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How to demonstrate that the circumference of a circle is 2pir?

  • Last Updated : 28 Nov, 2021

Circle is a closed two-dimensional figure where all the set of points in the plane is equidistant from a given point called “center”. Every line that passes through the circle forms the line of reflection symmetry. In addition to this, it has rotational symmetry around the center for every angle.

Circumference of a circle

The perimeter of the circle is known as the circumference of the circle. It is the measure of the total length of the boundary of the circle. It is measured as the product of the constant π and the diameter of the circle. For instance, circumference is measured in the sense of a person walking across a circular park, or a circular table to be bordered. The circumference is a linear value. It is measured in the units of measurement of length.

The measure of the boundary of the circle is known as the circumference of the circle. The circumference of the circle is measured in terms of units of length like centimeters, meters, or kilometers.

The three most important elements making up the circumference of the circle, 

  • Center: A fixed distance from any other point from the circumference is referred to as the center of the circle.
  • Diameter: The distance across the circle through the center is known as the diameter.
  • Radius: The distance from the center of a circle to any point on the circumference of the circle.

How to demonstrate that the circumference of a circle is 2 pi r?

Answer:

The value of Pi (π) is equivalent to the ratio of the circumference of a circle to its diameter. This value is approximately equivalent to nearly 3.14159.

Mathematically,

π = C/D

where, 

C is referred to as the circumference

D is referred to as the diameter

Upon rearranging the terms we obtain, 

C = π × D

Writing in terms of radius, 

2r = D

Hence,

C = 2πr

Hence the circumference of a circle is C = 2πr

Sample Questions

Question 1. Find the circumference of a circle with a radius of 3 cm? use π = 3.14.

Solution:

Here we have to find the circumference of the circle

Given:

Radius of the circle = 3 cm

As we know that

Circumference of the circle = 2πr

Circumference of the circle = 2 × π × r

Circumference of the circle = 2 × π × r 

Circumference of the circle = 2 × 3.14 × 3

Circumference of the circle = 18.84 cm

Question 2. If the diameter of a circle is 36 m the find its circumference? Use π = 3.14.

Solution:

To find the circumference of the circle

Given: 

Diameter of the circle = 36 m

As we know that

Radius of a circle = Diameter of a circle/2

Radius of a circle = 36/2

Radius of a circle = 18 m

Now,

Circumference of the circle = 2πr

Circumference of the circle = 2 × π × r

Circumference of the circle = 2 × π × r 

Circumference of the circle = 2 × 3.14 × 18

Circumference of the circle = 113.04 m

Question 3. Assume that circumference of a circle is 2200 cm, then find the radius of the circle? Use π = 22/7

Solution:

Here we need to find the radius of the circle when its circumference is given,

Given:

Circumference of a circle = 2200 cm

As we know that

Circumference of the circle = 2πr

2200 = 2 × π × r

2200 = 2 × 22/7 × r 

r = 2200 × 7/22 × 1/2

r = 350 cm

Therefore,

Radius of the circle is 350 cm when its circumference is 2200 cm.

Question 4. Find the cost of fencing a circular playground when its radius is 30 m at the rate of ₹25 per meter? Use π = 3.14.

Solution:

Here we need to find the cost of fencing the circular playground

Given:

Radius of the circle = 30 m

As we know that

Circumference of the circle = 2πr

Circumference of the circle = 2 × π × r

Circumference of the circle = 2 × π × r 

Circumference of the circle = 2 × 3.14 × 30

Circumference of the circle = 188.4 m

Now,

Cost of fencing the circular playground =  ₹25 × circumference of the circular playground

Cost of fencing the circular playground =  ₹25 × 188.4

Cost of fencing the circular playground =  ₹4710

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