Area of a Circle (original) (raw)

Last Updated : 22 Apr, 2026

The area of a Circle is the measure of the two-dimensional space enclosed within its boundaries. It is mostly calculated by the size of the circle's radius which is the distance from the center of the circle to any point on its edge. The area of a circle is proportional to the square of its radius.

area_of_circle

The area of the circle is calculated using the formula

Area of Circle = πr2

where, **r is the radius and π is the constant value

**Example: If the length of the radius of a circle is 3 units. Calculate its area.

**Solution:

We know that radius r = 3 units
So by using the formula: Area = πr2
r = 3, π = 3.14

Area = 3.14 × 3 × 3 = 28.26

Therefore, the area of the circle is 28.26 units2

**Area of Circle using Diameter

The diameter of a circle is double the length of the radius of the circle, i.e. 2r.

The area of the circle can also be found using its diameter

Area = (π/4) × d2

where,
d is the diameter of the circle.

**Example: If the length of the diameter of a circle is 8 units. Calculate its area.

**Solution:

We know that diameter = 8 units, so by using the formulas: Area = (π/4) × d2
d = 8, π = 3.14
Area = (3.14 /4) × 8 × 8 = 50.24 unit 2

Thus, the area of the circle is 50.24 units2

**Area of a Circle using Circumference

The circumference is defined as the length of the complete arc of a circle.

Area = C2/4π

Where, **C is the circumference

**Example: If the circumference of the circle is 4 units. Calculate its area.

**Solution:

We know that circumference of the circle = 4 units (given) so by using the above formula:
C = 4, π = 3.14

Area = 4 × 4 / (4 × 3.14) = 1.273 unit2

Therefore, the area of the circle is 1.273 unit2

Derivation

The area of a circle can be visualized and proved using two methods, namely

Circle Area Using Rectangles

The area of the Circle is derived by the method discussed below. For finding the area of a circle the diagram given below is used,

circle_2

After studying the above figure carefully, we split the circle into smaller parts and arranged them in such a way that they formed a parallelogram.

If the circle is divided into small and smaller parts, at last, it takes the shape of a rectangle.

Area of Rectangle = length × breadth

Comparing the length of a rectangle and the circumference of a circle we can see that, the length is = ½ the circumference of a circle
Length of a rectangle = ½ × 2πr = πr
Breadth of a rectangle = radius of a circle = r

Area of circle = Area of rectangle = πr × r = πr2

Area of the circle = πr2

Where **r is the radius of the circle.

Circle Area Using Triangles

The area of the circle can easily be calculated by using the area of a triangle. For finding the area of the circle using the area of the triangle consider the following experiment.

The figure so obtained is a triangle with base **2πr and height **r as shown in the figure given below,

circle_3

Thus the area of the circle is given as,

A = 1/2 × base × height
A = 1/2 × (2πr) × r
**A = πr 2

Area of a Sector of Circle

The area of a sector of a circle is the space occupied inside a sector of a circle’s border. A semi-circle is likewise a sector of a circle, where a circle has two equal-sized sectors.

A = (θ/360°) × πr2

Where,
**θ is the sector angle subtended by the arcs at the center (in degrees),
**r is the radius of the circle.

Area of Quadrant of circle

A quadrant of a circle is the fourth part of a circle. It is the sector of a circle with an angle of 90**°. So its area is given by the above formula.

A = (θ/360°) × πr2

Area of Quadrant = (90°/360°) × πr2
= πr2 / 4

Practice Problems

**Question 1: What is the area of a circle of radius 7 cm?
**Question 2: The diameter of a circle is 7 cm. Find its area.
**Question 3: Determine the area of a circle in terms of pi, if radius = 6 cm.
**Question 4: Calculate the area of a circle if its circumference is 88 cm.