Table of Contents

## 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:

Serial No | Description | Formula |

1 | In Terms of Radius | A = \pi \times {r^2} |

2 | In Terms of Diameter | A = \frac{{\pi \times {d^2}}}{4} |

3 | In terms of Circumference | A = \frac{{{C^2}}}{{4\pi }} |

## 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

Radius:

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 }}

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