# पृष्ठम्:गणितसारसङ्ग्रहः॒रङ्गाचार्येणानूदितः॒१९१२.djvu/३८५

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189
CHAPTER VII--MEASUREMENT OF AREAS.

of the annulus is 45. What is the calculated measure of the area of (this) annulus?

18. In the case of a figure resembling the crescent moon, the breadth is seen to be 2 hastas, the outer curve 68 hastas and the inner curve 82 hastas. Say what the (resulting) area is.

The rule for arriving at the (practically approximate value of the) area of the circle :-

19.[1] The (measure of the) diameter multiplied by three is the measure of the circumference ; and the number representing the square of half the dianmeter, if multiplied by three, gives the (resulting) area in the case of a complete circle. Teachers say that, in the case of a semicircle, half (of these) give (respectively) the measure (of the circumference and of the area).

Examples an illustration thereof.

20. In the case of a circle, the diameter is 18. What is the circumference, and what the (resulting) area (thereof)? In the case of a semicircle, the diameter is 18: tell me quickly what the calculated measure is (of the area as well as of the circumference).

The rule for arriving at (the value of) the area of an elliptical figure:-

21.[2] The longer diameter, increased by half of the (shorter) diameter and multiplied by two, gives the measure of the circumference of the elliptical figure. One-fourth of the (shorter) diameter, multiplied by the circumference, gives rise to the (measure of the) area (thereof).

19.^  The approximate character of the measure of the circumference as well as of the area as given here is due to the value of a being taken as 3.

21.^  The formula given for the circumference of an ellipse is evidently an approximation of a different kind. The area of an ellipse is ${\displaystyle \pi .a.b}$, where a and b are the semi-axes. If ${\displaystyle \pi }$ is taken to be equal to 3, then ${\displaystyle \pi .a.b=3a.b}$. But the formula given in the stanza makes the area equal to ${\displaystyle 2ab+b2}$.