Perimeter & Area of Circles | Mathematics (Maths) Class 7 PDF Download

Area of a Circle

The area of a circle is the space occupied by the circle in a two-dimensional 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 = πr², where r is the radius of the circle. The unit of area is the square unit, for example, m², cm², in², etc. Area of Circle = πr² or πd²/4 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 two-dimensional 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.

Circle and Parts of a Circle

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.

Parts of 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 below-given figure help you visualize the same. Circumference can be measured by using the given formula:
Circumference of a Circle = 2πR = πD

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/2r = C/d
C = 2πr
Let us understand the different parts of a circle using the following real-life example.
Consider a circular-shaped park as shown in the figure below. We can identify the various parts of a circle with the help of the figure and table given below.

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 Formulas

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 = πr² or πd²/4 in square units,  where π = 22/7 or 3.14, and d is the diameter.
Area of a circle, A = πr² square units
Circumference / Perimeter = 2πr units
Area of circle can be calculated by using the formulas:

• Area = π x r², where 'r' is the radius.
• Area = (π/4) x d², where 'd' is the diameter.
• Area = C²/4π, 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 = πd²/4. 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 Diameter

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)²/4π. 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 Circle using Circumference

Area of a Circle-Calculation

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.

Derivation of Area of a Circle

Why is the area of the circle is πr²? To understand this, let's first understand how the formula for area of a circle is derived.

Visualizing area of circle using area of rectangle

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 = πr². Therefore, the area of the circle is πr², 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 3-D shape. The surface of a sphere will be spherical in shape but a circle is a simple plane 2-dimensional 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 = πr² where 'r' is the radius of the circle and the value of π is approximately 3.14 or 22/7.

Example:

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 =  πr² = 3.14 x 15 x 15 = 706.5
Area of the Pizza = 706.5 sq. cm.