Area of a Circle – Definition, Formulas, Examples

发布时间：2025-08-26 19:42

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What is the Area of a Circle

The area of a circle is the space occupied by the circle in a two-dimensional plane. It is expressed in square units such as m2, cm2, etc.

The shaded region below shows the area of the circle.

Area of a Circle

Formulas

The area of a circle can be calculated using three different formulas. The formulas are used based on whether the radius, diameter, or circumference is known to us. Each of the three situations is discussed below with their formulas and solved examples.

How to Find the Area of a Circle Using the Radius

When the radius of a circle is known, the formula to determine the area is given below:

Area of a Circle Formula

Let’s solve some examples to understand the concept better.

Solved Examples

Find the area of a circle whose radius is 6 inches.

Solution:

As we know,
A = πr2, here π = 3.141, r = 6 inches
= (3.141 × 36) cm2
= 113.07 cm2

Solve the area of the given circle in terms of π.

Solution:

As we know,
A = πr2, here π = 3.141, r = 11 m
A = πr2, here π = 3.141, r = 11 m
= (π × 121) m2
= 121π m2

Find the radius of the circle whose measure of surface area is 330 mm2

Solution:

As we know,
A = πr2, here A = 330 mm2,π = 3.141
330= 3.141 × r2
r2 = 330/3.141 = 105.06
r = 10.24 mm

How to Calculate the Area of a Circle Using the Diameter

When the diameter of the circle is given, the formula to get the area is given below:

Area of a Circle using Diameter

Derivation

The above formula is obtained from the standard formulas A = πr2 …..(1)

As we know,

Diameter (d) = 2 × radius (r)

r = d/2

Putting the value of r in equation (1) we get,

A = π(d/2)2

A = πd2/4

Let’s solve some examples to understand the concept better.

Solved Examples

Find the area of a circle with a diameter of 14 inches.

Solution:

As we know,
Area (A) = πd2/4, here d = 14, and π = 3.141
= 3.141 × (14)2/4
= 153.90 inches2

What is the approximate area of the given circle.

Solution:

As we know,
Area (A) = πd2/4, here d = 25 cm, and π = 3.141
= 3.141 × (25)2/4
= 490.78 cm2

Calculate the diameter of a circle with an area of 170 m2.

Solution:

As we know,
Area (A) = πd2/4, here A = 170 m2,π = 3.141
170 = 3.141 × d2/4
680 = 3.141 × d2
d2 = 680/3.141 = 216.49 m
d = 17.79 m

How to Calculate the Area of a Circle Using the Circumference

Given the circumference of the circle, we can calculate the area using the given formula.

Area of a Circle with Circumference

Derivation

The above formula is obtained from the standard formulas A = πr2 …..(1)

As we know,

Circumference (C) = 2πr

r = C/2π

Putting the value of r in equation (1) we get,

A = π × (C/2π)2

Area (A) = C2/4π

Let’s solve some examples to understand the concept better.

Find the area of a circle whose circumference is 12.12 m.

Solution:

As we know,
Area (A) = C2/4π, here C = 12.12 m,π = 3.141
= (12.12)2/4 × 3.141
= 146.89/12.56 = 11.69 m2

Find the circumference of a circle whose area is 88 cm2

Solution:

As we know,
Area (A) = C2/4π, here A = 88 cm2, π = 3.141
88 = C2/4 × 3.141
C2 = 1105.28
C = 33.24 cm

Derivation of Area of a Circle

The formula for the area of a circle can be obtained and proved using two methods.

Derivation of Area of Circle

1) Using Area of Parallelogram

Let the given circle be divided into 16 equal sectors and then arranged approximately in the form of a parallelogram. The total area of the circle will be equal to the total areas of the parallelogram. Since each sector has an equal area, each sector will have an equal arc length. Taken together, half of the circle is represented using one color and the other half using another color. The higher the number of sectors, the more the figure looks like a parallelogram with length equal to πr and breadth equal to r.

Thus,

Area of the parallelogram = Area of the circle

Area of the circle = πr × r = πr2

2) Using Area of Triangle

The other way to derive the formula for the area of a circle is by dividing the circle with radius ‘r’ into several concentric circles and then spreading the lines, thus forming a triangle. The base of the triangle will be equal to the circumference of the circle, and the height equal to the radius of the circle.

Thus,

Area of the triangle = Area of the circle

Area of the circle = ½ × base × height

= ½ × (2πr) × r

= πr2

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