Question

A piece of paper in the shape of a sector of a circle is rolled up to form a right-circular cone. The value of the angle $\theta$ is :

• A. $\frac{10\pi }{13}$
• B. $\frac{9\pi }{13}$
• C. $\frac{5\pi }{13}$
• D. $\frac{6\pi }{13}$

Answer 1

The correct option is A $\frac{10\pi }{13}$

In the given right-circular cone,

By applying pythagoras theorem, we get slant height$\left(l\right)$,

$l^2$ $=$ $5^2$ $+$ ${12}^2$

$l$ $=$ $13$

Circumference$\left(2\pi r\right)$ of the circle(i.e., the base of the right-circular cone) is the arc length of the sector of the circle$\left(L\right)$

$L$ $=$ $2\pi r$

$L$ $=$ $10\pi$

The slant height$\left(l\right)$ of the right-circular cone is the Radius$\left(R\right)$ of the sector of the circle

$R$ $=$ $13$

The arc length$\left(L\right)$ of the sector is given by:

$L$ $=$ $\theta$ $\times$ $R$

$\theta$ $=$ $\frac{L}{R}$

$\theta$ $=$ $\frac{10\pi }{13}$

Therefore, $\theta$ $=$ $\frac{10\pi }{13}$