Circles and Pi:

Pi and the Circumference of a Circle

The perimeter of a shape is the outside edge of that shape. When we talk about circles, we call that the circumference. The ratio or fraction between the circumference and the diameter \(\left ( \frac{\text{circumference}}{\text{diameter}} \right )\) of any circle is the same no matter the size of the circle. That ratio is \({\text{π}}\) (pi).

The video in this lesson will go over how to find the circumference of a circle when given the diameter or radius. Here are some vocabulary words to help you with the lesson.

Circumference = the perimeter (or the outside edge) of a circle
Diameter = the distance across the circle at the widest part
Radius = the distance from the center of the circle to the edge, or half the diameter
Ratio = how two things are related through multiplication and division

Video Source (09:42 mins) | Transcript

Let \({\text{r}} = \text{radius}\) and \({\text{d}} = \text{diameter}\)

Circumference: \(2{\text{π}} {\text{ r}} = {\text{π}} {\text{ d}}\)

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. We 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, we will always multiply pi into our 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.

Additional Resources

Khan Academy: Labeling Parts of a Circle (2:00 mins; Transcript)
Khan Academy: Radius, Diameter, Circumference, & π (11:05 mins; Transcript)

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.

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.

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.

Solutions

75.4
Radius = 18
Circumference = 113 (Written Solution)

We are trying to find the circumference or perimeter of a circle with a diameter of 36.

We 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 we need to use the formula:

\({\text{C}} = 2 {\text{π}} {\text{ r}}\)

The first thing that we need to do is replace ‘r’ with 18

\({\text{C}} = 2 {\text{π}} 18\)

Since we can multiply in any order, we can multiply 2 times 18 which gives us:

\({\text{C}} = 36 {\text{π}}\)

This is an acceptable answer, but we can also multiply 36 by \({\text{π}}\) to get:

\({\text{C}} = 113.097335529...\)

So the circumference of the circle is about 113 when rounded to the nearest whole number.
62.8 mm (Written Solution)

To find the circumference or perimeter of a circle, we need to use the following formula:

\({\text{C}} = 2{\text{π}} {\text{ r}}\)

The first thing that we need to do is replace ‘r’ with 10

\({\text{C}} = 2{\text{π}} 10\)

Since we can multiply in any order, we can multiply 2 times 10 which gives us:

\({\text{C}} = 20{\text{π}}\)

This is an acceptable answer, but we can also multiply 20 by \({\text{π}}\) to get:

\({\text{C}} = 62.83185307...\)

So the circumference of the circle is about 62.8 mm when rounded to the nearest tenth.
69.12 in
Note: Depending on the number of digits after the decimal point you use for pi, you will get a slightly different answer. This first answer assumes you use at least 3 digits after the decimal point (3.142). If you only use two digits after the decimal point in your equation (3.14), your answer will be 69.08 in. The more digits you use, the more accurate your answer will be.
424 m (Solution Video | Transcript)
Length of spoke (radius) = 28.5 cm
Circumference = 179.07 cm (Solution Video | Transcript)