Elements of geometry and trigonometry . ome a radius of the circle, andany portion of the polj^gon, as ABCO, will become the sectorOAKBC. and the part of the perimeter AB-fBC, will beccmothe arc AKBG. PROPOSITION IX. THEOREM. The area of a regular polygon is equal to its perimeter, multi-plied by half the radius of the inscidbed circle. Let there be the regular polj-gonGHIK, and ON, OT, radii of the in-scribed circle. The triangle GOHwill be measured by G1Ï x ^OT; thetriangle OHI, by III x ^ON : butON=^OT; hence the two trianglestaken together will be measured by(GH + HI)xiOT. And, by con-tinu
Elements of geometry and trigonometry . ome a radius of the circle, andany portion of the polj^gon, as ABCO, will become the sectorOAKBC. and the part of the perimeter AB-fBC, will beccmothe arc AKBG. PROPOSITION IX. THEOREM. The area of a regular polygon is equal to its perimeter, multi-plied by half the radius of the inscidbed circle. Let there be the regular polj-gonGHIK, and ON, OT, radii of the in-scribed circle. The triangle GOHwill be measured by G1Ï x ^OT; thetriangle OHI, by III x ^ON : butON=^OT; hence the two trianglestaken together will be measured by(GH + HI)xiOT. And, by con-tinuing the same operation for theother triangles, it will appear thatthe sum of them all, or the wholepolygon, is measured by the sum of the bases GH, HI, & the perimeter of the polygon, multiplied into -JOT, or halfthe radius of the inscribed circle. Scholium. The radius OT of the inscribed circle is nothingelse than the perpendicular let fall from the centre on one ofthe sides : it is sometimes named the apothem of the BOOK V 119 PROPOSITION X. THEOREM.
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