Question

The central angle and radius of a sector of a circular disc ${180^ \circ }$ are and are 21cm respectively. If the edges of the sector of a disc are joined together to make a hollow cone, then find the radius of the cone.
A.10.5 cm
B.10 cm
C.12.5 cm
D.None of these

Answer 1

Hint: Since the edges of the sector of a circular disc are joined together to form the cone, the radius of the disc becomes the slant height of the newly formed cone. The curved surface area of the cone is also equal to the area of the sector of the disc.

Complete step-by-step answer:
The area of the sector of a circle with radius r and angle formed by the sector at the centre be $\theta $ is given by $\dfrac{\theta }{{360}} \times \pi {r^2}$.
Here the angle formed by the sector of a circular disc at the centre is given to be ${180^ \circ }$and radius of the disc is given to be 21 cm.
The area of the given sector can be calculated by substituting the value ${180^ \circ }$ for $\theta $ and 21 for r in the formula $\dfrac{\theta }{{360}} \times \pi {r^2}$
Area of sector of disc = $\dfrac{{\pi {{\left( {21} \right)}^2}}}{2}$
The curved surface area of the cone must also be equal to the area of the sector of the circular disc.
The curved surface area of a cone is given by $\pi Rl$, where $R$ is radius of the cone and $l$ is the slant height.
Since the edges of the sector of a circular disc are joined together to form the cone, the radius of the disc becomes the slant height of the newly formed cone. Therefore, $l = 21$.
Thus the curved surface area of the cone is $\pi R\left( {21} \right)$.
Equating the curved surface area of cone to the area of sector of circular disc,
$\dfrac{{\pi {{\left( {21} \right)}^2}}}{2} = \pi R\left( {21} \right)$
$R = 10.5$
Thus, option A is the correct answer.

Note: The curved surface of the cone is $\pi rl$ where $r$ is the radius of the cone, $l$ is the slant height of the cone. The area of the sector of the circular disc is $\dfrac{\theta }{{360}} \times \pi {r^2}$where $\theta $ is the angle formed at the centre and $r$ is there radius of the circle.