Question

A regular square pyramid has an altitude of 15 m and a volume of 125 m3.

1. What is the base edge of a square pyramid?
2. What is the slant height of the pyramid?
3. What is the lateral area in m2?

Answer 1

We have $h=15\ \text{m}, \ \ V=125 \ \text{m}^3.$

1 . Let $a$ be the base edge of this square pyramid.

Volume of the pyramid is $V=\frac{1}{3} B h$ , where $B=a^2$ is area of base.

$V=\frac{1}{3}a^2h$

$a^2=\frac{3V}{h}$

$a=\sqrt{\frac{3V}{h}}= \sqrt{\frac{3\times 125}{15}}=\sqrt{25}=5$

2.Let $l$ be the slant height of the pyramid.

Then $l=\sqrt{h^2+\left(\frac{a}{2}\right)^2}=\sqrt{15^2+2.5^2}=\sqrt{925/4}=5\sqrt{37}/2\approx 15.2\ \text{m}$

3.We can find the lateral area using formula $L=\frac{1}{2}Pl$ , where $P=4a$ is the perimeter of the base, $l$ is the slight height.

$L=\frac{4al}{2}=\frac{4\times 5\times 5\sqrt{37}}{2\times 2}=25\sqrt{37}\approx 152\ \text{m}^2$ .

Answer: 1. $a=5\ \text{m}$ , 2. $l=5\sqrt{37}/{2}\approx 15.2\ \text{m}$ , 3. $L=25\sqrt{37}\approx 152\ \text{m}^2$ .