# Area of a Circle

Sub Topics
Area is the space measured inside the boundary of the circle.

Circle

A locus of a point in a plane such that its distance from a fixed point in the plane is always the same is called a circle.
The fixed point is the centre and the fixed distance is called the radius of the circle.

A circle with centre O and the radius r is denoted by C(O, r)

Circumference is the perimeter of a circle.

### Area of Circle

The area of a circle is the area inside the circle enclosed by its circumference.

## How to Find The Area of a Circle

Area of a circle can be calculated using different methods. Using the given radius or given diameter of the circle or given circumference.

1. Given the radius: Area of the circle is the product of $\pi$ and $r^{2}$, read as ‘pi  r square’.

2. Given the diameter: Using the given diameter the radius is calculated

r =$\frac{d}{2}$ (radius is half the diameter)

Area is then calculated by multiplying $\pi$ and $r^{2}$

3. Given the circumference:
Using the formula for circumference, radius is calculated.

Circumference = 2 $\pi$ r

r = $\frac{circumference}{2 \pi}$ ,  r = $\frac{C}{2\pi}$

Area of the circle
= $\frac{C^{2}}{4\pi}$

## Area of a Circle Formula

If you are given the radius :

Area of a circle = $\pi r^{2}$   ($\pi$ = pi taken as $\frac{22}{7}$ or 3.142, r = radius of the circle)

## Surface Area of a Circle

Surface area is considered for solid figures such as sphere, cylinder, prism etc.

Solid 3-dimensional figure of a circle is a sphere.

Surface Area of a sphere =  $4 \pi r^{2}$  or  $\pi d^{2}$ ($\pi$ = pi taken as $\frac{22}{7}$ or 3.142, r = radius  d = diameter).

## Calculate Area of a Circle

Area of a circle = $\pi r^{2}$Let us consider an example,

Given radius of the circle = 4 units

Area of the circle =$\pi \times (4)^{2}$
=  (3.14) $\times$ 16  = 50.24 sq units.

Given the diameter of the circle,

Diameter = 28 units

Radius = $\frac{d}{2}$ = $\frac{28}{2}$ = 14 units

Area of the circle  = $\pi \times (14)^{2}$

= ($\frac{22}{7}$) $\times 14 \times 14$    (here $\pi$ = $\frac{22}{7}$)

=  22 $\times$ 2 $\times$ 14 =  616 sq units.

## Area of a Semi-Circle

A diameter in a circle divides a circle into two equal arcs. The two arcs is called a semi circle. In simple words a semi-circle is a half circle.

Area of a semi circle is half the area of the given circle.

Area of a semi circle = $\frac{(Area\ of\ the\ circle)}{2}$

= $\frac{(\pi r^{2})}{2}$.

## Area of a Circle Using Diameter

Diameter is the chord in circle passing through the centre of the circle. It is denoted by the letter ‘d’. It is the longest chord in a circle.
Diameter= d = 2 $\times$ radiusi.e. diameter is twice the radius of a circle.

Given the diameter, the area of the circle can be calculated using the formula:

Area of the circle = $\frac{\pi d^{2}}{4}$ ($\pi$ = pi taken as 3.142 approx, d=diameter of the circle)

Let us consider an example:

Given the diameter of a circle 8 units calculate the area of the circle.

Solution:
Given, diameter = 8 units

Area of the circle = $\frac{\pi d^{2}}{4}$

= $\frac{(3.142)\times (8)^{2}}{4}$

= 50.272 sq units.

## Area of a Sector of a Circle

Sector of a circle:
A sector of a circle is the part of the circle enclosed by the two radii and the intercepted arc. It looks like the shape of a circular pizza piece.

If the arc is the minor arc then the sector formed is the minor sector. The remaining part of the circle is called the major sector of the circle.

The area inside this part is the area of the sector.
The angle at the centre of a circle is 360°, in other words ‘$2\pi$’ ($\pi$ = 180°)
Area of the circle is $\pi r^{2}$ , so the area of a sector with some angle ‘$\theta$’ less than $2\pi$ can be written as:

($\frac{\theta}{2\pi}) \times \pi r^{2}$

On further simplification, we get    ($\frac{\theta}{2}) \times r^{2}$

Area of a sector =  $(\frac{1}{2}) \times \theta \times r^{2}$ (when $\theta$ given in radians).

Area of a sector
= $(\frac{1}{2}) \times (\theta \times \frac{\pi}{180}) \times r^{2}$  (when $\theta$ given in degrees).

## Area of a Segment of a Circle

Segment of a circle: The part of the circular region enclosed by an arc and a chord, this region including the arc and the chord is the segment of the circle.

The segment containing the minor arc is called the minor segment and the segment containing the major arc is called the major segment. The centre of the circle lies in the major segment.

Area of a segment= $(\frac{r^{2}}{2}) \times (\theta - \sin \theta)$

(r= radius of the circle, $\theta$ is in radians)

Area of a segment = $(\frac{r^{2}}{2}) \times [(\theta \times \frac{\pi}{180}) - \sin \theta]$

(r= radius of the circle, $\theta$ is in degrees)