# Area of a Circle

## Area of a Circle

The space occupied by a circle in a two-dimensional plane is called the Area of the Circle.

Let us consider a Circle as shown in the figure below. From the figure, $r$ is the Radius, $d$ is the Diameter and $C$ is the Circumference or Perimeter of the circle. The formula for finding the Area of the Circle can be represented in three different forms in terms of Radius, Diameter and Circumference as discussed below.

## Area of a Circle Formula

### Area of a Circle in terms of Radius

The Radius of a circle is the distance from the centre to any point on the circle. The formula for finding the Area of a Circle whose Radius is $r$ as shown in the above figure is given by,

$A = \pi \times {r^2}$

Where, $\pi = \frac{{22}}{7}$ or $\pi = 3.141592….$

### Area of a Circle in terms of Diameter

The Diameter of a circle is twice the Radius of the circle, therefore,

$d = 2 \times r$

$\therefore r = \frac{d}{2}$

$Area,\,\,A = \pi {r^2}$

$A = \pi {\left( {\frac{d}{2}} \right)^2}$

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

### Area of a Circle in terms of Circumference

The Circumference or Perimeter of a circle is given by,

$C = 2\pi r$ or $C = \pi d$

$r = \frac{C}{{2\pi }}$ or $d = \frac{C}{\pi }$

Therefore, the Area of the Circle in terms of Circumference will be,

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

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

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

## List of Formulas for Area of a Circle

The above mentioned formulas on Area are tabulated below:

## Solved Examples on Area of a Circle

1. If the radius of a circle is 7 cm then find the area of the circle?

Solution:
Given,
Radius, $r= 7 cm$
We know that,
$A = \pi {r^2}$
$A = \frac{{22}}{7} \times {7^2}$
$A = \frac{{22}}{7} \times 7 \times 7$
$A = 22 \times 7$
$A = 154c{m^2}$
Therefore, the area of the circle is $154c{m^2}$.

2. If the diameter of a circle is 21 cm then find the area of the circle?

Solution:
Given,
$d=21cm$
Method 1: Converting diamater to radius:
$r = \frac{d}{2} = \frac{{21}}{2} = 10.5$
$A = \pi {r^2}$
$A = 3.14 \times {\left( {10.5} \right)^2}$
$A = 346.1c{m^2}$
Method 2: Calculating area from diameter:
$A = \frac{{\pi {d^2}}}{4}$
$A = \frac{{3.14 \times {{\left( {21} \right)}^2}}}{4}$
$A = \frac{{1384.74}}{4}$
$A = 346.1c{m^2}$
Therefore, the area of the circle is $346.1c{m^2}$

3. If the circumference of a circle is 40 cm then find the area of the circle?

Solution:
Given,
Circumference, $C=40 cm$
We know that,
$A = \frac{{{C^2}}}{{4\pi }}$
$A = \frac{{{{\left( {40} \right)}^2}}}{{4 \times 3.14}}$
$A = \frac{{1600}}{{12.56}}$
$A = 127.3c{m^2}$
Therefore, the area of the circle is $127.3c{m^2}$

4. If the area of a circle is $400 c{m^2}$, then find the circumference, radius and diameter of the circle?

Solution:
Given, Area, $A=400 c{m^2}$
Circumference:
We know that,
$A = \frac{{{C^2}}}{{4\pi }}$
$C = \sqrt {\frac{A}{{4\pi }}}$
$C = \sqrt {\frac{{400}}{{4 \times 3.14}}}$
$C = \sqrt {\frac{{400}}{{12.56}}}$
$C = \sqrt {31.84}$
$C = 5.64cm$
$A = \pi {r^2}$
$r = \sqrt {\frac{A}{\pi }}$
$r = \sqrt {\frac{{400}}{{3.14}}}$
$r = \sqrt {127.3}$
$r = 11.28cm$
Diameter:
$d=2\times r$
$d=2\times 11.28$
$d=22.56 cm$

5. The area of a circle is 49π. what is its circumference?

Solution: The formula for finding the area of a circle is $A=\pi\times r^2$
Given, $A=49\pi$
Therefore,
$49\pi=\pi\times r^2$
$\therefore\,\,\,\,r=7$
The circumference of the Circle will be:
$C=2\times\pi\times r$
$C=2\times\frac{22}{7}\times 7$
$C=44$
Therefore, the circumference of the circle is $44$ units.

6. The radius of a circle is increased by 1%. find how much % does its area increases?

Solution: Let the radius of the circle is $r$.
Therefore, the area of the circle will be, $A=\pi\times r^2$
If the radius of increased by $1%$, then the new radius will be,
$r_1=\frac{1}{100}\times r$
$r_1=1.01r$
The new area will be:
$A_1=\pi\times r_1^2$
$A_1=\pi\times (1.01r)^2$
$A_1=1.0201\pi r^2$
Difference between the new and the old area,
$A_1-A=1.0201\pi r^2-\pi\times r^2$
$A_1-A=.0201$
Therefore, the percentage increase in area is 2.01%

7. Find the area of a quadrant of a circle whose circumference is 22cm?

Solution: The formula for finding the area of a circle in terms of circumference is given by
$A = \frac{{{C^2}}}{{4\pi }}$
$A=\frac{{{22^2}}}{{4\pi }}$
$A=\frac{{{121}}}{{pi }}$
$A=38.51\,cm^2$
The area of a quadrant is $= \frac{1}{4}\times Total Area$
$=\frac{1}{4}\times 38.51$
$=9.62\,cm^2$

## Frequently Asked Questions (FAQ)

What is the formula of area of a circle?

The formula for area of a circle is $A = \pi \times {r^2}$ or $A = \frac{{\pi \times {d^2}}}{4}$.

What is the area and perimeter of a circle?

The area of a circle is the space occupied by the circle in a plane. The formula for finding area of a circle is $A = \pi \times {r^2}$. The perimeter of a circle is the length of a complete circle. The formula for finding perimeter is $2\pi r$.

How to you find the area with diameter?

The formula for finding the area of a circle when diameter is given is $A = \frac{{\pi \times {d^2}}}{4}$, where, d is the diameter.

What is the area and radius?

The area of a circle is the space occupied by the circle in a plane and the radius is the distance from the centre of a circle to any point in the circle.

What is a circle in your own words?

A circle is a collection of points which are at a fixed distance from a fixed point called Centre.

What are the types of circle?

The types of circle are: Concentric circle, Contact of circle and Orthogonal Circle.

How do you find the area of a circle using circumference?

The area of a circle can be found out using circumference by the following formula:
$A = \frac{{{C^2}}}{{4\pi }}$