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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 = 12πr2

Perimeter of a semicircle = (π + 2)r.

Area and Perimeter of Semicircle

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

Area of a quadrant of a circle = 14πr2.

Perimeter of a quadrant of a circle = (π2 + 2)r.

Area and Perimeter of Quadrant of a Circle

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 = 227.)

Solution:

Let r be the radius. Then,

area = 12 ∙ πr^2

⟹ 308 cm^2 = 12227 ∙ r^2

⟹ 308 cm^2 = 2214 ∙ r^2

2214 ∙ r^2 = 308 cm^2

⟹ r^2 = 1422 ∙ 308 cm^2

⟹ r^2 = 711 ∙ 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

                       = (227 + 2) ∙ 14 cm

                       = 367  ×  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 = 227.)

Solution:

Let the radius be r. 

Perimeter and Area of Quadrant of a Circle

Then,

perimeter = (π2 + 2)r

⟹ 75 cm = (12 ∙ π + 2)r

⟹ 75 cm = (12227  + 2)r

⟹ 75 cm = (117  + 2)r

⟹ 75 cm = 257r

257r = 75 cm

⟹ r = 75 × 725 cm

⟹ r = 3 × 7 cm

⟹ r = 21 cm.

Therefore, the radius of the circle is 21 cm.

Now, area = 14πr^2

                = 14 ∙  227 ∙ 21^2 cm^2

                = 14 ∙  227 ∙ 21 ∙ 21 cm^2

                = 6932 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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