Figures Inside One Another

Geometry Level 2

The square of the figure is inscribed in a semicircle and the circle is inscribed in the square. The circle has an area of 10 cm 2 10\text{ cm}^2 . What is the area of the semicircle?

25 cm 2 25 \text{ cm}^2 30 cm 2 30 \text{ cm}^2 35 cm 2 35\text{ cm}^2 40 cm 2 40 \text{ cm}^2

Jádson Bráz by Jádson Bráz , 22, Brazil

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2 solutions

Pranshu Gaba
Sep 16, 2016

Let r r denote the radius of the small circle, and let R R denote the radius of the large semicircle, then by Pythagorean theorem , we can express R R in terms of r r , R 2 = r 2 + ( 2 r ) 2 = 5 r 2 R^2 = r^2 + (2r)^2 = 5r^2 .

So the area of the semicircle is 1 2 π R 2 = 1 2 π ( 5 r 2 ) = 5 2 π r 2 = 5 2 ( 10 cm 2 ) = 25 cm 2 \dfrac 12 \pi R^2= \dfrac12 \pi (5r^2) = \dfrac52 \pi r^2 = \dfrac52 (10\text{ cm}^2) = \boxed{25\text{ cm}^2} .

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The diameter of the small circle is the radius of the large circle.

Brenda Daniel - 4 years, 7 months ago

From the area of a circle, we can compute for the diameter of the circle.

d = 3.568 cm

by Pythagorean Theorem, we can compute for the radius of the semi-circle

R = sqr root (3.568^2 + 1.784^2) = 3.99 cm Area of semi circle = 1/2 * pi * 3.99^2 = 25 cm^2

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