Area and Perimeter of a Semicircle

Here we will discuss about the area and perimeter of a semicircle with some example problems.

Area and Perimeter of a Semicircle

Area of a semicircle = \(\frac{1}{2}\) πr\(^{2}\)

Perimeter of a semicircle = (π + 2)r





Solved example problems on finding the area and perimeter of a semicircle:

1. Find the area and perimeter of a semicircle of radius 7 cm. (Use π = \(\frac{22}{7}\)).

Solution:

Given, radius = r = 7 cm.

Problems on Area and Perimeter of a Semicircle

Then, area of semicircle = \(\frac{1}{2}\) πr\(^{2}\)

                                   = \(\frac{1}{2}\) × \(\frac{22}{7}\) × 7\(^{2}\) cm\(^{2}\)

                                   = 11 × 7 cm\(^{2}\)

                                   = 77 cm\(^{2}\)

Perimeter of a semicircle = (π + 2)r

                                     = (\(\frac{22}{7}\) + 2) × 7 cm

                                     = \(\frac{36}{7}\) × 7 cm

                                     = 36 cm


2. Find the area and perimeter of the figure in which PQRS is a square of side 28 cm and STR is a semicircle. (Use π = \(\frac{22}{7}\)).

Area and Perimeter of Semicircular Figure

Solution:

The required area = Area of the square PQRS + Area of the semicircle STR

                          = a\(^{2}\) + \(\frac{1}{2}\) πr\(^{2}\)

                          = 28\(^{2}\) cm\(^{2}\) + \(\frac{1}{2}\) × π × (\(\frac{1}{2}\) × 28)\(^{2}\) cm\(^{2}\)

                          = (28\(^{2}\) + \(\frac{1}{2}\) × \(\frac{22}{7}\) × 14\(^{2}\)) cm\(^{2}\)

                          = (28\(^{2}\) + \(\frac{1}{2}\) × \(\frac{22}{7}\) × 14 × 14) cm\(^{2}\)

                          = (28\(^{2}\) + 11 × 2 × 14) cm\(^{2}\)

                          = (28\(^{2}\) + 11 × 28) cm\(^{2}\)

                          = 28(28 + 11) cm\(^{2}\)

                          = 28 × 39 cm\(^{2}\)

                          = 1092 cm\(^{2}\)

The required perimeter = PQ + PS + QR + semicircular arc STR

                                  = 28 cm + 28 cm + 28 cm + π × (\(\frac{1}{2}\) SR)

                                  = 84 cm + \(\frac{22}{7}\) × \(\frac{1}{2}\) × 28 cm

                                  = 84 cm + 11 × 4 cm

                                  = 84 cm + 44 cm

                                  = 128 cm





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