The perimeter of a geometric figure is defined as the measurement of the boundary around a given area. When calculating the perimeter of a standard polygon, the perimeter can often be found by adding the side lengths of a given shape. In the special case of a parallelogram, the perimeter can be calculated from just two sides, and when finding the perimeter of a square it is just four times one side length.

The real challenge comes with the circle. The ‘perimeter’ of a circle has no straight lines that are easily measured, so they require a special formula to determine this value. The perimeter of the circle actually has special terminology to describe it; we call this measurement the circumference of a circle. We define the circumference to be the total distance around the boundary of the circle or the distance around the area of the circle.

#### TL;DR (Too Long; Didn't Read)

The perimeter of a circle, known as the circumference in math, can be found using the radius or diameter of a circle and the following formula:

## The Circumference Formula

Circles make use of an iconic irrational number: pi (π). With any circle, when the circumference is divided by diameter of the circle, we get the mathematical constant pi, roughly equivalent to 3.141592653589. Since pi is irrational (a decimal number that continues forever without repeating), it is often approximated to 3.14, and this value of pi serves to be accurate enough for most calculations.

#### Tips

Only 39 digits of pi would be needed to calculate the circumference of the observable to within the width of a hydrogen atom!

### Deriving the Formula for Circumference

If we start with the ratio that determines pi, then we can calculate the general formula for the circumference of a circle.

to get this formula in terms of diameter and pi, we can multiply diameter by both sides of the equation leading to:

We can simplify the formatting for this formula by representing the circumference with the variable *C* and the diameter with the variable *d*. This is similar to representing the quantity pi wth its greek letter π.

So the formula for circumference is:

### Circumference in Terms of Radius of a Circle

The circumference a circle can also be represented in terms of the radius of the circle. We can define the radius of a circle in terms of the diameter through the relationship:

Substituting this into the previous equation to find circumference of the circle, we get:

### Further Uses

Now that we can easily calculate the circumference from either diameter or radius, we can use this formula to find the perimeter of a semi circle or arc length.

Arc length can be explained as simply the length of a section of the boundary around a circle. When trying to find the circumference of a semi circle or a section of a curved surface this can be very useful.

The derivation of this formula can be complicated; here are additional resources to understand arc length.

## Reversing the Formula for Circumference of a Circle

If a real world situation or high school worksheet provides the circumference of a circle as a starting value, we can still use this value to quickly find other measurements of these important geometric shapes. By solving for a circle’s radius or diameter in terms of circumference we can find these useful formulas.

These simple line segments can then be used to find the area of a circle, surface area, or volume using other circle formulas.

Tips

- Calculators can help you out when figuring the circumference, but it is better to exercise that brain muscle and work it out by hand.

About the Author

Andrea Coventry has been writing online since 2007. Her expertise includes teaching, working with children and Montessori schools. Her work has appeared in various online publications. Coventry is certified by the American Montessori Society.

Photo Credits

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