes.

Two solids with the same height and the same cross-sectional area at every altitude have the same volume. This means that, as long as their heights are equal, skewed versions of the same solid have the same volume.

If

V_{1}

and

V_{2}

are the volumes of the above solids, then they are equal.

$V_{1} = V_{2}$

Proof

Informal Justification

This principle will be proven by using a set of identical coins. Consider a stack in which each of these coins is placed directly on top of each other. Consider also another stack where the coins lie on top of each other, but are not aligned.

The first stack can be considered as a right cylinder. Similarly, the second stack can be considered as an oblique cylinder, which is a skewed version of the first cylinder. Because the coins are identical, the cross-sectional areas of the cylinders at the same altitude are the same.

Since the coins are identical, they have the same volume. Furthermore, since the height is the same for both stacks, they both have the same number of coins. Therefore, both stacks — cylinders — have the same volume. This reasoning is strongly based on the assumption that the face of the coins have the same area.

Discussion

Definition of a Cylinder

A cylinder is a three-dimensional figure that has two circular bases that are parallel and equal in size, connected by a curved surface.

A cylinder both of the bases are depicted

The axis of a cylinder is the segment that connects the center of the bases. The height of a cylinder is the perpendicular distance between the bases. The radius of the cylinder is the radius of one of the bases.

Cylinder

If the axis of a cylinder is not perpendicular to the bases, it is called an oblique cylinder.

Oblique Cylinder

If the dimensions of a cylinder are known, then its volume and surface area can be calculated.

Rule

Volume of a Cylinder

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

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

$V = π r^{2} h$

Proof

Informal Justification

Consider a rectangular prism and a right cylinder that have the same base area and height.

In this case, the cross-sections of the prism and the cylinder are congruent to their bases. Therefore, their cross-sectional areas at every altitude are equal.

By the Cavalieri's Principle, two solids with the same height and the same cross-sectional area at every altitude have the same volume. Therefore, the volume of the cylinder is the same as the volume of the prism.

V_{C} = V_{P}

Furthermore, the volume of a prism can be calculated by multiplying the area of its base by its height.

V_{P} = B h

By the Transitive Property of Equality, a formula for the volume of the cylinder can be written.

{V_{C} = V_{P} V_{P} = B h \Rightarrow V_{C} = B h

Finally, not only is

B

the area of the base of the prism, but also the area of the base of the cylinder. Since the base of the cylinder is a circle, its area is the product of

π

and the square of its radius

r .

Therefore,

B = π r^{2}

can be substituted in the above formula.

V_{C} = B h substitute V_{C} = π r^{2} h

This formula applies to all cylinders because there is always a prism with the same base area and height. Also, by the Cavalieri's principle, this formula still holds true for oblique cylinders.

Example

Finding the Volume of a Cylinder

LaShay loves golfing. She is about to buy a new golf bag.

golf court

The the bag she is thinking about buying is a shaped like a cylinder with a height of

132

centimeters and its radius is

15

centimeters. For all of her golf clubs to fit in the bag, its volume must be at least

90000

cubic centimeters. Therefore, she wants to calculate the volume of the bag. Help her do this, approximating the answer to three significant figures.

Hint

The volume of a cylinder is the product of $π,$ the square of its radius, and its height.

Solution

The golf bag is in the shape of a cylinder. Therefore, to calculate its volume, the formula for the volume of a cylinder can be used.

It is known that

h = 132

and

r = 15 .

These two values can be substituted into the formula.

V = π r^{2} h

$h = 132$ , $r = 15$

V = π (15^{2}) (132)

Evaluate right-hand side

Calculate power

V = π (225) (132)

Multiply

V = 29700 π

Use a calculator

V = 93305.30181 \dots

Round to $3$ significant digit(s)

V \approx 93300

The volume of LaShay new golf bag is about

93300

cubic centimeters. This is enough to keep all of her golf clubs!

Example

Finding the Radius of a Cylinder

Kriz is making an experiment to complete a Chemistry project. He is using a test tube in the shape of a cylinder with a height of $75$ millimeters and a volume of $20000$ cube millimeters.

tube

For this tube to suit the experiment, it radius must not be greater than

9

millimeters. Help Kriz find the radius! Approximate the answer to one decimal place.

Hint

The formula for the volume $V$ of a cylinder is $V = π r^{2} h,$ where $r$ and $h$ are the radius of the base and the height of the cylinder, respectively.

Solution

The volume of a cylinder is the product of

π,

the square of the base's radius, and the height of the cylinder.

V = π r^{2} h

It is known that the height and the volume of the cylinder are

75

millimeters and

20000

cube millimeters, respectively. These two values can be substituted into the formula for volume of a cylinder, which can then be solved for

r .

V = π r^{2} h

$h = 75$ , $V = 20000$

20000 = π r^{2} (75)

Solve for

r

CommutativePropMult

Commutative Property of Multiplication

20000 = 75 π r^{2}

$LHS / 75 π = RHS / 75 π$

\frac{2 0 0 0 0}{7 5 π} = r^{2}

$LHS = RHS$

\frac{2 0 0 0 0}{7 5 π} = r

Rearrange equation

r = \frac{2 0 0 0 0}{7 5 π}

Use a calculator

r = 9.213177 \dots

Round to $1$ decimal place(s)

r \approx 9.2

When solving the equation, only the principal root was considered. This is because the radius of a circle is always positive. Therefore, the radius of the cylinder's base is about

9.2

millimeters. Since the radius is greater than

9

millimeters, the tube does not suit the experiment.

Discussion

Surface Area of a Cylinder

Besides the radius, height, and volume, another essential characteristic of a cylinder is its surface area. The surface area of a solid is the measure of the total area that the surface of the solid occupies.

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.

$S = 2 π r h + 2 π r^{2}$

Proof

The cylinder's surface area can be seen as three separate parts that are top, bottom, and side. The area of the side has the shape of a rectangle of width $h .$ The top and the bottom are circles of radius $r .$

Since the top and the bottom are congruent circles, they have the same area.

Area of One Circle A = π r^{2} Area of Two Circles A = 2 π r^{2}

To find the area of the rectangle, its length must be found first. The length of the rectangle that forms the lateral part of the cylinder is the circumference of the base, which is

2 π r .

Therefore, the area of the rectangle is the product of

h

and

2 π r .

Area of the Rectangle A = 2 π r h

The surface area of the cylinder

S

is the sum of the areas of the rectangle and the circles.

$S = 2 π r h + 2 π r^{2}$

Example

Finding the Surface Area of a Cylinder

Jordan's father is giving her a new baseball bat for her birthday. To avoid damage to the bat, Papa Jordan will store the bat in a box that has the shape of a cylinder with a height of $30$ inches and a radius of $1.6$ inches. Since this is a present for Jordan, Papa will use wrapping paper to make it even more special.

cylinder

Ignoring any paper overlapping, what is the minimum area of paper that Jordan's father needs? Approximate the answer to one decimal place.

Hint

The area of wrapping paper Jordan's father needs is the same as the surface area of the cylinder.

Solution

To find the minimum area of paper that Jordan's father needs, the surface area

S

of the cylinder needs to be calculated.

S = 2 π r h + 2 π r^{2}

Here,

r

and

h

are the radius and the height of the cylinder, respectively. The radius is

1.6

inches and the height is

30

inches. Therefore, these values can be substituted in the above formula, which can then be simplified.

S = 2 π r h + 2 π r^{2}

$r = 1.6$ , $h = 30$

S = 2 π (1.6) (30) + 2 π (1.6^{2})

Evaluate right-hand side

Calculate power

S = 2 π (1.6) (30) + 2 π (2.56)

CommutativePropMult

Commutative Property of Multiplication

S = 2 (1.6) (30) π + 2 (2.56) π

Multiply

S = 96 π + 5.12 π

Add terms

S = 101.12 π

Use a calculator

S = 317.677849 \dots

Round to $1$ decimal place(s)

S \approx 317.7

Pop Quiz

Practice Finding the Surface Area and Volume of a Cylinder

Find the volume or surface area of the cylinder.

cylinders

Closure

Volume of Skewed Solids

The challenge presented at the beginning of this lesson asked whether the volume of a cylinder changes if the solid is skewed.

cylinder

The Cavalieri's Principle was studied in this lesson. This principle states that two solids with the same height and the same cross-sectional area at every altitude have the same volume. Therefore, if their heights are equal, skewed versions of the same solid have the same volume. This means that the volume of a cylinder does not change even if the cylinder is skewed.

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