Area of a Circle: Formula, Derivation, Examples

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 radius of the circle.

The area of the circle is calculated using the formula,

Area of Circle = πぱいr²
Area of Circle = πぱいd² / 4

Where,

• r is radius,
• d is diameter, and
• πぱい = 22/7 or 3.14

Area of circle formula is useful for measuring areas of circular fields or plots. It is also useful to measure the area covered by circular furniture and other circular objects.

Parts of Circle

A circle is a closed curve in which all the points are equidistant from one fixed point i.e. centre. Examples of circles as seen in everyday life are clocks, wheels, pizzas, etc.

Radius

The distance of a point from the boundary of the circle to its centre is termed its radius. Radius is represented by the letter 'r' or 'R'. The area and circumference of a circle are directly dependent on its area.

Diameter

The longest chord of a circle that passes through its centre is termed its diameter. It is always twice its radius.

Diameter formula: The formula for the diameter of a circle is Diameter = 2 × Radius

d = 2×r or D = 2×R

also, conversely, the radius can be calculated as:

r = d/2 or R = D/2

Circumference

The circumference of the circle is the total length of its boundary i.e. perimeter of a circle is termed its circumference. The Circumference of a circle is given by the formula C = 2πぱいr.

Area of Circle Formulas

The formula for finding an area of a circle is directly proportional to the square of its radius. It can also be found out if the diameter or circumference of a circle is given. The area of a circle is calculated by multiplying the square of the radius by πぱい.

Area of Circle with Radius

Area = πぱいr²

where, r is the radius and πぱい 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 = πぱいr²
r = 3, πぱい = 3.14

Area = 3.14 × 3 × 3 = 28.26

Therefore, the area of the circle is 28.26 units²

Area of Circle in Terms of 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ぱい/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ぱい/4) × d²
d = 8, πぱい = 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ぱい

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ぱい = 3.14

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

Therefore, the area of the circle is 1.273 unit²

Area of Circle Derivation

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

• Circle Area Using Rectangles
• Circle Area Using Triangles

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,

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 × 2πぱいr = πぱいr
Breadth of a rectangle = radius of a circle = r

Area of circle = Area of rectangle = πぱいr × r = πぱいr²

Area of the circle = πぱいr²

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.

• Let us take a circle with a radius of r and fill the circle with concentric circles till no space is left inside the circle.
• Now cut open each concentric circle and arrange them in a triangular shape such that the shortest length circle is placed at the top and the length is increased gradually.

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

Thus the area of the circle is given as,

A = 1/2 × base × height
A = 1/2 × (2πぱいr) × r
A = πぱいr²

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.

of a sector of a circle formula is given below:

A = (θしーた/360°) × πぱいr²

Where,
θしーた 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しーた/360°) × πぱいr²

Area of Quadrant = (90°/360°) × πぱいr²
               = πぱいr² / 4

Difference Between Area and Circumference of Circle

The basic difference between the area and the circumference of the circle is discussed in the table below,

Circle Real World Examples

We come across various examples which resemble circular shapes in our daily life.

Some of the most common examples of the real-life circular things which we observe in our daily life are shown in the image below.

Read More:

• Area of Square
• Area of Triangle
• Area of Rectangle
• Area of Quadrilateral
• Area of Rhombus
• Area of Trapezium
• Area of Parallelogram

Area of Circle Examples

Let's solve some example questions on the area of circle concepts and formulas you learnedⁱⁿ so far :

Example 1: A large rope is in a circular shape. Its radius is 5 units. What is its area?

Solution:

A large rope is in circular shape means it is similar to circle, so we can use circle formulae to calculate the area of the large rope.

given, r = 5 units, πぱい = 3.14

Area = 3.14 × 5 × 5
        =  78.50 unit²

Thus, the area of the circle is 78.50 units²

Example 2: If the rope is in a circular shape and its diameter is 4 units. Calculate its area.

Solution:

We know that rope is in circular shape, and its diameter = 4 units
πぱい = 3.14

Area = (3.14 /4) × 4 × 4 
        = 12.56 units²

Therefore, the area of the rope is 12.56 units²

Example 3: If the circumference of the circle is 8 units. Calculate its area.

Solution:

Circumference of the circle = 8 units (given), πぱい = 3.14

Area = 8 × 8 / (4 × 3.14) 
        = 5.09 units²

Therefore, the area of the circle is 5.09 units²

Example 4: Find the circumference and the area of the circle if the radius is 21 cm.

Solution:

 Radius, r = 21 cm, circumferencer of the circle = 2πぱいr cm.

Now, substituting the value, we get

C = 2  × (22/7)× 21
C = 2×22×3
C = 132 cm

Thus, circumference of the circle is 132 cm.

Now, area of the circle = πぱいr² cm²

A = (22/7) × 21 × 21
A = 22 × 63
A = 1386 cm² 

Thus, area of the circle is 1386 cm² 

Example 5: Find the area of the quadrant of a circle if its radius is 14 cm.

Solution:

Given r = 14 cm, πぱい = 22 / 7

Area of quadrant =  πぱいr² / 4
                            = 22 / 7 × 14² × 1/4
                            = 154 cm²

Thus, the required area of quadrant = 154 cm²

Example 6: Find the area of the sector of a circle that subtends 60° angle at the center, and its radius is 14 cm.

Solution:

Given r = 14 cm, πぱい = 22 / 7

Area of sector = (θしーた/360°) × πぱいr² 
                       = (60° /  360°) × 22 / 7 × 14²
                       = 102.67 cm²

Thus, the required area of quadrant = 102.67 cm²

Practice Problems on Area of Circle

Here are some practice problems on the area of a circle for you to solve:

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.

Answer Key

Answer 1: Area of a circle with radius 7 cm is 154 cm².
Answer 2: Area of a circle with diameter 7 cm is 38.5 cm².
Answer 3: Area in terms of πぱい, radius = 6 cm is 36πぱい cm².
Answer 4: Area of a circle with circumference 88 cm is 616 cm².