Area and Perimeter of a Semicircle

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

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.

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 = $\frac{22}{7}$).

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

9th Grade Math

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