Question

The lateral area of the frustum of right circular cone is calculated using the formula
A. \[\dfrac{1}{2}(2\pi R + 2\pi r)L\]
B. \[\dfrac{1}{2}(2\pi R - 2\pi r)L\]
C. \[\dfrac{1}{2}(2\pi R + 2\pi r)\]
D. \[\dfrac{1}{3}(2\pi R + 2\pi r)L\]

Answer 1

Hint: Frustum of a right circular cone is that portion of right circular cone included between the base and a section parallel to the base not passing through the vertex. And the altitude in the frustum is the perpendicular distance between the two bases.

Complete step-by-step answer:
Properties of Frustum of Right Circular Cone
i. The altitude of a frustum of a right circular cone is the perpendicular distance between the two bases. It is denoted by h.
ii. All elements of a frustum of a right circular cone are equal. It is denoted by L.
                         

By ratio and proportion:
\[\begin{array}{l}
\dfrac{{{L_1}}}{R} = \dfrac{L}{{R - r}}\\
{L_1} = \dfrac{{LR}}{{R - r}}
\end{array}\]
From the figure:
\[\begin{array}{l}
{L_2} = {L_1} - L\\
{L_2} = \dfrac{{RL}}{{R - r}} - L\\
{L_2} = \dfrac{{RL - (R - r)L}}{{R - r}}\\
{L_2} = \dfrac{{rL}}{{R - r}}
\end{array}\]
The length of arc is the circumference of the base:
\[\begin{array}{l}
{s_1} = 2\pi R\\
{s_2} = 2\pi r
\end{array}\]
From the figure:
\[\begin{array}{l}
\therefore A = \dfrac{1}{2}{s_1}{L_1} - \dfrac{1}{2}{s_2}{L_2}\\
 \Rightarrow A = \dfrac{1}{2}(2\pi R)\left( {\dfrac{{RL}}{{R - r}}} \right) - \dfrac{1}{2}(2\pi r)\left( {\dfrac{{rL}}{{R - r}}} \right)\\
 \Rightarrow A = \dfrac{{\pi {R^2}L}}{{R - r}} - \dfrac{{\pi {r^2}L}}{{R - r}}\\
 \Rightarrow A = \dfrac{{\pi {R^2}L - \pi {r^2}L}}{{R - r}}\\
 \Rightarrow A = \dfrac{{\pi \left( {{R^2} - {r^2}} \right)L}}{{R - r}}\\
 \Rightarrow A = \dfrac{{\pi (R + r)(R - r)L}}{{R - r}}\\
 \Rightarrow A = \pi (R + r)L
\end{array}\]

So, the correct answer is “Option A”.

Note: The actual definition of the whole thing can be given as; The lateral area of the frustum of a right circular cone is equal to one-half the sum of the circumference of the bases multiplied by slant height.