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Discussion
Definition of a Circle
Before moving on, the definition of a circle and some of its parts will be reviewed.
Next, the formula for calculating the circumference of a circle will be discussed.
Finally, the formula for calculating the area of a circle will be seen.
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.
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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 r2. Since the radius is 6.5 meters, the circumference and area can be calculated.
| Formula | Substitute | Simplify | Approximate | |
|---|---|---|---|---|
| Circumference | C=2πr | C=2π(6.5) | C=13π | C≈40.8 |
| Area | A=πr2 | A=π(6.52) | A=42.25π | A≈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.
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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.
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.
C=2πr
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=2πr
Substitute
C=20
20=2πr
Solve for r
DivEqn
LHS/2=RHS/2
10=πr
DivEqn
LHS/π=RHS/π
π10=r
RearrangeEqn
Rearrange equation
r=π10
UseCalc
Use a calculator
r=3.183098…
RoundSigDig
\RoundSigDig{3}
r≈3.18
A=πr2
To do so, 3.18 will be substituted for r in the formula for the area of a circle.
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.
Discussion
Definition of a Cylinder
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.
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Solution
The volume of a cylinder is the product of π, the square of the base's radius, and the height of the cylinder.
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.
V=πr2h
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=πr2h
SubstituteII
h=75, V=20000
20000=πr2(75)
Solve for r
CommutativePropMult
\CommutativePropMult
20000=75πr2
DivEqn
LHS/75π=RHS/75π
75π20000=r2
SqrtEqn
LHS=RHS
75π20000=r
RearrangeEqn
Rearrange equation
r=75π20000
UseCalc
Use a calculator
r=9.213177…
RoundDec
Round to 1 decimal place(s)
r≈9.2
Discussion
Surface Area of a Cylinder
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.
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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=2πrh+2πr2
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.
Pop Quiz
Practice Finding the Surface Area and Volume of a Cylinder
Find the volume or surface area of the cylinder.
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.
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.