# पृष्ठम्:गणितसारसङ्ग्रहः॒रङ्गाचार्येणानूदितः॒१९१२.djvu/३९४

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198
GAṆITASĀRASAṄGRAHA.

The rule for arriving at the minutely accurate measurement of the area (of trilateral and quadrilateral figures):-

50.[1] Four quantities represented (respectively) by half the sum of the sides as diminished by (each of) the sides (taken in order) are multiplied together; and the square root (of the product so obtained) gives the minutely accurate measure (of the area of the figure). Or the measure of the areas may be arrived at by multiplying by the perpendicular (from the top to the base) half the sum of the top measure and the base measure. (The latter rule does) not (hold good) in the case of an inequi-lateral quadrilateral figure.

Examples in illustration thereof.

51. In the case of an equilateral triangle, 8 daṇdas and give the measure of the base as also of each of the two sides. You, who know calculation, tell me the accurate value of the area (thereof) and also of the perpendicular (to the base) as well as of the segments (of the base caused thereby).

52. In the case of an isosceles triangle (each of the two (equal) sides measures 13 daṇdas, and the base measures 10. (What is) the accurate measure of the area thereof, and of the perpendi.

50.^  Algebraically represented:-

Area of the trilateral figure = ${\displaystyle {\sqrt {s(s-a)(s-b)(s-c)}}}$; where s is half the sum of the sides, a,b,c, the respective measures of the sides of the trilateral figure;
or ${\displaystyle {\tfrac {c}{2}}\times p}$, where p is the perpendicular distance of the vertex from the base.
Area of a quadrilateral figure ${\displaystyle ={\sqrt {(s-a)(s-b)(s-c)(s-d)}}}$ where s is half the sum of the sides, and a,b,c,d the measures of the respective sides of the quadrilateral figure;
or ${\displaystyle ={\tfrac {b+d}{2}}\times p}$ (except in the case of an iequilateral quadrilateral) where p is the measure of either of the perpendiculars drawn to the base from the extremities of the top side.

The formulas here given for trilateral figures are correct; but those given for quadrilateral figures hold good only in the case of cyclic quadrilaterals, as in these formulas sight is lost of the fact that for the same measure of the sides the value of the area as well as of the perpendicular may vary.