Question

From a circle of radius $15cm$, a sector central angle ${216}^o$ is cut and its bounding radii are joined without overlap so as to form a cone. Find its volume.

• A. $1081.3c{m}^3$
• B. $1071.3c{m}^3$
• C. $1018.3c{m}^3$
• D. $1061.9c{m}^3$

Answer 1

The correct option is C $1018.3c{m}^3$

Radius of the circle $=r=15cm.$

angle of the sector $=\theta =216°.$

Length of the circular arc cut off from the circle

$=\frac{2\pi r\times \theta }{360°}$

$=\frac{2\times 22\times 15\times {216}^{\circ }}{360°\times 7}$

$=\frac{396}{7}cm$

As the piece is bent into a right circular cone.

So, the slanting height $h$ of the cone $=radiusofcircle=R=15cm.$

$h=15cm$

circumference of the base of the cone = arc length $=\frac{396}{7}cm$

Base radius $R$ of the cone $=R=\frac{396}{7}\times \frac{1}{2\pi }=9cm$

Height of the cone $=H=\sqrt{(h^2-R^2)}=\sqrt{({15}^2-9^2)}=12cm$

Volume of the cone $=V=\frac{1}{3}\pi \times R^{2}H$

$V=\frac{22}{7}\times \frac{1}{3}\times 9^2\times 12c{m}^3$

$V=\frac{7128}{7}c{m}^3$

$V=1018.29c{m}^3.$