How to calculate the length of a circle

Geometry of a Circle

Definition

The distance around the boundary of a circle is called the circumference.

The distance across a circle through the centre is called the diameter.

The distance from the centre of a circle to any point on the boundary is called the radius. The radius is half of the diameter; $2r=d$.

The line segment that joins two points on the circle is a chord. Every diameter is a chord, but not every chord is a diameter.

The area that a chord cuts off is called a segment.

The area inside a circle and bounded by two radii is a sector.

The length between two points around the circumference of a circle is an arc.

Circumference

Definition

The formula for calculating the circumference is \[C = \pi d \qquad\text{or } \qquad C = 2\pi r\] where $d$ is the diameter and $r$ is the radius.

Worked Examples

Example 1

The radius of a given circle is $r=4$cm. Calculate the circumference.

Solution

\begin{align} C &= 2\pi r\\ &= 2 \times \pi \times 4\\ &= 8 \pi \\ &\approx 25.1 \text{cm (to 1 decimal place)} \end{align}

Example 2

Find the diameter of a circle with circumference $18$cm.

Solution

\begin{align} C&=\pi d\\ 18 &= \pi d \end{align} Divide both sides by $\pi$: \begin{align} \frac{18}{\pi} &= d \\ d &= \frac{18}{\pi}\\ d &\approx 5.7 \text{cm (to 1 decimal place)} \end{align}

Area

Definition

The area of a circle with radius $r$ is \[\text{Area }=\pi r^2\]

Worked Examples

Example 1

The radius of a given circle is $2.5$cm. Find the area of the circle.

Solution

\begin{align} \text{Area }&=\pi r^2\\ &=\pi \times 2.5^2\\ &= 6.25\pi \\ &\approx 19.6 \text{cm² (to 1 decimal place)} \end{align}

Example 2

The area of a circle is $50$cm². Find the radius.

Solution

\begin{align} \text{Area }&=\pi r^2\\ 50 &= \pi \times r^2 \end{align} Divide both sides by $\pi$. \begin{align} r^2 &= \frac{50}{\pi}\\ r &= \sqrt{\frac{50}{\pi} }\\ r &\approx 4.0 \text{cm (to 1 decimal place)} \end{align}

Workbook

This workbook produced by HELM is a good revision aid, containing key points for revision and many worked examples.

• The circle

See Also

• Arc Length and Area of a Sector

External Resources

• The geometry of a circle workbook at mathcentre.

Test Yourself

Test yourself: Area of Geometric Shapes

Please provide any value below to calculate the remaining values of a circle.

While a circle, symbolically, represents many different things to many different groups of people including concepts such as eternity, timelessness, and totality, a circle by definition is a simple closed shape. It is a set of all points in a plane that are equidistant from a given point, called the center. It can also be defined as a curve traced by a point where the distance from a given point remains constant as the point moves. The distance between any point of a circle and the center of a circle is called its radius, while the diameter of a circle is defined as the largest distance between any two points on a circle. Essentially, the diameter is twice the radius, as the largest distance between two points on a circle has to be a line segment through the center of a circle. The circumference of a circle can be defined as the distance around the circle, or the length of a circuit along the circle. All of these values are related through the mathematical constant π, or pi, which is the ratio of a circle's circumference to its diameter, and is approximately 3.14159. π is an irrational number meaning that it cannot be expressed exactly as a fraction (though it is often approximated as 22/7) and its decimal representation never ends or has a permanent repeating pattern. It is also a transcendental number, meaning that it is not the root of any non-zero, polynomial that has rational coefficients. Interestingly, the proof by Ferdinand von Lindemann in 1880 that π is transcendental finally put an end to the millennia-old quest that began with ancient geometers of "squaring the circle." This involved attempting to construct a square with the same area as a given circle within a finite number of steps, only using a compass and straightedge. While it is now known that this is impossible, and imagining the ardent efforts of flustered ancient geometers attempting the impossible by candlelight might evoke a ludicrous image, it is important to remember that it is thanks to people like these that so many mathematical concepts are well defined today.

Circle Formulas

D = 2R C = 2πR A = πR2

where: R: Radius D: Diameter C: Circumference A: Area π: 3.14159

What is the length across a circle?

How do you calculate length?