How to prove that the area of a circle is pi r squared?

Circle is a closed two-dimensional figure which has a center where all the points in the plane are equidistant from it. Every line passing through the circle forms the line of reflection symmetry. In addition to this, it has rotational symmetry around the center for every angle. Some of the examples of circles are wheels, pizzas, circular ground, etc.

Properties of Circles

A circle is characterized by the following set of properties: 

• The outer line of the circle is situated at equal distances from the center.
• The diameter of the circle divides it into two equal parts.
• Circles with equal radii are congruent to each other.
• Circles with different radii are similar to each other.
• The diameter of the circle is referred to as the largest chord and is considered to be double the radius.

Parts of Circle

A circle is a collection of points that are situated at a fixed distance from the center of the circle. The area of the circle is considered to be the measure of the space or region enclosed inside the circle.

• Radius: The distance from the center to a point on the boundary. It is denoted by the letter ‘r’ or ‘R’. It is used for the determination of the circumference of the circle. 
• Diameter: A line passing through the center and the endpoints lying on the circle. It is denoted by the letter ‘d’ or ‘D’.

Diameter formula: The diameter formula of a circle is represented as twice its radius. 

Diameter = 2 × Radius

In other words, 

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

How to prove that the area of a circle is pi r squared?

Proof:

A circle can be easily segregated into 16 equal sectors which are arranged in the following form. All of the sectors are equal in area. This implies that all the sectors have equal arc length. In case the number of sectors cut from the circle is increased, the parallelogram will eventually look like a rectangle with a length equivalent to πr and breadth equivalent to r.

The area of a rectangle (A) is also considered to be the area of a circle. Therefore, 

• A = πr×r
• A = πr²

Sample Questions

Question 1. Find the area of the circle when its radius is 10 cm? Use π = 3.14.

Solution:

Here we need to find the area of the circle,

Given:

Radius of the circle = 10 cm

As we know that

Area of the circle = πr²

Area of the circle = 3.14 × 10 × 10

Area of the circle = 314 cm²

Therefore,

Area of the circle is 314 cm².

Question 2. If the diameter of a circle is 24 m then find the area of the circle? Use π = 3.14

Solution:

Here we need to find the area of the circle,

Given:

Diameter of the circle = 24 m

Radius of the circle = 24/2

Radius of the circle = 12 m

As we know that

Area of the circle = πr²

Area of the circle = 3.14 × 12 × 12

Area of the circle = 452.16 m²

Therefore,

Area of the circle is 452.16 m².

Question 3. If the area of the circle is 3850 cm² then find the radius of the circle? Use π = 22/7.

Solution:

Here we have to find the radius of the circle using its area.

Given:

Area of the circle = 3850 cm²

As we know that

Area of the circle = πr²

Area of the circle = 22/7 × r²

3850 = 22/7 × r²

r² = 3850 × 7/22

r² = 1225

r = √1225

r = 35 cm

Therefore,

Radius of the circle is 35 cm when the area of the circle is 3850 cm².

Question 4. Find the cost of carpeting a circular gymnastic hall with a radius of 33 m at the rate of ₹350 per m²? Use π = 3.14.

Solution:

Here we need to find the cost of carpeting gymnastic hall,

Given:

Radius of the circular gymnastic hall = 33 m

As we know that

Area of the circle = πr²

Area of the circle = 3.14 × 33²

Area of the circle = 3.14 × 33 × 33

Area of the circle = 3419.46 m²

Now,

Cost of carpeting = ₹350 × area of the circular gymnastic hall

Cost of carpeting = ₹350 × 3419.46

Cost of carpeting = ₹1196811

Therefore,

Cost of carpeting circular gymnastic ground is ₹1196811.