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