Table of Contents
Last modified on August 3rd, 2023
The area of a circle is the space occupied by the circle in a two-dimensional plane. It is expressed in square units such as m2, cm2, etc.
The shaded region below shows the area of the circle.
The area of a circle can be calculated using three different formulas. The formulas are used based on whether the radius, diameter, or circumference is known to us. Each of the three situations is discussed below with their formulas and solved examples.
When the radius of a circle is known, the formula to determine the area is given below:
Let’s solve some examples to understand the concept better.
Solved Examples
Find the area of a circle whose radius is 6 inches.
As we know,
A = πr2, here π = 3.141, r = 6 inches
= (3.141 × 36) cm2
= 113.07 cm2
Solve the area of the given circle in terms of π.
As we know,
A = πr2, here π = 3.141, r = 11 m
A = πr2, here π = 3.141, r = 11 m
= (π × 121) m2
= 121π m2
Find the radius of the circle whose measure of surface area is 330 mm2
As we know,
A = πr2, here A = 330 mm2,π = 3.141
330= 3.141 × r2
r2 = 330/3.141 = 105.06
r = 10.24 mm
When the diameter of the circle is given, the formula to get the area is given below:
Derivation
The above formula is obtained from the standard formulas A = πr2 …..(1)
As we know,
Diameter (d) = 2 × radius (r)
r = d/2
Putting the value of r in equation (1) we get,
A = π(d/2)2
A = πd2/4
Let’s solve some examples to understand the concept better.
Solved Examples
Find the area of a circle with a diameter of 14 inches.
As we know,
Area (A) = πd2/4, here d = 14, and π = 3.141
= 3.141 × (14)2/4
= 153.90 inches2
What is the approximate area of the given circle.
As we know,
Area (A) = πd2/4, here d = 25 cm, and π = 3.141
= 3.141 × (25)2/4
= 490.78 cm2
Calculate the diameter of a circle with an area of 170 m2.
As we know,
Area (A) = πd2/4, here A = 170 m2,π = 3.141
170 = 3.141 × d2/4
680 = 3.141 × d2
d2 = 680/3.141 = 216.49 m
d = 17.79 m
Given the circumference of the circle, we can calculate the area using the given formula.
Derivation
The above formula is obtained from the standard formulas A = πr2 …..(1)
As we know,
Circumference (C) = 2πr
r = C/2π
Putting the value of r in equation (1) we get,
A = π × (C/2π)2
Area (A) = C2/4π
Let’s solve some examples to understand the concept better.
Find the area of a circle whose circumference is 12.12 m.
As we know,
Area (A) = C2/4π, here C = 12.12 m,π = 3.141
= (12.12)2/4 × 3.141
= 146.89/12.56 = 11.69 m2
Find the circumference of a circle whose area is 88 cm2
As we know,
Area (A) = C2/4π, here A = 88 cm2, π = 3.141
88 = C2/4 × 3.141
C2 = 1105.28
C = 33.24 cm
The formula for the area of a circle can be obtained and proved using two methods.
Let the given circle be divided into 16 equal sectors and then arranged approximately in the form of a parallelogram. The total area of the circle will be equal to the total areas of the parallelogram. Since each sector has an equal area, each sector will have an equal arc length. Taken together, half of the circle is represented using one color and the other half using another color. The higher the number of sectors, the more the figure looks like a parallelogram with length equal to πr and breadth equal to r.
Thus,
Area of the parallelogram = Area of the circle
Area of the circle = πr × r = πr2
The other way to derive the formula for the area of a circle is by dividing the circle with radius ‘r’ into several concentric circles and then spreading the lines, thus forming a triangle. The base of the triangle will be equal to the circumference of the circle, and the height equal to the radius of the circle.
Thus,
Area of the triangle = Area of the circle
Area of the circle = ½ × base × height
= ½ × (2πr) × r
= πr2
Last modified on August 3rd, 2023
This is an informative and well-explained post on the area of a circle. It provides clear definitions, formulas, and examples that help readers understand this mathematical concept in depth.