Question

Three semicircles are drawn inside a big circle as shown in the figure. If the radius of the two identical smaller semi-circles is 1/4th of that of the big circle and the radius of the bigger semi-circles is twice that of the small circle. what proportion of the big circle's area is shaded?

a) 11/12
b) 12/16
c) 13/16
d) 13/14​

Answer 1

$\bold{\huge{\underline{\underline{\sf{\blue{AnsWer:}}}}}}$

$\bold{\large{\mathtt{\red{Proportion\:of\:the\:bigger\:circle\:shaded\:is\:{\dfrac{13}{16}}}}}}$

$\bold{\huge{\underline{\underline{\sf{\blue{Explanation:}}}}}}$

Given :

• Three semicircles are drawn inside a big circle.
• The radius of the two identical smaller semi-circles is 1/4th of that of the big circle.
• The radius of the bigger semi-circles is twice that of the small semi circle.

To Find :

• Shaded proportion of the big circle's area.

Solution :

Let the radius of the bigger circle be R.

$\bold{\tt{\underline{\underline{As\:per\:the\:given\:statements:}}}}$

Radius of semi circle = $\tt{\dfrac{R}{4}}$

Radius of bigger semi circle = $\tt{2\:\times\:{\dfrac{R}4}}$

➜ $\tt{\dfrac{R}{2}}$

Calculate the area of the bigger circle :

FORMULA :

$\tt{Area\:of\:a\:circle\:=\:\pi\:r^2}$

➜ $\tt{\pi\:\times\:R^2}$ ----> (i)

Calculate the area of the smaller semi circles :

FORMULA :

$\tt{Area\:of\:semi\:circle\:=\:{\dfrac{\pi\:R^2}{2}}}$

➜ $\tt{\dfrac{\pi\:(R/4)^2}{2}}$ ----> (ii)

We have two semi circle with same radius. Therefore area of bigger circle would constitute sum of 2 semi circle with radius equals as $\tt{\dfrac{R}{4}}$

Area of bigger semi circle :

➜ $\tt{\dfrac{\pi\:(R/2)^2}{2}}$ ----- > (iii)

Area of three semi circles :

➜ $\tt{\dfrac{1}{2}}$ ($\tt{2\:\times\:{\dfrac{\pi\:R^2}{4^2}}}$ + $\tt{\dfrac{\pi\:R^2}{2^2}}$)

➜ $\tt{\dfrac{1}{2}}$ ($\tt{2\:\times\:{\dfrac{\pi\:R^2}{16}}}$ + $\tt{\dfrac{\pi\:R^2}{4}}$)

➜ $\tt{\dfrac{1}{2}}$ × $\tt{\dfrac{\pi\:R}{4}}$ ($\tt{\dfrac{2R}{4}\:+\:{\dfrac{R}{1}}}$)

➜ $\tt{\dfrac{1}{2}}$ $\tt{\dfrac{\pi\:R}{4}}$ × ($\tt{\dfrac{2R\:+\:4R}{4}}$)

➜$\tt{\dfrac{1}{2}}$ × $\tt{\dfrac{\pi\:R}{4}}$ ($\tt{\dfrac{6R}{4}}$)

➜ $\tt{\dfrac{\pi\:R}{2\:\times\:4}}$ × $\tt{\dfrac{6R}{4}}$

➜ $\tt{\dfrac{6\:\pi\:R^2}{2\:\times\:4\:\times\:4}}$

➜ $\tt{\dfrac{3\:\pi\:R^2}{4\:\times\:4}}$

➜ $\tt{\dfrac{3\:\pi\:R^2}{16}}$

Area of shaded region :

Area of shaded region is the difference between the bigger circle in which the three semi circles are inscribed and the sum of the three semi circles.

➜ $\tt{\pi\:R^2}$ - $\tt{\dfrac{3\:R^2}{16}}$

➜ $\tt{\dfrac{16(\:\pi\:R^2)\:-\:3\:\pi\:R^2}{16}}$

➜ $\tt{\dfrac{16\:\pi\:R^2\:-\:3\:\pi\:R^2}{16}}$

➜ $\tt{\dfrac{13\:\pi\:R^2}{16}}$

Now we have area of the shaded region as well as the area of the bigger circle.

Calculating the ratio :

➜ $\tt{\dfrac{Area\:of\:shaded\:region}{Area\:of\:bigger\:circle}}$

➜ $\tt{\dfrac{13\:\pi\:R^2/16}{\pi\:R^2}}$

➜ $\tt{\dfrac{13}{16}}$

The proportion of the big circle's area shaded is 13/16.