<Tr> <Th> Rectangular pyramid </Th> <Th> Rhombic pyramid </Th> </Tr> <P> The volume of a pyramid (also any cone) is V = 1 3 b h (\ displaystyle V = (\ tfrac (1) (3)) bh), where b is the area of the base and h the height from the base to the apex . This works for any polygon, regular or non-regular, and any location of the apex, provided that h is measured as the perpendicular distance from the plane containing the base . In 499 AD Aryabhata, a mathematician - astronomer from the classical age of Indian mathematics and Indian astronomy, used this method in the Aryabhatiya (section 2.6). </P> <P> The formula can be formally proved using calculus . By similarity, the linear dimensions of a cross-section parallel to the base increase linearly from the apex to the base . The scaling factor (proportionality factor) is 1 − y h (\ displaystyle 1 - (\ tfrac (y) (h))), or h − y h (\ displaystyle (\ tfrac (h-y) (h))), where h is the height and y is the perpendicular distance from the plane of the base to the cross-section . Since the area of any cross-section is proportional to the square of the shape's scaling factor, the area of a cross-section at height y is b (h − y) 2 h 2 (\ displaystyle b (\ tfrac ((h-y) ^ (2)) (h ^ (2)))), or since both b and h are constants, b h 2 (h − y) 2 (\ displaystyle (\ tfrac (b) (h ^ (2))) (h-y) ^ (2)). The volume is given by the integral </P> <Dl> <Dd> b h 2 ∫ 0 h (h − y) 2 d y = − b 3 h 2 (h − y) 3 0 h = 1 3 b h . (\ displaystyle (\ frac (b) (h ^ (2))) \ int _ (0) ^ (h) (h-y) ^ (2) \, dy = (\ frac (- b) (3h ^ (2))) (h-y) ^ (3) (\ bigg) _ (0) ^ (h) = (\ tfrac (1) (3)) bh .) </Dd> </Dl>

What shape is a pyramid without a top