Question

The circle above has an area of $25\pi$ and is divided into $8$ congruent regions. What is the perimeter of one of these regions?

• A. $10-25\pi$
• B. $10+\frac{5}{8}\pi$
• C. $10+\frac{5}{4}\pi$
• D. $10+5\pi$
• E. $10+25\pi$

Answer 1

Answer: (C)

$10+\frac{5}{4}\pi$

Let ${}^{\prime }{r}^{\prime }$ be the radius of the given circle

Given, area of the circle $=$ $25\pi$

$\Rightarrow \pi r^2$ $=$ $25\pi$

$\Rightarrow \pi$ $\times$ $r^2$ $=$ $25$ $\times$ $\pi$

$\Rightarrow r^2$ $=$ $\frac{25\times \pi }{\pi }$

$\Rightarrow r^2$ $=$ $5^2$

$\Rightarrow r$ $=$ $\sqrt{5^2}$

$\Rightarrow r$ $=$ $5$

Let $AB$ be an ${}^{\prime }ar{c}^{\prime }$ of one of those $8$ congruent regions,

As the length of the circumference is also divided into $8$ equal arcs,

Length of arc $AB$ $=$ $\frac{1}{8}$ $\times$ Circumference of the circle

$=$ $\frac{1}{8}$ $\times$ $2\pi r$

$=$ $\frac{5\pi }{4}$

$\Rightarrow AB$ $=$ $\frac{5\pi }{4}$

As $A$ and $B$ are points on the circumference of the circle,

$OA$ $=$ $OB$ $=$ $r$ $=$ $5$

Perimeter of the region $=$ $OA$ $+$ $AB$ $+$ $BO$

$=$ $5$ $+$ $\frac{5\pi }{4}$ $+$ $5$

$=$ $10$ $+$ $\frac{5\pi }{4}$

Therefore, Perimeter of one of those regions is ${}^{\prime }$$10$ $+$ $\frac{5}{4}\pi$${}^{\prime }$.