Question Number 134713 by bramlexs22 last updated on 06/Mar/21
$$\mathrm{geometry} \\ $$All the edges of a regular square pyramid have a length of 8. What is the volume?
Answered by john_santu last updated on 06/Mar/21
$$\mathcal{R}{eguler}\:{square}\:{pyramid}\:{already} \\ $$$${a}\:{regular}\:{quadrilateral}. \\ $$$${An}\:{equilateral}\:{triangle}\:{is}\:{two}\:{of} \\ $$$${mirrored}\:{acroos}\:{the}\:{long}\:{leg} \\ $$$$\mathrm{30}°−\mathrm{60}°−\mathrm{90}°\:{right}\:{the}\:{triangles} \\ $$$${with}\:{sides}\:{in}\:{the}\:{ratio}\:\mathrm{1}\::\:\sqrt{\mathrm{3}}\::\:\mathrm{2}\:,\: \\ $$$${The}\:{altitude}\:{of}\:{face}\:{triangle}\:{is} \\ $$$$\frac{\sqrt{\mathrm{3}}}{\mathrm{2}}\:×\:\mathrm{8}\:=\:\mathrm{4}\sqrt{\mathrm{3}}\:.\:{That}\:{triangle}\:{is}\: \\ $$$${tilted}\:{in}\:{so}\:{the}\:{top}\:{vertex}\:{is}\:{over}\: \\ $$$${the}\:{center}\:{of}\:{square}\:\mathrm{4}\:{units}\:{from}\:{the} \\ $$$${edge}\::\:{h}^{\mathrm{2}} \:+\:\mathrm{4}^{\mathrm{2}} \:=\:\left(\mathrm{4}\sqrt{\mathrm{3}}\right)^{\mathrm{2}} \Rightarrow{h}=\mathrm{4}\sqrt{\mathrm{2}} \\ $$$${so}\:{the}\:{Volume}\:=\:\frac{\mathrm{1}}{\mathrm{3}}{A}.{h}\:=\:\frac{\mathrm{1}}{\mathrm{3}}×\mathrm{8}^{\mathrm{2}} \:×\:\mathrm{4}\sqrt{\mathrm{2}} \\ $$$${Vol}\:=\:\frac{\mathrm{256}\sqrt{\mathrm{2}}\:}{\mathrm{3}}\:{units}^{\mathrm{3}} \:\bullet\: \\ $$