# How to calculate circumference

Circles are a very common shape. If you think about it, we are surrounded by circular figures in our everyday life: most glassware we use to drink water, coffee mugs, bicycle and car tires, engagement rings, roundabouts, our pupils, among many other.

Since ancient times, humans have been attracted to this simple but puzzling geometry for many different reasons: it has no straight sides, it possesses an infinite number of symmetry planes, and it is closely related to the mathematical constant π (keep reading to learn more about this!).

An important characteristic of circles is their circumference, or the distance covered around it. Let’s learn how to calculate it without the need of flexible measuring tape, and let’s discover why so many great mathematicians are so attracted to it.

## How to calculate circumference

To calculate a circle’s circumference first find its radius, r, which equals half of its diameter. Then, apply the following equation to find its circumference, C:

Where π = 3,14159.

## What is the circumference?

Circumference is defined as the perimeter of a circle. But what exactly is the perimeter? Its name comes from Greek, as many concepts in geometry do, and is formed by περί (peri), which means “around”, and μέτρον (metron), which means “to measure”. Together, they form an expression that means “to measure around” something, in this case, around the space enclosed by geometrical figures.

Ancient Greeks loved to experiment with different shapes they called polygons. These are shapes built with straight segments called sides, which form a closed figure. You have probably seen many polygons. For example, triangles are 3-sided polygons, while rectangles —and squares among them— are 4-sided polygons. Every polygon has a specific perimeter, which depends on how long its sides are.

Another figure Greeks were very fond of is the circle. In this case, the perimeter is referred to as circumference, which in turn comes from the Latin circum, which means “around”, and ferre, which means “to carry”. This is because, as with any type of perimeter, it refers to the length of the path that outlines a shape. Since, in this case, the shape is circular, the perimeter will revolve around it, and thus the Latin expression to “carry around” something makes perfect sense.

A circle’s circumference can be measured as any other length using a measuring tape. Nevertheless, this is not practical at times. For example, imagine you need to determine the perimeter of a circular piece of land to install a fence of a proper length. If the land is too big, going around it to measure its perimeter in sections would simply take a lot of time and effort.

Another way to measure a circle’s circumference could be by extending its perimeter in a straight line, and then measuring it with any suitable device. This is, of course, not possible in many scenarios, but it could look something like the following image. Here, C represents the circle’s circumference.

Finally, a more practical way to determine a circle’s circumference is by use of its enigmatic properties. Let’s see how:

A very interesting property of circles is the relationship between their perimeter and their diameter. The latter simply refers to how wide a circle is. It is alternately defined as the length of any line that connects two points on its perimeter, and which at the same time passes through its center. The following image shows both quantities:

What is interesting about circles is that no matter its size, the ratio of its circumference and its diameter will always result in the same number: 3,14159265359… We call this number pi and represent it with the Greek letter π. This simple but puzzling phenomenon has been known for thousands of years, and can be written as:

It is important to remember that this equation holds for any circle in the universe. Since it relates two basic characteristics of a circle, it can be used to calculate them if the other one is known. If we solve the equation for the circumference we get:

This way, a new method to find a circle’s circumference can be extracted from the definition of π. A third important characteristic of a circle is its radius, which simply equals half of its diameter:

By replacing equation 4 in equation 3, we get equation 1, which is the most common mathematical definition of the circumference. Now, to calculate this value for any circle, you only need to find its radius, multiply it by 2 and then by the mathematical constant π. Since π is unitless, the circumference inherits the units of length from the radius.

Equation 3 implies that, for any circle with diameter d, its circumference will have a value of pi times d. If we choose a circle of diameter equal to 1 m, its circumference will be pi meters, as the following animation shows:

Taken from: Wikimedia Commons, John Reid, 2006.

Example 1:

Find the circumference of a circle with a diameter equal to 1,5 m.

Example 2:

What is the diameter of a circle with a circumference of 10,3 m?

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