Area of a Circle: Formula, Derivation, Examples

Area of a Circle is the measure of the two-dimensional space enclosed by a circle. It is mostly calculated by the size of the circle’s radius.

Let’s learn how to find the area of the circle using the formulas, with the help of examples.

Area of Circle

Area of circle is the measure of the space enclosed by the circular shape. It is the total region occupied by the circle within its boundaries.

Area of circle is calculated using the formula,

Area of Circle = πr²

Area of Circle = πd² / 4

where,                                                                                      
r is radius
d is diameter
π = 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.

What is Circle

Circle is a collection of points that are at a fixed distance from a particular point. The distance from the center to the circle is known as the radius.

It has rotational symmetry around the center for every angle. Some examples of circles are wheels, pizzas, circular ground, etc.

Illustration of Circle and its Parts

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• Circles

Parts of Circle

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.

Various terms related to the circle are discussed below:

1. 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.

2. Diameter: 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

3. 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 = 2πr.

Circumference of Circle

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. Area of a circle is calculated by multiplying the square of the radius by π.

Formulas for finding the area of a circle are,

• Area = πr²
• Area = (π/4) × d²
• Area  = C²/4π

where,

π is the constant with a value of 3.14 (approx),
r is the radius of circle,
d is the diameter of circle,
C is the circumference of circle.

Area of Circle with Radius

Area = 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 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/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 formulae:

C = 4, π = 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

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

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

Derivation of Circle Area Using Rectangles

After studying the above figure carefully, we split the circle into smaller parts and arranged them in such a way that they form 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 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 2πr 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 × (2πr) × r

A = πr²

How to Find Area of Circle

Various steps required to find the area of the circle are given below:

Step 1: Mark the radius of the circle.

Step 2: Put the value of the radius in the formula A = πr², where r is the radius and π is the constant with a value of 3.14 (approx)

Step 3: Obtained the answer in step 2 is the required area of the circle. It is measured in square units.

If the diameter of a circle is given, it is first changed to radius using the relation,

Diameter = Radius / 2

Area of a Sector of Circle

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.

Area of a sector of a circle formula is given below:

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

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,

	Circumference (C)	Area (A)
Area vs. Perimeter of Circle
Definition	The length of the boundary of the circle is called the circumference of the circle.	The total space occupied by the boundary of the circle is called the area of the circle.
Formula	C = 2πr	A = πr2
Units	Circumference is measured in m, cm, etc.	Area is measured in m2, cm2
Radius Dependence	Radius is directly proportional to the circumference of the circle.	Area is directly proportional to the square of the radius of the circle.
Diameter Dependence	Diameter is directly proportional to the circumference of the circle.	Area is directly proportional to the square of the diameter of the circle.

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• Circumference of Circle

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 Trapezium
• Area of a Rhombus

Area of Circle Examples

Let’s solve some example questions on the area of circle concepts and formulas you learnt 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 = 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 = 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 = 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 = 2πr 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 = πr² 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 = 22 / 7

Area of quadrant =  πr² / 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 = 22 / 7

Area of sector = (θ/360°) × πr²
                       = (60° /  360°) × 22 / 7 × 14²
                       = 102.67 cm²

Thus, the required area of quadrant = 102.67 cm²

Area of Circle Practice Problems

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

1. What is the area of a circle of radius 7 cm?

2. The diameter of a circle is 7 cm. Find its area.

3. Determine the area of circle in terms of pi, if radius = 6 cm.

4. Calculate the area of a circle if its circumference is 88 cm

Circle Area Formula- FAQs

1. How to find the area of circle?

Area of a circle can be determined by using the formulas:

• Area = π x r², where, r is radius of circle
• Area = (π/4) x d²,where, d is diameter of circle
• Area = C²/4π, where, C is circumference of circle

2. Write the formula for circumference of a circle.

Circumference of circle is the boundary of the circle. Circumference can be calculated by multiplying the radius of circle with twice π. i.e. Circuference = 2πr.

3. What is area of circle in terms of diameter ?

The formula of the area of the circle, using the diameter of the circle is π/4 × diameter².

4. What is area of circle when the circumference is given?

When circumference of the circle is given then its area is calculated easily using the formula,

Area = C²/4π

where,
C is the circumference of the circle