Introduction
In this lesson, you will learn how to find the circumference (perimeter) of a circle.
This video illustrates the lesson material below. Watching the video is optional.
Circumference
Understanding these terms will help you with the rest of the lesson.
- Circumference (C): The perimeter (outside length) of a circle, or the distance around the circle
- Diameter (d): The distance across the circle at the widest part that passes through the center of the circle
- Radius (r): The distance from the center of the circle to the edge, or half of the diameter
- Ratio: How two things are related through multiplication and division
The ratio, or fraction, between the circumference and the diameter of a circle is always the same, no matter the size of the circle. That ratio is \(\mathbf{\pi}\) (pi).
\begin{align*} \pi = \frac{Circumference}{diameter}\end {align*}
Pi or \(\mathbf{\pi}\)
Pi is a Greek number, and it represents the number 3.141592…. The number \(\pi\) goes on for infinity and doesn’t seem to have a stopping point. For this lesson, remember the first three digits: 3.14.
\begin{align*} \pi = 3.14 = \frac{Circumference}{diameter} \end{align*}
Finding the Circumference of a Circle
There are two simple formulas for finding the circumference of a circle where C is the circumference, d is the diameter, and r is the radius of the circle.
\begin{align*} C&=\pi d &\text{(when the diameter is given)}\\\\ C&=2\pi r &\text{(when the radius is given)}\\\\ \end{align*}
Example 1
Find the circumference of a circle that has a diameter of 10 units.
\begin{align*} C &= \pi d &\color{red}\small\text{Formula when diameter is given}\\\\ C &= (\pi)(10) &\color{red}\small\text{Substitute given terms}\\\\ C &= 10 \pi &\color{red}\small\text{Exact value with \(\pi\)}\\\\ \end{align*}
\(10\pi\) would be an acceptable answer, but you can simplify the answer further by multiplying 10 by 3.14. Your calculator may also have the pi symbol \(\pi\), in which case you can multiply 10 by \(\pi\) for a more accurate calculation.
\begin{align*} C &= \pi d &\color{red}\small\text{Formula when diameter is given}\\\\ C &= (3.14)(10) &\color{red}\small\text{Substitute given terms}\\\\ C &= 31.4 &\color{red}\small\text{Multiply}\\\\ \end{align*}
The circumference of the circle is 31.4 units.
Example 4
Find the circumference of a circle with a radius of 3 units.
\begin{align*} C &= 2\pi r &\color{red}\small\text{Formula when radius is given}\\\\ C &= 2(3.14)(3) &\color{red}\small\text{Substitute given terms}\\\\ C &= 18.84 &\color{red}\small\text{Multiply}\\\\ \end{align*}
The circumference of the circle is 18.84 units.
Things to Remember
- The relationship between the circumference and diameter of a circle is \(\pi\).
- To find the circumference of a circle, you can use two different formulas:
- \(C=\pi d\) (when the diameter is given)
- \(C=2\pi r\) (when the radius is given)
Standard Mathematical Formats with Pi
When pi is part of a solution there are two ways you can display the solution. The first way is to write the number part of the solution multiplied to pi such as \( 13 \pi\; ft \) or \( 5.3 \pi\; cm \). You generally write the number then pi and then the units.
The second way to show your solution is to multiply the number portion of the solution to pi and then round to an appropriate place value. (Example: \( 13 \pi\; ft = 40.84\; ft \) rounded to the nearest hundredth)
In this course, you will always multiply pi into the solution and round to an appropriate place value. Just know that the other way is commonly used and you may see it in textbooks or other classes as a standard way to write solutions when pi is involved.
Practice Problems
- A circle has a radius of 12. Use the formula for the circumference of a circle to find the circumference of this circle. Round to the nearest tenth. (
- A circle has a diameter of 36. Given that the diameter of a circle is equal to 2 times the radius, find the radius of this circle and then use the formula for the circumference of a circle to find the circumference of this circle. Round to the nearest whole number. (
Radius = 18
Circumference = 113
Details:
You are trying to find the circumference or perimeter of a circle with a diameter of 36.
You know that the radius is half the length of the diameter.
Since the diameter is 36, that means the radius is 18 because the radius is half of the diameter. To find the circumference you need to use the formula:
\({\text{C}} = 2 {\text{π}} {\text{ r}}\)
The first thing to do is replace ‘r’ with 18.
\({\text{C}} = 2 {\text{π}} 18\)
Since you can multiply in any order, you can multiply 2 times 18 which gives you:
\({\text{C}} = 36 {\text{π}}\)
This is an acceptable answer, but you can also multiply 36 by \({\text{π}}\) to get:
\({\text{C}} = 113.097335529...\)
The circumference of the circle is about 113 when rounded to the nearest whole number. - A coin has a radius of 10 mm. Use the formula for the circumference of a circle to find the circumference of the circle defined by this coin. Round to the nearest tenth. (
Details:
To find the circumference or perimeter of a circle, use the following formula:
\({\text{C}} = 2{\text{π}} {\text{ r}}\)
The first thing to do is replace ‘r’ with 10.
\({\text{C}} = 2{\text{π}} 10\)
Since you can multiply in any order, you can multiply 2 times 10 which gives you:
\({\text{C}} = 20{\text{π}}\)
This is an acceptable answer, but you can also multiply 20 by \({\text{π}}\) to get:
\({\text{C}} = 62.83185307...\)
The circumference of the circle is about 62.8 mm when rounded to the nearest tenth. - A clock has a radius of 11 inches. Use the formula for the circumference of a circle to determine the circumference of the circle defined by this clock. Round to the nearest hundredth. (
- One of the world’s largest Ferris wheels can be found in London and is known as the London Eye. The glass enclosures where the passengers ride are known as cabins. The distance from the center of the London Eye to one of the cabins (radius) is 67.5 meters. Use the formula for the circumference of a circle to find the circumference of the London Eye. Round to the nearest whole number. (
- A bicycle tire has numerous spokes that radiate out from the center of the wheel to provide support to the tire. The diameter of a particular tire is 57 cm. Find the length of one of the spokes (radius) and then use this information to calculate the circumference of the circle defined by this tire. Round to the nearest hundredth. Note: Be sure to use either the pi button on your calculator or at least 4 digits after the decimal point for pi (3.1416) if you type it in. See the note on question #4 for why this is important. (
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