What is a circle?

A circle is a set of points in a given plane, all equally distanced from one point – the center. The definition Euclid gave in the first book of the Elements is: A circle is a plane figure bounded by one curved line, and such that all straight lines drawn from a certain point within it to the bounding line, are equal. The bounding line is called its circumference and the point, its center.

There are 4 important values for the circle: the diameter, radius, surface area and, of course, the circumference.

The circle is divided into two regions: the interior and exterior. However, we should make a clear distinction between a circle and a disc. Generally, we use the term circle to define either the entire shape or just the outline. In reality, the circle is just the perimeter, while the entire shape is called a disc. Generally in geometry the perimeter is the outer boundary of a two-dimensional shape. For the disc, the perimeter is the circumference. The disc possesses something known as circular symmetry. This means that, whatever angle you view it from, it will always appear the same.

History of the circle

The circle has great historical value. It got its name from the Greek word κίρκος (kirkos), which means hoop or ring. While it is a basic shape, humans have been observing it for millions of years because it is widely present in nature. Because of its unique properties, many scholars in the medieval times believed it to be a divine shape. But long before that, the circle served as an inspiration for the invention of the wheel.

The circle signifies many sacred and spiritual concepts, including, but not limited to unity, infinity, wholeness, the universe, divinity, balance, stability and perfection, as the circle was always considered a perfect shape.

What is the circumference of a circle?

The simplest way to define the circumference of a circle is as the arc length of a circle. In other words, if you took the circle and straightened it out into a line, the length of the line would equal the circumference of the circle. The formula for calculating the length of the circumference of a circle is:

C = 2 \cdot r \cdot \pi

r is the radius of a circle, which is the distance between the circle and the center point, while π is a mathematical constant, roughly equal to 3.14.

From here, it’s easy to see how mathematicians found the value of π, and you can even do it yourself. Using the formula above, we can get the formula for π, which would be:

\pi\ = \frac {C}{2r}

For a sphere, which is a three dimensional shape, the formula for calculating the circumference is the same as for the circle. The Earth is a sphere, and its circumference is 40,075 km or 24901.451 miles.

How to calculate the circumference of a circle?

Calculating the circumference of a circle is very easy. We mentioned the formula before, so let’s do a quick example now. Let’s say we have a circle with a diameter of 2 m. What would its circumference be? Don’t let the diameter confuse you. As we saw, the radius is the distance from the center point to the border, while the diameter is essentially a line that goes through the center and connects two points in the circle. So, the radius is equal to half the diameter:

r= \frac {D}{2} \\ r= \frac {2m}{2} = 1m

Now we can calculate the circumference:

C = 2 \cdot 1m \cdot 3.14 \\ C= 6.28m

And while we’re at it, let’s calculate the surface area as well. The formula for calculating it is:

S= r^2 \cdot \pi

So, the calculation would go as follows:

S= (1m)^2 \cdot 3.14 \\ S= 1m^2 \cdot 3.14 \\ S = 3.14 m^2

If you need to calculate the surface area of a circle, or any other shape, check out our Square Footage Calculator!

Circumference to diameter

To calculate the diameter of a circle from its circumference, all we have to do is reverse engineer the formula from before, which was:

C=2 \cdot r \cdot \pi \\ 2 \cdot r = D \\ C= D \cdot \pi

So, the length of the diameter would be equal to:

D=\frac {C}{\pi}

Squaring the circle

Other than being an idiom for “doing the impossible“, this is a problem proposed by ancient mathematicians. The aim of this problem is to construct a square that has the same surface area (a2) as a given circle (r2*π). This task was thought to be possible because it was still not known that π was a transcendental number, which is a number that is not the root of any non-zero polynomial. Because π falls in the realm of transcendental numbers, it is impossible to construct it with a compass and straightedge with a finite number of moves. If we remove the last requirement, it is in some sense possible to square a circle. However, this is unimportant as it can’t be realistically done by a human.