Answer to Question #112089 in Geometry for Alexis Ebuka

Question

A 216 sector of a circle of radius 5cm is bent to form a cone .what is the base radius. What is the vertical angle. What is the curved surface area

Answer 1

Radius of the circle = $R=5cm,$ angle of the sector = $\theta=216\degree.$

Length of the circular arc cut off from the circle 

$$L=2\pi R\times{\theta \over 360\degree}=2\pi(5){216\degree \over 360\degree}=6\pi(cm)$$

When the sector is cut and its bounding radii is bent to form a cone, slant height of the cone, $l=R=5cm.$

Let $r$  and $h$ be the radius and height of the cone formed.

Circumference of the base of the cone $=2\pi r=6\pi.$

$$r=3cm.$$

The base radius is $3cm.$

Height of the cone

$$h=\sqrt{l^2-r^2}=\sqrt{5^2-3^2}=4(cm)$$

Let $\varphi$ be a vertical angle. Then

$$\sin(\varphi/2)={r \over l}={3\over 5}=0.6$$

$$\varphi=2\arcsin(0.6)\approx73.74\degree$$

Find the curved surface area

$$\pi rl=\pi(3)(5)=15\pi (cm^2)\approx47.12(cm^2)$$