Area of Circle: Definitions, Examples, Formulas, History, Terminology, Derivations, and How to Calculate

Area of Circle Overview

The area of a circle formula can be used to calculate the area occupied by a circular field or plot. A circle is a two-dimensional form with no volume. A circle has only one area and one perimeter/circumference. The space filled by a circle on a two-dimensional plane is defined as its area. Alternatively, the area of a circle is the space occupied inside the boundary/circumference of the circle. The area of a circle is calculated as A = πr², where r is the radius of the circle. The unit of area is the square unit, such as m², cm², in², and so on.

What is a Circle?

A circle is a closed geometric object with a collection of numerous points at a certain distance from its center. The components of a circle are explored in detail below-

Parts of a Circle

The following are the parts of the circle-

1. Diameter: The diameter of a circle is defined as a line that goes through the center and has its ends on the circle. The letter "d" or 'D' is used to symbolize it.
2. Radius: The radius of a circle is the distance from the center to a point on the edge. The letter 'r' or 'R' is used to symbolize it. Radius is significant in the formula for calculating the area and circumference of a circle, which we shall study later.
3. Circle Diameter Formula: A circle's diameter formula is twice its radius. Radius = 2 diameters. d = 2r, or D = 2R. If you know the diameter of a circle, you can determine its radius as r = d/2 or R = D/2.
4. Circumference: The circumference of a circle is equal to the length of its border. This means that the circumference of a circle is also known as its perimeter. The circumference of the circle will be equal to the length of the rope that neatly wraps around its perimeter. The diagram below might help you visualize this. The circumference may be calculated using the following formula-

Circumference of a circle  = 2πR x πD

Where,

'r' is the radius of the circle and is a mathematical constant with an approximate value of 3.14 or 22/7. For a circle with a radius of 'r' and a circumference of 'C', the formula is:

π = Circumference/Diameter

π = C/2r = C/d

C = 2πr

Example of a Circle

A circle is a spherical form without boundaries or edges. A circle is a closed form, a two-dimensional shape, and a curved shape in geometry. A car tire, a time-telling wall clock, and lollipops are a few objects in our immediate environment that have a circular form.

Read more about the Segment of a Circle.

History of the Area of Circle 

The area can be calculated using integral calculus or its more complex child, real analysis, in modern mathematics. The Ancient Greeks, on the other hand, investigated the area of a disc. In the fifth century B.C., Eudoxus of Cnidus discovered that the area of a disc is proportional to its radius squared. In his book Measurement of a Circle, Archimedes utilized Euclidean geometry techniques to demonstrate that the area inside a circle is equal to that of a right triangle whose base has the length of the circle's circumference and whose height matches the circle's radius. The diameter is 2r, and the area of a triangle equals half the base times the height; hence, the area of the disc is r². Prior to Archimedes, Hippocrates of Chios demonstrated in his quadrature of the lune of Hippocrates that the area of a disc is proportionate to the square of its diameter, but he did not define the constant of proportionality.

Area of Circle Terminology

Although the area of a circle is frequently used in casual situations, the word disc refers to the inner region of the circle. In contrast, a circle is designated for the border alone, which is a curve and covers no area. As a result, the area of a disc is a more accurate expression for the area surrounded by a circle.

Read more about the Father of Mathematics.

What is the Area of Circle?

The area of a circle is the amount of space encompassed within the circle's edge. The area filled by the circle is the territory within the circle's perimeter. It is also known as the total number of square units included within that circle. Circle area = πr²or πd²/4 in square units, where

Pi (π) = 22/7 or 3.14

r = radius of the circle

d = diameter of the circle

Pi (π) is the circumference-to-diameter ratio of any circle. It's a unique mathematical constant.

Area of Circle Formulas 

The area of a circle may be computed in stages using the diameter and circumference of the circle. We can calculate the radius and area of a circle using the diameter and circumference. However, these formulas give the quickest way to calculate the area of a circle. If a circle has a radius 'r,' the area of the circle is equal to ᴨr² or ᴨd²/4  in square units, where = 22/7 or 3.14 and d is the diameter.

Area of a circle, A = πr²square units

Circumference / Perimeter = 2πr units

The following formulae can be used to compute the area of a circle-

Area = π × r², where 'r' is the radius.

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

Area = C²/4π, where 'C' is the circumference (sometimes referred to as the perimeter).

What is the Area of Circle Derivation?

Divide a circle into several triangles such that they may be united in the shape of a rectangle, as illustrated in the figure, to calculate the area of the circle. The more portions there are, the clearer the rectangular form will be. We already know that the area of a rectangle equals the sum of its length and width. According to the figure, the length of the rectangle is half the diameter of the circle, marked by r. The radius is equal to the width of the rectangle. As a result, the rectangle's area equals length x breadth = r x r = r, which is the area of the circle formula.

Read more about the Area of Parallelogram and Area of Square.

How to Calculate the Area of Circle?

As we know, the area of a circle is equal to pi times the square of its radius, i.e.,πx r². To calculate the area of a circle, we must first determine its radius or diameter. For example, if the radius of a circle is 7 cm, its area will be-

Area of a circle with 7 cm radius = r² 

                                                = (7)²

                                                = 22/7 x 9 x 9

                                                = 22 x 7 

                                                = 154 sq.cm.

Area of Circle using Diameter

In terms of diameter, the area of the circular formula is-

1. The area of a circle = πd²/4. 'd' represents the circle's diameter. 
2. The circle's diameter = twice its radius. d = 2r. 
3. In general, we must first determine the radius of the circle and then the area of the circle based on the diameter. Using this technique, we can easily calculate the area of a circle given the diameter of the circle.

Area of Circle using Circumference

The circumference of a circle can be used to calculate its area. The formula (circumference)² /4π gives the area of a circle in terms of circumference. Finding the area of a circle from its circumference generally involves two easy steps. The circumference of a circle is used to calculate the radius of the circle first. This radius is also useful for calculating the area of a circle. However, utilizing this method, we will be able to directly calculate the area of a circle from its circumference.