Area of Circle Formula, Definition, Derivation & Solved Questions

Area of Circle Formula: In geometry, a circle is a closed plane which is a locus of a point revolving around a certain point at a certain distance away from that particular point. A circle is a closed curved shape having an equidistant outer line from its centre. The length of the line connecting the central point to the outer line is called the radius of the circle. In your surroundings, you can notice various circular-shaped things like wheels, hand watches, coins, sun, etc. In this article, you will learn more about the circle and its parts, the area of the circle formula and the derivation of the area of a circle with some solved examples.

What is Circle?

A circle is defined as a closed geometric shape in which the collection of many points is present at a certain distance from its centre. The parts of a circle are discussed below in detail.

Radius of Circle: It refers to the distance of a fixed point on the outline of a circle from the central point. It is expressed by 'r' or 'R'. The radius plays a vital role in calculating the area and circumference of a circle.

Diameter of Circle: It refers to a line that passes through the central point of a circle and its boundary points placed on the outer line of that circle. It is expressed by 'd' or 'D'. The diameter of a circle is considered two times its radius. 

Diameter = 2 × Radius = 2 r.

Circumference: It refers to the length of the boundary of a circle. In geometry, the perimeter of a circle is called the circumference of a circle. It is measured by calculating the length of the rope that moves around the boundary of a circle properly. The circumference formula is mentioned below.

Circumference of a Circle (C) = 2πr

Area of Circle Formula

The area of a circle is defined as the measure of the space or region occupied by the circle. It can also be measured by finding the total number of square units placed inside that particular circle. The area occupied by one complete round of the radius of the circle on the plane of 2-D is called the area of a circle.  It is expressed in square units like metre². The formula for the area of a circle is explained below. 

Area of Circle = πr² or πd² / 4  where π (Pi) = 22 / 7 or 3.14

The formula for the area of a circle is A = πr², where r is the radius of the circle. The unit of area of a circle is the square unit, for example, m², cm², etc. Pi (π) is defined as the ratio of the circumference of a circle to the diameter of a circle. In mathematics, it is a special constant.

The area of a circle can be written in the following ways: 

Area of Circle = π × r², where 'r' is the radius of that circle.

Area of Circle = (π/4) × d², where 'd' is the diameter of that circle.

Area of Circle = C²/4π, where 'C' is the circumference of that circle.

Derivation of the Area of Circle Formula

For checking the mathematical derivation of the area of a circle formula, you need to see the given figures carefully. 

After observing the above figure, you can notice that if you fragment the circle into smaller sections or shapes and organize them properly then it makes a parallelogram from a circle. If the circle is further fragmented into smaller sections then it gradually makes a rectangle. As you can see that the more sections it has, the more it inclines to have a rectangular shape.

As we know that the area of a rectangle = length × breadth

The width of a rectangle = radius of a circle (r)

If you compare the length of a rectangle and the circumference of a circle then you can observe that the length of a rectangle is half times the circumference of a circle.

Since the Area of circle = Area of the rectangle formed out of that circle = ½ (2πr) × r

Hence the area of the circle is πr², where the value of π is 22/7 or 3.14.

Area of Circle Formula Examples

Question 1: Calculate the area of a circular wheel whose radius is 12 cm.

Solution: Given that the radius of the wheel = 12 cm

By using the area of a circle (A),

A = πr² = 3.14 x 12 x 12 = 452.57 square cm.

Question 2: The ratio of the area of 2 circular coins is 16:25. By using the area of circle formula find the ratio of the radii of both coins.

Solution: Suppose the radius of the first coin = r1

Area of the first coin = a1

The radius of the 2nd coin = r2

Area of the 2nd coin = a2

As per the question, given that a1: a2 = 16: 25

The area of a circle = πr²

πr1² : πr2²= 16 : 25

Taking square roots on both sides of the above expression, we get

r1 : r2 = 4 : 5

Hence the ratio of the radii of given two coins = 4: 5

Question 3: A swimming pool is in the form of a circular shape. The inner radius of the pool is 25 metres and the outer radius is 32 metres. Find the area of that swimming pool. 

Solution: Given that outer radii r1 = 32 m, and inner radii r2 = 25 m.

Suppose the area of an outer circle is a1 and the area of an inner circle is a2

Area of swimming pool = a1 - a2 = πr1² - πr2² = π(32² - 25²) = (22/7) × (1024 - 625) = 1254 square metres.

Hence the area of the swimming pool is 1254 square metres.

Question 4: A cable is in the shape of an equilateral triangle. Each side of that triangle has a length of 9 inches. The cable is turned into a circular shape. Find the area of the circle that is formed by bent cables.

Solution: As we know the perimeter of the equilateral triangle = 3 × sides = 3 × 9 = 27 inches.

Hence the perimeter of the triangle is 27 inches.

As per the question, the perimeter of the equilateral triangle = circumference of the circle formed.

So, Circumference of a Circle = 2πr = 2 × 22/7 × r = 27 

r = (27 × 7) / (44) = 4.295

Hence the radius of the circle is 4.295 inches. 

Area of a circle = πr² = 22/7 ×(4.295)² = 57.98 square inches.

Therefore, the area of a circle is 57.98 square inches.

Question 5: In a circular handwatch, the time is 3:00 pm. The length of the minute needle of the watch is 7 units. Calculate the distance covered by the tip of the minute needle if the time is shown as 3:30 pm.

Solution: If the minute needle is at 3:30 pm, then it travels half of the circular watch. Hence the distance travelled by the minute needle of the watch is exactly half of the circumference. 

Distance travelled by minute of watch = 2 π r/ 2 = π r

where r is the length of the minute needle 

So the distance travelled = 22/7 × 7 = 22 units.