Area of a Circle
The area of a circle is the space occupied by the circle in a twodimensional plane. Alternatively, the space occupied within the boundary/circumference of a circle is called the area of the circle. The formula for the area of a circle is A = πr^{2}, where r is the radius of the circle. The unit of area is the square unit, for example, m^{2}, cm^{2}, in^{2}, etc. Area of Circle = πr^{2} or πd^{2}/4 in square units, where (Pi) π = 22/7 or 3.14. Pi (π) is the ratio of circumference to diameter of any circle. It is a special mathematical constant.
The area of circle formula is useful for measuring the region occupied by a circular field or a plot. Suppose, if you have a circular table, then the area formula will help us to know how much cloth is needed to cover it completely. The area formula will also help us to know the boundary length i.e., the circumference of the circle. Does a circle have volume? No, a circle doesn't have a volume. A circle is a twodimensional shape, it does not have volume. A circle only has an area and perimeter/circumference. Let us learn in detail about the area of a circle, surface area, and its circumference with examples.
Area of Circle Definition
A circle is a collection of points that are at a fixed distance from the center of the circle. A circle is a closed geometric shape. We see circles in everyday life such as a wheel, pizzas, a circular ground, etc. The measure of the space or region enclosed inside the circle is known as the area of the circle.
Radius: The distance from the centre to a point on the boundary is called the radius of a circle. It is represented by the letter 'r' or 'R'. Radius plays an important role in the formula for area and circumference of a circle, which we will learn later.
Diameter: A line that passes through the centre and its endpoints lie on the circle is called the diameter of a circle. It is represented by the letter 'd' or 'D'.
Diameter: The diameter of a circle is twice its radius. Diameter = 2 × Radius
d = 2r or D = 2R
If the diameter of a circle is known, its radius can be calculated as:
r = d/2 or R = D/2
Circumference: The circumference of the circle is equal to the length of its boundary. This means that perimeter of a circle is equal to its circumference. The length of rope that wraps around circle's boundary perfectly will be equal to its circumference. The belowgiven figure help you visualize the same. Circumference can be measured by using the given formula:
where 'r' is the radius of the circle and π is the mathematical constant whose value is approximated to 3.14 or 22/7. The circumference of a circle can be used to find the area of that circle.
For a circle with radius ‘r’ and circumference ‘C’:
π = Circumference/Diameter
π = C/2rπ = C/d
C = 2πr
What is the Area of Circle?
The area of a circle is the amount of space enclosed within the boundary of a circle. The region within the boundary of the circle is the area occupied by the circle. It may also be referred to as the total number of square units inside that circle.
Area of Circle Formula
The area of a circle can be calculated in intermediate steps from the diameter, and the circumference of a circle. From the diameter and the circumference, we can find the radius and then find the area of a circle. But these above formulae provide the shortest method to find the area of a circle. Suppose a circle has a radius 'r' then the area of circle = πr^{2} or πd^{2}/4 in square units, where π = 22/7 or 3.14, and d is the diameter.
Area of a circle, A = πr^{2} square units
Circumference / Perimeter = 2πr units
Area of circle can be calculated by using the formulas:
 Area = π x r^{2}, where 'r' is the radius.
 Area = (π/4) x d^{2}, where 'd' is the radius.
 Area = C^{2}/4π, where 'C' is the circumference.
Area of a Circle using Diameter
The Area of the circle in terms of the diameter is: Area of a Circle = πd^{2}/4. Here 'd' is the diameter of the circle. The diameter of the circle is twice the radius of the circle. d = 2r. Generally from the diameter, we need to first find the radius of the circle and then find the area of the circle. With this formula, we can directly find the area of the circle, from the measure of the diameter of the circle.
Area of a Circle using Circumference
The area of a circle in terms of the circumference is given by the formula. (Circumference of a Circle)^{2}/4\(\pi \). There are two simple steps to find the area of a circle from the given circumference of a circle. The circumference of a circle is first used to find the radius of the circle. This radius is further helpful to find the area of a circle. But in this formulae, we will be able to directly find the area of a circle from the circumference of the circle.
Area of a CircleCalculation
The area of the circle can be conveniently calculated either from the radius, diameter, or circumference of the circle. The constant used in the calculation of the area of a circle is pi, and it has a fractional numeric value of 22/7 or a decimal value of 3.14. Any of the values of pi can be used based on the requirement and the need of the equations. The below table shows the list of formulae if we know the radius, the diameter, or the circumference of a circle.
Area of a Circle when the radius is known.  πr^{2} 
Area of a Circle when the diameter is known.  πd^{2}/4 
Area of a Circle when the circumference is known.  C^{2}/4π 
Derivation of Area of a Circle
Why is the area of the circle is πr^{2}? To understand this, let's first understand how the formula for area of a circle is derived.
The circle can be cut into a triangle with the radius being the height of the triangle and the perimeter as its base which is 2πr. We know that the area of the triangle is found by multiplying its base by the height, and then dividing by 2, which is 1/2 x 2πr x r = πr^{2. }Therefore, the area of the circle is πr^{2}, where r, is the radius of the circle and the value of π is 22/7 or 3.14.
Surface Area of Circle Formula
Surface area of a circle is the same as the area of a circle. In fact, when we say the area of a circle, we mean nothing but its total surface area. Surface area is the area occupied by the surface of a 3D shape. The surface of a sphere will be spherical in shape but a circle is a simple plane 2Dimensional shape.
If the length of the radius or diameter or even the circumference of the circle is given, then we can find out the surface area. It is represented in square units. Surface area of a circle = πr^{2} where 'r' is the radius of the circle and the value of π is approximately 3.14 or 22/7.
RealWorld Examples
Ron and his friends ordered a pizza on Friday night. Each slice was 15 cm in length.
Calculate the area of the pizza that was ordered by Ron. You can assume that the length of the pizza slice is equal to the pizza’s radius.
Solution:
A pizza is circular in shape. So we can use the formula of a circle to calculate the area of the pizza.
Radius is 15 cm
Area of the Circle = πr^{2} = 3.14 x 15 x 15 = 706.5
Area of the Pizza = 706.5 sq. cm.
Solved Examples on Area of Circle

Example 1: Find the circumference and the area of a circle with radius 14 cm.
Solution:
Given: Radius of the circle = 14 cm
Circumference of the Circle = 2πr
= 2 x 22/7 x 14
= 2 x 22 x 2
= 88 cm
Area of a Circle = πr^{2}
= 22/7 x 14 x 14
= 22 x 2 x 14
= 616 sq. cm.
Area of the Circle = 616 sq. cm.

Example 2: The ratio of the area of 2 circles is 4:9. Find the ratio of their radii.
Solution:
Let us assume the following:
Radius of the 1st circle = R1
Area of the 1st circle = A1
Radius of the 2nd circle = R2
Area of the 2nd circle = A2
It is given that A1 : A2 = 4 : 9
Area of a Circle = πr^{2}
πR1^{2 }: πR2^{2 }= 4 : 9
Taking square roots of both sides,
R1^{ }: R2^{ }= 2 : 3
Therefore, ratio of the radii = 2:3

Example 3: A race track is in the form of a circular ring. The inner radius of the track is 58 yd and the outer radius is 63 yd. Find the area of the race track.
Solution:
Given: R = 63 yd, r = 56 yd.
Let the area of outer circle be A_{1} and the area of inner circle be A_{2}
Area of race track = A_{1}  A_{2} = πR^{2}  πr^{2} = π(63^{2}  56^{2}) = 22/7 × 833 = 2,618 square yards.
Therefore, the area of the race track is 2618 square yards.

Example 4: A wire is in the shape of an equilateral triangle. Each side of the triangle measures 7 in. The wire is bent into the shape of a circle. Find the area of the circle that is formed.
Solution:
Perimeter of the Equilateral Triangle: Perimeter of the triangle = 3 × side = 3 × 7 = 21 inches.
Since the perimeter of the equilateral triangle = Circumference of the circle formed.
Thus, the perimeter of the triangle is 21 inches.
Circumference of a Circle = 2πr = 2 × 22/7 × r = 21. r = 21 × 7/22 × 2 = 3.34.
Therefore, the Radius of the circle is 3.34 cm. Area of a circle = πr^{2} = r^{2 } = 22/7 ×(3.34)^{2 }= 35.06 square inches.
Therefore, the area of a circle is 35.06 square inches.
The area of the race track is 2618 square yards.

Example 5: The time shown in a circular clock is 3:00 pm. The length of the minute hand is 21 in. Find the distance traveled by the tip of the minute hand when the time is 3:30 pm.
Solution:
When the minute hand is at 3:30 pm, it covers half of the circle. So, the distance traveled by the minute hand is actually half of the circumference. Distance \(= \pi\) (where r is the length of the minute hand). Hence the distance covered = 22/7 × 21 = 22 × 3 = 66 inches. Therefore, the distance traveled is 66 inches.
FAQs on Area of Circle
How to Calculate the Area of a Circle?
Area of circle can be calculated by using the formulas:
Area = π x r^{2}, where 'r' is the radius.
Area = (π/4) x d^{2}, where 'd' is the radius.
Area = C^{2}/4π, where 'C' is the circumference.
What is the Formula of Area of Circle?
Area = π x r^{2}. The area of a circle is π multiplied by the square of the radius. Area of a Circle when the radius 'r' is given is πr^{2}. Area of a circle when the diameter 'd' is known is πd^{2}/4. π is approx 3.14 or 22/7.
Area(A) could also be found using the formulas A = (π/4) x d^{2}, where 'd' is the radius and A= C^{2}/4π, where 'C' is the given circumference.
What is the Area and Perimeter of a Circle?
The circumference of the circle is equal to the length of its boundary. This means that the perimeter of a circle is equal to its circumference. The area of a Circle is πr^{2} and perimeter/circumference is 2πr when the radius is 'r'. π is approx 3.14 or 22/7. Circumference or radius length of a circle are important to find the area of that circle.
For a circle with radius ‘r’ and circumference ‘C’:
π = Circumference/Diameter
π = C/2r
Therefore, C = 2πr
Why is the Area of a Circle πr^{2}?
A circle can be divided into many small sectors which can then be rearranged accordingly to form a parallelogram. When the circle is divided into even smaller sectors, it gradually becomes the shape of a rectangle. We can clearly see that one of the sides of the rectangle will be the radius and the other will be half the length of the circumference, i.e, π. As we know that the area of a rectangle is its length multiplied to the breadth which is π multiplied to 'r'. Therefore, the area of the circle is πr^{2}
What is the Value of π?
The value of pi/π is approximately 3.14. Pi is an irrational number. This means that its decimal form neither ends (like 1/5 = 0.2) nor becomes repetitive (like 1/3 = 0.3333...). Pi is 3.141592653589793238... (to only 18 decimal places).
How do you Find the Circumference and Area of a Circle?
The area and circumference of a circle can be calculated using the following formulas. Circumference = 2πr ; Area = πr^{2}. The circumference of the circle can be taken as π times the diameter of the circle. And the area of the circle is π times the square of the radius of the circle.
How to Calculate the Area of a Circle with Diameter?
The diameter of the circle is double the radius of the circle. Hence the area of the circle formula using the diameter is equal to π/4 times the square of the diameter of the circle. The formula for the area of the circle, using the diameter of the circle π/4×diameter^{2}.
How Do you Find the Area of a Circle Given the Circumference?
The area of a circle can also be found using the circumference of the circle. The radius of the circle can be found from the circumference of the circle and this value can be used to find the area of the circle. Assume that the circumference of the circle is 'C'. We have C = 2πr, or r = C/2π. Now applying this 'C' value in the Area formula we have A = πr^{2} = π × (C/2π)^{2}= C^{2}/4π.
What is the Area of Circle with Radius 3 m?
The area of a circle is π multiplied by the square of the radius. Area of a circle(A) when the radius 'r' is given is πr^{2}. π is approx 3.14 or 22/7. Therefore, area = 3.14 x 3 x 3 = 28.26 sq. m.
Circumference of a Given Circle is 16 cm. What will be its Area?
Circumference of a circle = 16 cm
We know the formula of circumference, C =2πr
So,
2πr = 16
or r = 16/2π = 8/π
Substituting the value of 'r' in the area formula, we get:
A = πr^{2}
A = π(8/π)^{2} = 64/π
On solving,
Area = 20.38 sq. cm.