Chapter 12.3, Problem 25PPS

To determine

To find:

The slant height of the pyramid.

Answer to Problem 25PPS

The slant height is $16\text{ cm}$ .

Explanation of Solution

Given information:

The surface area of a triangular pyramid is $532$ square centimeters and the base is $24$ centimeters with a hypotenuse of $25$ centimeters.

Calculation:

First find the missing side using Pythagorean theorem:

  $p=\sqrt{h^2-b^2}$

Substitute $h=25$ and $b=24$ in the above formula:

  $\begin{array}{c}p=\sqrt{h^2-b^2}\\ =\sqrt{{25}^2-{24}^2}\\ =\sqrt{625-576}\\ =\sqrt{49}\\ =7\end{array}$

So, the third side is $7\text{ cm}$ .

Perimeter of the triangle is:

  $\begin{array}{c}P=7+25+24\\ =56\text{ cm}\end{array}$

Find the base area:

  $\begin{array}{c}B=\frac{1}{2}pb\\ =\frac{1}{2}7\cdot 24\\ =84{\text{ cm}}^2\end{array}$

The surface area $S$ of regular pyramid is:

  $\begin{array}{c}S=\frac{1}{2}Pl+B\\ 532=\frac{1}{2}\left(56\right)l+84\\ 532=28l+84\\ 28l=532-84\\ 28l=448\\ l=16\end{array}$

Hence, the slant height is $16\text{ cm}$ .