Question

Two square pyramids are similar. If the ratio of a pair of corresponding edges is a : b, what is the ratio of their volumes? What is the ratio of their surface areas?

Answer 1

Ratio of their volumes = a³ : b³

Ratio of their surface areas =$\frac{a^{2} + a\sqrt{4h^{2}+a^{2}}}{b^{2} + b\sqrt{4h^{2}+b^{2}}}$

Step-by-step explanation:

Two square pyramids are similar with their edges are in the ratio of a : b.

Volume of a square pyramid with edge and h = a  is given by the formula,

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

Volume of a square pyramid with edge b  and h = b is given by the formula,

= $\frac{b^{2}\times h }{3}$ = $\frac{b^{3} }{3}$

Ratio of their volumes = a³ : b³ since h/3 gets cancelled.

Total surface area of square pyramid with the edge a =

$a^{2} + a\sqrt{4h^{2}+a^{2}}$

Total surface area of square pyramid with the edge b =

$b^{2} + b\sqrt{4h^{2}+b^{2}}$

Ratio of the surface area = $\frac{a^{2} + a\sqrt{4h^{2}+a^{2}}}{b^{2} + b\sqrt{4h^{2}+b^{2}}}$