# पृष्ठम्:गणितसारसङ्ग्रहः॒रङ्गाचार्येणानूदितः॒१९१२.djvu/४३४

एतत् पृष्ठम् परिष्कृतम् अस्ति
238
GAṆITASĀRASAṄGRAHA.

In the case of (the quadrilateral figure with) three equal sides, the square root (of the difference between the two area-squares above noted) is added to the approximate measure of the area. (On treating the resulting sum as the optional quantity and) on adding and subtracting (the said square root as before), the base and the top-side are obtained so as to have to be divided by the square root of (such) optional quantity . (Here also), the approximate measure of the area, on being divided by the square root of (this) optional quantity, gives rise to the measure of the other sides.

An example in illustration thereof.

166${\displaystyle {\tfrac {1}{2}}}$. The accurate measure of the area is 5; the approximate measure of the area is 13; and the optionally chosen quantity is 16. What are the values of the base, the top-side, and the (other) side in the case of a quadrilateral figure with two equal sides ?

An example relating to a quadrilateral figure with three equal sides.

167${\displaystyle {\tfrac {1}{2}}}$. The accurate measure of the area is 5; and the approximate measure of the area is 13. Think out and tell me, O friend, the values of the sides of the quadrilateral figure with three equal sides.

The rule for arriving, when the approximate and the accurate measures of an area are known, at the equilateral triangle and also at the diameter of the circle, having those same approximate and accurate measures (for their area):-

168${\displaystyle {\tfrac {1}{2}}}$. That which happens to be the square root of the square root of the difference between the squares of the (approximate measure and of the accurate measure of the given) area is to be

${\displaystyle R={\tfrac {a(b+d)}{2}};p=\left({\tfrac {b+d}{2}}\right)^{2};{\text{ and }}r={\tfrac {b+d}{2}}\times {\sqrt {a^{2}-{\tfrac {\left(b-d\right)^{2}}{4}}}}}$

The formulas given above for the base and the top-side can be easily verified by substituting these values of R, r and p therein. Similarly the rule may be seen to hold good in the case also of a quadrilateral figure with three equal side.

168${\displaystyle {\tfrac {1}{2}}}$. For the approximate and accurate values of an equilateral triangle see rules in stanza 7 and 50 of this chapter.