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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