Area and Circumference of a Circle

Here we will discuss about the area and circumference (Perimeter) of a circle and some solved example problems.

The area (A) of a circle or circular region is given by

A = πr$^{2}$

where r is the radius and, by definition,

π = $\frac{\textrm{circumference}}{\textrm{diameter}}$ = $\frac{22}{7}$ (approximately).

The circumference (P) of a circle with radius r is given by, P = 2πr

                                                or,

The perimeter (circumference) of a circular region, with radius r is given by, P = 2πr

Solved example problems on finding the area and circumference (Perimeter) of a circle:

1. The radius of a circular field is 21 m, find its perimeter and area. (Use π = $\frac{22}{7}$)

Solution:

According to the question, given r = 21 m.

Then, perimeter of a circular field = 2πr

                                                 = 2 × $\frac{22}{7}$ × 21 m

                                                 = 2 × 22 × 3 m

                                                 = 132 m

Area of a circular field = πr$^{2}$

                                 = $\frac{22}{7}$ × 21$^{2}$ m$^{2}$

                                 = $\frac{22}{7}$ × 21 × 21 m$^{2}$

                                 = 22 × 3 × 21 m$^{2}$

                                 = 1386 m$^{2}$

2. The perimeter of a circular plate is 132 cm, find its area. (Use π = $\frac{22}{7}$)

Solution:

Let the radius of the plate be r.

Then, perimeter of a circular plate = 2πr

or, 132 cm = 2 × $\frac{22}{7}$ × r

or, r = $\frac{132 \times 7}{2 \times 22}$ cm

       = $\frac{6 \times 7}{2}$

       = 21 cm

Therefore, area of a circular plate = πr$^{2}$

                                                  = $\frac{22}{7}$ × 21$^{2}$ cm$^{2}$

                                                  = $\frac{22}{7}$ × 21 × 21 cm$^{2}$

                                                  = 22 × 3 × 21 cm$^{2}$

                                                  = 1386 cm$^{2}$

3. If the area of a circle is 616 cm$^{2}$ then, find its circumference. (Use π = $\frac{22}{7}$)

Solution:

Let the radius of the circle be r cm.

Area of the circle = πr$^{2}$

or, 616 cm$^{2}$ = $\frac{22}{7}$ × r$^{2}$

or, r$^{2}$ = $\frac{616 \times 7}{22}$ cm$^{2}$

 or, r = $\sqrt{\frac{616 \times 7}{22}}$ cm

        = $\sqrt{28 \times 7}$ cm

        = $\sqrt{2 \times 7 \times 2 \times 7}$ cm

        = $\sqrt{14 \times 14}$ cm

        = 14 cm

Therefore, radius of the circle = 14 cm.

Therefore, circumference of the circle = 2πr

                                                       = 2 × $\frac{22}{7}$ × 14

                                                       = 2 × 22 × 2 cm

                                                       = 88 cm

9th Grade Math

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