In this tutorial, we’ll focus on how to find the area of a circle. We’ll look at the important formulas, and learn how to use them to solve problems.

So let’s get started. First, a quick recap of a few important concepts and terms

## Circles - A Quick Intro

A circle is the set of all points in a plane that are at a fixed distance $\hspace{0.2em} (r) \hspace{0.2em}$ from a certain point $\hspace{0.2em} (O) \hspace{0.2em}$.

Before we move ahead, here are a few important terms you’ll come across when working with circles or finding their areas.

##### Radius

The radius $\hspace{0.2em} (r) \hspace{0.2em}$ of a circle is the distance of a point (any point) on the circle from its center.

By definition, all points on a circle are at the same distance from the center. And that distance is the radius.

##### Diameter

The diameter $\hspace{0.2em} (d) \hspace{0.2em}$ is the length of a line segment that passes through the center of the circle and touches it on both ends.

It is twice the radius.

$d = 2r$

##### Circumference

Circumference $\hspace{0.2em} (C) \hspace{0.2em}$ refers to the perimeter of a circle. It is the length of the circle’s boundary.

$C = 2 \pi r$

$\hspace{0.2em} \pi \hspace{0.2em}$ (read as pi) is an irrational constant with an approximate value of $\hspace{0.2em} \frac{22}{7} \hspace{0.2em}$ or $\hspace{0.2em} 3.14 \hspace{0.2em}$. However, it’s there on most calculators. So you have access to much more accurate values of it without having to remember them.

## Area of a Circle

The area of a circle refers to a measure of the region (in the 2-dimensional plane) occupied by the circle.

So the area of the circle above is a quantitative measure of the red region.

1. Area of a circle using its radius $\hspace{0.2em} r \hspace{0.2em}$,

$A = \pi r^2$

2. Area using a circle’s diameter $\hspace{0.2em} d \hspace{0.2em}$,

$A = \frac{\pi d^2}{4}$

3. Area using a circle’s circumference $\hspace{0.2em} C \hspace{0.2em}$,

$A = \frac{C^2}{4 \pi}$

4. Area of a sector with radius $\hspace{0.2em} r \hspace{0.2em}$ and arc length $\hspace{0.2em} l \hspace{0.2em}$,

$A = \frac{lr}{2}$

5. Area of a sector with radius $\hspace{0.2em} r \hspace{0.2em}$ and central angle $\hspace{0.2em} \theta \hspace{0.2em}$,

$A = \pi r^2 \cdot \frac{\theta}{360 \degree}$ $\hspace{0.2em} \theta \hspace{0.2em}$ in degrees

$A = \frac{r^2 \theta}{2}$ $\hspace{0.2em} \theta \hspace{0.2em}$ in radians

Skip to examples ### Deriving the Main Formula (W/ Radius)

Here are a couple of ways we can obtain the formula for the area of a circle.

##### Method 1

Consider slicing the circle into a number of equal parts (as shown in the top half of the image below) and then joining them side by side (as shown in the bottom half).

As you can see, what we end up with is an approximation of a parallelogram. Also, as we increase the number of slices, the result would be closer and closer to a perfect parallelogram.

So if we can find the area of the parallelogram, we would have found the area of the circle.

Now, the base of the parallelogram would be equal to half the circumference of the circle (the other half is on the other side – the top). And the height would be equal to the radius.

Using the formula for the area of a parallelogram, we have

$\begin{align*} A &= \text{base} \cdot \text{height} \\[1em] &= \pi r \cdot r \\[1em] &= \pi r^2 \end{align*}$

And that’s it. We have derived the area of the circle.

##### Method 2

This time, consider dividing the circle into thin circular strips (as shown in the top half of the figure below), cutting open each strip, and stacking them one over another (as shown in the bottom half).

That gives us something of a triangular shape. Also, as we make those strips thinner, the result becomes increasingly closer to an actual triangle.

That means we can get the area of the circle by finding the area of the triangle.

Now, the base of the triangle would be equal in length to the outer-most strip. Or equal to the circumference of the circle. And the height would be equal to the radius.

Using the formula for the area of a triangle, we have –

$\begin{align*} A &= \frac{1}{2} \cdot \text{base} \text{height} \\[1.3em] &= \frac{1}{2} \cdot 2 \pi r \cdot r \\[1.3em] &= \pi r^2 \end{align*}$

Sometimes, the radius of the circle may not be directly available to you. So, here are the formulas for the area of a circle using the diameter or circumference.

##### 1. Using Diameter $\hspace{0.2em} (d) \hspace{0.2em}$

Here’s how we get the formula.

First, we find the radius in terms of the diameter.

$\begin{align*} d = 2r \\[1em] r = \frac{d}{2} \end{align*}$

And then, we substitute this value of r in our standard formula.

$\begin{align*} A &= 2 \pi r^2 \\[1.3em] &= \pi \left ( \frac{d}{2} \right ) ^2 \\[1.3em] &= \frac{\pi d^2}{4} \end{align*}$

##### 2. Using Circumference $\hspace{0.2em} (C) \hspace{0.2em}$

Here’s how we get this formula.

This time we start by finding the radius in terms of the circumference.

$\begin{align*} C &= 2 \pi r \\[1em] r &= \frac{C}{2 \pi} \end{align*}$

And then, substituting the value of r in our formula for area, we have –

$\begin{align*} A &= \pi r^2 \\[1.3em] &= \pi \left ( \frac{C}{2 \pi} \right ) ^2 \\[1.3em] &= \frac{C^2}{4 \pi} \end{align*}$

We'lll look at the formulas for the area of a sector later in the tutorial.

## How to Find the Area of a Circle | Examples

Alright. Time to solve some examples using the formula we have learned so far.

Example

Find the area of the circle shown below.

Solution

A simple and easy question. We just need to plug in the value of the radius (r = 22 ft) in the formula for the area of a circle.

$\begin{align*} A &= \pi r^2 \\[1em] &= \pi \cdot 22^2 \\[1em] &\approx 1520.53 \end{align*}$

So the area of the circle is $\hspace{0.2em} 1520.53 \text{ ft}^2 \hspace{0.2em}$.

Example

Find the area of a circle with a diameter of $\hspace{0.2em} 28 \hspace{0.2em}$ cm.

Solution

Again, we’ll be using the same formula.

$A = \pi r^2$

However, this time the question gives us the diameter ($\hspace{0.2em} d = 28 \hspace{0.2em}$cm ) and not the radius. So, the first step is to get the radius.

$\begin{align*} r &= \frac{d}{2} \\[1.3em] &= \frac{28}{2} = 14 \end{align*}$

Now substituting the value of $\hspace{0.2em} r \hspace{0.2em}$ in our formula, we have –

$\begin{align*} A &= \pi \cdot 14^2 \\[1em] &\approx 615.75 \end{align*}$

That’s it. The area of the circle is $\hspace{0.2em} 615.75 \text{ cm}^2 \hspace{0.2em}$.

Example

The circumference of a circle is $\hspace{0.2em} 15 \hspace{0.2em}$. Find its area.

Solution

Here the question gives us the circumference of the circle $\hspace{0.2em} (C = 15) \hspace{0.2em}$. So we’ll use the relation between $\hspace{0.2em} C \hspace{0.2em}$ and the radius, $\hspace{0.2em} r \hspace{0.2em}$, to obtain $\hspace{0.2em} r \hspace{0.2em}$.

$C = 2 \pi r$

Substituting the value of $\hspace{0.2em} C \hspace{0.2em}$ and solving for $\hspace{0.2em} r \hspace{0.2em}$ –

$r = \frac{15}{2 \pi}$

Finally, we use the formula for the area.

$\begin{align*} A &= \pi r^2 \\[1.3em] &= \pi \cdot \left ( \frac{15}{2 \pi} \right )^2 \\[1.3em] &\approx 1744.1 \end{align*}$

## Area of a Sector

A sector of a circle is the region enclosed between any two radii and the arc connecting them. For example, the yellow region in the figure below.

##### Sector's Area Using $\hspace{0.2em} r \hspace{0.2em}$ and $\hspace{0.2em} l \hspace{0.2em}$

The area of a sector with a radius $\hspace{0.2em} r \hspace{0.2em}$ and arc length $\hspace{0.2em} l \hspace{0.2em}$ is given by –

$A = \frac{lr}{2}$

Here's the derivation.

Because of the symmetry of a circle, the area of a sector in a circle is proportional to its arc length. So if $\hspace{0.2em} A_c \hspace{0.2em}$ is the area of the whole circle, the sector's area $\hspace{0.2em} (A) \hspace{0.2em}$ would be -

$A \hspace{0.2em} = \hspace{0.2em} A_c \cdot \frac{\text{arc length (sector)}}{\text{arc length (circle)}}$

Now we know $\hspace{0.2em} A_c = 2 \pi r^2 \hspace{0.2em}$ and arc length of a full circle is its circumference $\hspace{0.2em} 2 \pi r \hspace{0.2em}$. Substituting these into the equation above, we get

$\begin{align*} A \hspace{0.2em} &= \hspace{0.2em} \pi r^2 \cdot \frac{l}{2 \pi r} \\[1.5em] &= \hspace{0.2em} \frac{lr}{2} \end{align*}$

##### Sector's Area Using $\hspace{0.2em} r \hspace{0.2em}$ and $\hspace{0.2em} \theta \hspace{0.2em}$

The area of a sector with a radius $\hspace{0.2em} r \hspace{0.2em}$ and central angle $\hspace{0.2em} \theta \hspace{0.2em}$ is given by –

$A = \pi r^2 \cdot \frac{\theta}{360 \degree}$ $\hspace{0.2em} \theta \hspace{0.2em}$ in degrees

$A = \frac{r^2 \theta}{2}$ $\hspace{0.2em} \theta \hspace{0.2em}$ in radians

To derive these formulas, again we start from the idea that the area of a sector is proportional to its central angle (because of the symmetry). So,

$A \hspace{0.2em} = \hspace{0.2em} A_c \cdot \frac{\text{central angle (sector)}}{\text{central angle (circle)}}$

Now the central angle of a full circle is $\hspace{0.2em} 360 \degree$ or $\hspace{0.2em} 2 \pi \hspace{0.2em}$ radians. And of course, $\hspace{0.2em} A_c = 2 \pi r^2 \hspace{0.2em}$. Substituting the values into the above equation, we get the two formula (one for central angle in degrees and the other for that in radians).

Example

Find the area of the sector.

Solution

The formula for the sector of a circle when we know the radius $\hspace{0.2em} r \hspace{0.2em}$ and central angle $\hspace{0.2em} \theta \hspace{0.2em}$ is –

$A \hspace{0.2em} = \hspace{0.2em} \pi r^2 \cdot \frac{\theta}{360 \degree}$

Remember, this formula assumes $\hspace{0.2em} \theta \hspace{0.2em}$ is in degrees.

Remember, this formula assumes $\hspace{0.2em} \theta \hspace{0.2em}$ is in degrees.

Substituting the values into the formula, we have

$\begin{align*} A \hspace{0.2em} &= \hspace{0.2em} \pi r^2 \cdot \frac{\theta}{360 \degree} \\[1.5em] &= \hspace{0.2em} \pi \cdot 10^2 \cdot \frac{60 \degree}{360 \degree} \\[1.5em] &= \hspace{0.2em} 52.36 \end{align*}$

So the area of the sector is $\hspace{0.2em} 52.36 \hspace{0.2em}$ sq. units.

Example

Find the area of the sector.

Solution

The formula of the area of a sector when we know the arc length $\hspace{0.2em} l \hspace{0.2em}$ and radius $\hspace{0.2em} r \hspace{0.2em}$ is –

$A \hspace{0.2em} = \hspace{0.2em} \frac{lr}{2}$

Plugging the values of $\hspace{0.2em} l \hspace{0.2em}$ and $\hspace{0.2em} r$ ,

$\begin{align*} A \hspace{0.2em} &= \hspace{0.2em} \frac{lr}{2} \\[1.5em] &= \hspace{0.2em} \frac{12 \times 18}{2} \\[1.5em] &= \hspace{0.2em} 108 \end{align*}$

Done.

And with that, we come to the end of this tutorial. Until next time.