Question

The shaded portion (a quarter of the circle) is cut out from the circle, and then the remaining portion is folded to make a cone. Now, if the bottom of the cone is covered with paper to form the base, what will be the ratio of the area of original circle to the total surface area of the cone made?

• A. $21:16$
• B. $9:16$
• C. $1:1$
• D. $16:21$

Answer 1

The correct option is D $16:21$

Let r be the radius of the circle.
Area of the circle = $\pi r^2$

A quarter of the circle is removed and folded into a cone.
Lateral surface area of the cone formed = $\pi r^2-\frac{1}{4}\pi r^2$
= $\frac{3}{4}\pi r^2$

Let b be the radius of the base of the cone made.
The circumference of the base of the cone is equal to $\frac{3}{4}th$ of the original circle since the cone is made by folding the $\frac{3}{4}th$ of the original circle.
$\therefore 2\pi b=\frac{3}{4}\left(2\pi r\right)$
$\Rightarrow b=\frac{3}{4}r$

Area of the base of the cone = $\pi b^2$
= $\pi (\frac{3}{4}r{)}^2$
= $\frac{9}{16}\pi r^2$

Total surface area of the cone = lateral surface area + area of the base
= $\frac{3}{4}\pi r^2+\frac{9}{16}\pi r^2$
= $\frac{12}{16}\pi r^2+\frac{9}{16}\pi r^2$
= $\frac{21}{16}\pi r^2$

Ratio of the area of the original circle to the total surface area of the cone made = $\pi r^2:\frac{21}{16}\pi r^2$ = $16:21$