Mathematics in Ancient and Medieval India c6.s9.100

c6.s9.100

For a *volume of a frustum of a pyramid* the sides of whose upper face are a, b and the corresponding sides of whose lower face are a', b', the approximate value will be $A = ((a + a')/2) ((b + b')/2) × h, G = ½ (ab + a'b') h$ Then the accurate volume of the frustum of the wedge will be $V = ⅓ [½ (ab + a'b') h - ((a + a')/2) ((b + b')/2) h]$ $+ ((a + a')/2) ((b + b')/2) h . . . . . (1)$ On reduction we easily obtain, $V = ⅙ [ab + a'b' + (a + a')(b + b')] × h ... ... (2)$ The formula reappears in this reduced form in the works of Āryabhaṭa II¹, Śrīpati² and Bhāskara II³ the kinds of grain. Nemicandra has considered the case (for finer grains) in which the height is one-eleventh of the circumference of the base.¹ So there is absolutely no doubt that the formula was intended only for a rough calculation.