Mensuration on Circle

Measuring Circumference of  a Circle

In our daily life, we come across circular shapes in various places.  The diameter of a circle is twice the length of its radius (2r). i.e., d = 2r.  For any circle with a given ‘r’ or ‘d’, we can calculate its area and circumference using the formula given.

Area of the circle is calculated using the formula πr², whereas the circumference is calculated using 2πr. The value of π is ²²/₇.

Using the formula for finding the circumference of the circle, we can derive the following:

Circumference = 2πr
            C = 2r x π
            C = πd (since 2r = d)
            d = ^C/π or
            π = ^C/d or
            π = ^C/2r

From this, we can conclude that the ratio of the circumference to that of diameter is a constant (π).

Circumference / Diameter = π [²²/₇] or ^C/d = π 

Example 1: The diameter of a circular well is 4.2 m. What is its circumference?

Example 2: What is the circumference and area of the circular disc of radius 14 cm?

Given:
radius  = 14 cm

Solution:
circumference = 2πr

Thus, The circumference of the circular disk is 88 cm and the area is 616 square cm.

Example 3:  The diameter of a circular room is 6.3 m.  Find its circumference and area.

Given:
diameter  = 6.3 cm

Solution:

Example 4: If the circumference of the circle is 132 m. Then calculate the radius and diameter.

Given:
Circumference = 132 m

Solution:

d = 2r
d = 2 x 21
d = 42 cm

Thus, the radius and diameter are 21 cm and 42 cm respectively.

Example 5.  What is the distance travelled by the tip of the seconds hand of a clock in 1 minute, if the length of the hand is 56 mm.

Given:
radius = 56 mm

Solution:
circumference = 2πr
c = 2 x π x 56
c = 2 x 22 x 8 [π = 22/7]
c = 352 mm

Thus the distance travelled by the tip of the seconds hand of a clock in 1 minute is 352mm.

Example 6.  The radius of a tractor wheel is 77 cm.  Calculate the distance covered by it in 35 rotations?

Given:
radius = 77 cm

Solution:
The distance covered in one rotation (circumference) = 2πr
c = 2 x π x 77
c = 2 x 22 x 11 [π = 22/7]
c = 484 cm

The distance covered in 35 rotations = 35 x 484 = 16940 cm

Example 7:  A farmer wants to fence his circular poultry farm with barbed wire whose radius is 420 m.  The cost of fencing is ₹12 per meter.  How much more amount will be needed to fence his farm? 

Given:
radius = 420 m

Solution:
The length of the fence (circumference) = 2πr
c = 2 x π x 420
c = 2 x 22 x 60 [π = 22/7]
c = 2640 m

Cost of fencing 1 meter = 12
Cost of fencing 2640 meter = 12 x 2640 = 31,680.

Thus the cost of fencing the farm is RS. 31,680.

Example 8:  A ground is in the form of a circle whose diameter is 350 m. An athlete makes 4 revolutions. Find the distance covered by the athlete.

Given:
diameter (d) = 350 m

Solution:

Now, calculate the distance covered by the athlete:
Distance covered of 1 revolution = 1100 m
Distance covered of 4 revolution = 4 x 1100 = 4400

Thus, the distance covered by the athlete is 4400 m.

Example 9:  The diameter of the bullock cart wheel is 1.4 m.  Find the distance covered by it in 150 rotations?