Area And Perimeter Of A Semicircle And Quadrant Of A Circle

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Area and Perimeter of a Semicircle and Quadrant of a Circle

We will learn how to find the Area and perimeter of a semicircle and Quadrant of a circle.

Area of a semicircle = \(\frac{1}{2}\)πr2

Perimeter of a semicircle = (π + 2)r.

because a semicircle is a sector of sectorial angle 180°.

Area of a quadrant of a circle = \(\frac{1}{4}\)πr2.

Perimeter of a quadrant of a circle = (\(\frac{π}{2}\) + 2)r.

because a quadrant of a circle is a sector of the circle whose sectorial angle is 90°.

Here r is the radius of the circle.

Solved Examples on Area and perimeter of a semicircle and Quadrant of a circle:

1. The area of a semicircular region is 308 cm^2. Find its perimeter. (Use π = \(\frac{22}{7}\).)

Solution:

Let r be the radius. Then,

area = \(\frac{1}{2}\) ∙ πr^2

⟹ 308 cm^2 = \(\frac{1}{2}\) ∙ \(\frac{22}{7}\) ∙ r^2

⟹ 308 cm^2 = \(\frac{22}{14}\) ∙ r^2

⟹ \(\frac{22}{14}\) ∙ r^2 = 308 cm^2

⟹ r^2 = \(\frac{14}{22}\) ∙ 308 cm^2

⟹ r^2 = \(\frac{7}{11}\) ∙ 308 cm^2

⟹ r^2 = 7 × 28 cm^2

⟹ r^2 = 196 cm^2

⟹ r^2 = 14^2 cm^2

⟹ r = 14 cm.

Therefore, the radius of the circle is 14 cm.

Now, perimeter = (π + 2)r

                       = (\(\frac{22}{7}\) + 2) ∙ 14 cm

                       = \(\frac{36}{7}\)  ×  14 cm

                       = 36 × 2 cm

                       = 72 cm.

2. The perimeter of a sheet of paper in the shape of a quadrant of a circle is 75 cm. Find its area. (Use π = \(\frac{22}{7}\).)

Solution:

Let the radius be r. 

Then,

perimeter = (\(\frac{π}{2}\) + 2)r

⟹ 75 cm = (\(\frac{1}{2}\) ∙ π + 2)r

⟹ 75 cm = (\(\frac{ 1 }{2}\) ∙ \(\frac{22}{7}\)  + 2)r

⟹ 75 cm = (\(\frac{11}{7}\)  + 2)r

⟹ 75 cm = \(\frac{25}{7}\)r

⟹ \(\frac{25}{7}\)r = 75 cm

⟹ r = 75 × \(\frac{7}{25}\) cm

⟹ r = 3 × 7 cm

⟹ r = 21 cm.

Therefore, the radius of the circle is 21 cm.

Now, area = \(\frac{1}{4}\)πr^2

                = \(\frac{1}{4}\) ∙  \(\frac{22}{7}\) ∙ 21^2 cm^2

                = \(\frac{1}{4}\) ∙  \(\frac{22}{7}\) ∙ 21 ∙ 21 cm^2

                = \(\frac{693}{2}\) cm^2

                = 346.5 cm^2.

Therefore, area of the sheet of paper is 346.5 cm^2.

10th Grade Math

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