Question

The central angle of 2 sectors of a circle is in the ratio of 4:5 and are used to make 2 cones. What is the ratio of their curved surface area?

• A. 20
• B. 16
• C. 18
• D. 12

Answer 1

The correct option is C

18

Given the 2 sectors are from the same circle. Hence, only the arc lengths vary.

These arcs form the circumference of the base of the cones. The slant height will be equal to the radius of the circle.

Let the base radius of the cones be $r_1$ and $r_2$

Ratio of arcs of the 2 sectors is given by

$\frac{C_1}{C_2}=\frac{\frac{4x}{360}\times 2\pi r}{\frac{5x}{360}\times 2\pi r}$

$\frac{C_1}{C_2}=\frac{4}{5}$

Since, arc length is equal to the circumference, the ratio of the radii of the cone bases is given by

$\frac{2\pi r_1}{2\pi r_2}=\frac{4}{5}\Rightarrow \frac{r_1}{r_2}=\frac{4}{5}$

Ratio of the curved surface areas of the cones is given by

$\frac{\pi r_{1}l}{\pi r_{2}l}\Rightarrow \frac{r_1}{r_2}=\frac{4}{5}$