Chapter 12.3, Problem 38HP

To determine

To explain:

According to the given statement a cone and a square pyramid have the same slant height.

Answer to Problem 38HP

They are not equal.

Explanation of Solution

Given information:

A cone and a square pyramid have the same surface area. If the areas of their bases are also equal.

The surface area of the cone is given by:

  $S_{cone}=\pi r{l}_c+\pi r^2$ , where $r$ is radius and $l_c$ is slant height of cone.

The surface area of square pyramid is given by:

  $S_{pyramid}=\frac{1}{2}P{l}_{pyramid}+B^2$ , where $r$ is radius, $l_{pyramid}$ is slant height of pyramid and $B$ is the base area.

The area of the base of the cone and the pyramid are the same.

  $\begin{array}{c}{a}^2=\pi r^2\\ a=\sqrt{\pi r^2}\\ =r\sqrt{\pi }\end{array}$

The cone and the square pyramid have the same surface areas.

  $\begin{array}{c}\pi r{l}_c+\pi r^2=\frac{1}{2}P{l}_{pyramid}+B^2\\ \pi r{l}_c=\frac{1}{2}P{l}_{pyramid}\\ \pi r{l}_c=\frac{1}{2}\left(4a\right)l_{pyramid}\\ l_c=\frac{2a{l}_{pyramid}}{\pi r}\\ =\frac{2r\sqrt{\pi }{l}_{pyramid}}{\pi r}\\ =\left(\frac{2\sqrt{\pi }}{\pi }\right)l_{pyramid}\\ \approx 1.13l_{pyramid}\end{array}$

From the above calculation it is clear that the slant height of cone is greater than the slant height of pyramid.

Hence, they are not equal.