Problems on Area and Circumference of a Circle

Perimeter is associated with any closed figure like triangle, quadrilateral, polygons or circles. It is the measure of distance covered while going around the closed figure on its boundary.

For example, the perimeter of a square of side 2 cm = 8 cm, as we know that the square comprises of 4 sides having equal lengths, thus the total distance covered will be \(4 \times 2\), which will be the total length (i.e. Perimeter)

Area means the actual space enclosed by a closed figure (or within the perimeter). It means all the points within the closed figure and not the boundary. This is actually measured as the square of length.

Now, coming to Perimeter of a circle, as explained above, it is the measure of distance going around the boundary of a circle. This distance is difficult to calculate exactly. Looking at various circles having different radii, it is easy to visualize that the distance to go around is larger in a circle of larger radius than a smaller radius. Hence, perimeter is a function of the radius of a circle. In case of circle we generally use the term CIRCUMFERENCE instead of perimeter.

It is given by \(\large p = 2 \pi r\)

(Here r is a radius and π is a constant and actually defined as the ratio of circumference to the diameter of a circle).

The value of \(\large \pi\) is 3.1416

Circumference Formula of a Circle is given by

\(\large A = \pi r\)

Let us understand the concepts related to circles along with following questions-

  1. To cover a distance of 10 km a wheel rotates 5000 times. Find the radius of the wheel.

Solution: No. of rotations = 5000.

Total distance covered = 100 km and we have to find out the radius of the circle.

Let ‘r’ be the radius of the wheel

Circumference of the wheel = Distance covered in 1 rotation = 2πr.

In 5000 rotations, the distance covered = 100 km = 10 x 103x 102 cm.

Hence, in 1 rotation, the distance covered = \(\frac{1000000}{5000}cm=200\: cm\)

But this is equal to the circumference. Hence, 2πr = 2000

  • r = 200/2π
  • r = 100/π
  • Taking approximate value of π as 22/7, we get
  • r = 100 x 7/22
  • r = 31.82 cm approx.
  1. The diameter of the given semi-circular slice of watermelon is 14 cm. What will be the perimeter of the slice of watermelon?

Solution: Given diameter=14 cm

Radius=d/2= 7 cm

Perimeter (p) =2πr

p = 2 x 22/7 x 7= 44 cm.

An interesting fact to note here is that the shape of the slipe is semi-circular along with a line.

Thus perimeter of semi-circular slice is

\( P = \frac{p}{2} + 2r\)

\( P = \frac{44}{2} + 14\)

\(P = 36\) cm

3.The difference between the circumference and the diameter of a circular bangle is 5 cm. Find the radius of the bangle. (Take \(\pi = \frac{22}{7}\))

Solution: Let the radius of the bangle be’r’

According to the question:

Circumference – Diameter=5 cm

We know, Circumference of a circle = 2πr

Diameter of a circle = 2r

Therefore, 2πr – 2r =5 cm

2r(π-1) = 5cm

\(2r(\frac{22}{7}-1)=5cm\\ \\ 2r\times \frac{15}{7}=5\\ \\ r=\frac{5\times 7}{15\times 2}\\ \\ r=1.166cm\)

The radius of bangle is 1.166cm.

4: A girl wants to make a square shaped figure from a circular wire of radius 49 cm. Determine the sides of a square.

Solution: Let the radius of the circle be ’r’.

Length of the wire=circumference of the circle=\(2\pi r\)

\(2\times \frac{22}{7}\times 49=2\times 22\times 7\\ \\ =308\: cm\)

Let the side of the square be ‘s’.

Perimeter of the square = length of the wire = 4s

\(s=\frac{308}{4}\\ \\ s=77\:cm\)<

Therefore, the sides of the square is 77 cm.