Question

All edges of a square pyramid are equal.Total length of all edges is 90 cm .find the volume of square pyramid

Answer 1

Answer:

$\bold{237.3046875\sqrt{2}\ cm^3}$

Step-by-step explanation:

$Base\ edge = a \\ \\ Lateral\ edge = e \\ \\ Height = h \\ \\ Base\ diagonal = d \\ \\ \\ Here, a = e \\ \\ \\ Total\ length\ of\ all\ edges \\ \\ = 4a + 4e = 90\ cm \\ \\ = 4a + 4a = 90\ cm \\ \\ = 8a = 90\ cm \\ \\ a = e = \frac{90}{8} = 11.25\ cm \\ \\ \\ d = a\sqrt{2} = 11.25\sqrt{2}\ cm$

$\\ \\ \\ h = \sqrt{e^2 - (\frac{d}{2})^2} \\ \\ = \sqrt{(11.25)^2 - (\frac{11.25\sqrt{2}}{2})^2} \\ \\ = \sqrt{126.5625 - \frac{253.125}{4}} \\ \\ = \sqrt{\frac{506.25 - 253.125}{4}} \\ \\ = \sqrt{\frac{253.125}{4}} \\ \\ = \frac{11.25\sqrt{2}}{2}$

$\\ \\ \\ Volume \\ \\ = \frac{1}{3}a^2h \\ \\ = \frac{1}{3} \times (11.25)^2 \times \frac{11.25\sqrt{2}}{2} \\ \\ = \frac{1}{3} \times 126.5625 \times \frac{11.25\sqrt{2}}{2} \\ \\ = \frac{126.5625 \times 11.25\sqrt{2}}{6} \\ \\ = \frac{1423.828125\sqrt{2}}{6} \\ \\ = \bold{237.3046875\sqrt{2}\ cm^3}$