Question

Let X, Y, Z be respectively the areas of a regular pentagon, regular hexagon and regular heptagon which are inscribed in a circle of radius $1$. Then.

• A. $\frac{X}{5}<\frac{Y}{6}<\frac{Z}{7}$ and $X<Y<Z$
• B. $\frac{X}{5}<\frac{Y}{6}<\frac{Z}{7}$ and $X>Y>Z$
• C. $\frac{X}{5}>\frac{Y}{6}>\frac{Z}{7}$ and $X>Y>Z$
• D. $\frac{X}{5}>\frac{Y}{6}>\frac{Z}{7}$ and $X<Y<Z$

Answer 1

Answer: (C)

$\frac{X}{5}>\frac{Y}{6}>\frac{Z}{7}$ and $X>Y>Z$

The area of a regular polygon is given by this formula :-

area $=\frac{1}{2}\times$(apothem) $\times$ (perimeter)

On solving this general formula in terms of various parameter for regular polygon, we get

Area $=n\left(r^2\right)\times \tan \left(180/n\right)$

where $n=$ number of sides;

$r=$ apothem (radius of inscribed circle);

In the given question data, $r=1$ and

For regular pentagon, $n=5,$ then

$X=5\times \tan \left(36\right)=3.63$ and

$\frac{X}{5}=\tan \left(36\right)=0.726$

For regular hexagon, $n=6,$ then

$Y=6\tan \left(30\right)=3.46$ and

$\frac{Y}{6}=\tan \left(30\right)=0.577$

For regular heptagon, $n=7,$ then

$Z=7\times \tan \left(25.71\right)=3.37$ and

$\frac{Z}{7}=\tan \left(25.71\right)=0.481$

So, as we can see that $X>Y>Z$ and $\left(X/5\right)>\left(Y/6\right)>\left(Z/7\right)$

Therefore, Option (C) is the correct answer.