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circle, equal, angles and perimeter

POLYGON (from 70..vrovog, formed from Tro).vg, many, and ywvta, angle) a multilateral figure, or one whose peri meter consists of more than tbur sides and angles.

It' the sides and angles be equal, the figure is called a regular polygon. For similar polygons see SIMILAR.

Polygons are distinguished according to the number of their sides. Those of five sides are called pentagons; those of six, hexagons ; those of seven, heptagons ; those of eight, octagons, &C.

PoLynoss, General properties of: Euclid demonstrates the following :-1. That every polygon may be divided into as many triangles as it bath sides. 2. The angles of any polygon, taken together, make twice as many right angles, abating four, as the figure bath sides. Thus, if the polygon hash five sides, the double of that is ten ; whence, subtracting four, there remains six right angles. 3. Every polygon, cir cumscribed about a circle, is equal to a right-angled triangle, one of whose legs is the radius of the circle, and the other the perimeter, or sum of all the sides of the polygon. Hence every regular polygon is equal to a right-angled-triangle, one of whose legs is the perimeter of the polygon; and the other a perpendicular, drawn from the centre to one of the sides of the polygon.

Hence, also, every polygon circumscribed about a circle is larger than it ; and every polygon inscribed is less than the circle. The same likewise appears hence, that the thing containing is ever greater than the thing contained. Hence, again, the perimeter of every polygon circumscribed about a circle, is greater than the circumference of that circle ; and the perimeter of every polygon inscribed, less ; whence it follows, that a circle is equal to a right-angled triangle, whose base is the circumference of the circle, and its height the radius ; since this triangle is less than any polygon circum scribed, and greater than any inscribed.

Nothing therefore is wanting to the quadrature of the circle, but to find a right line equal to the circumference of a circle.

The subjoined table gives the areas of polygons and their perpendiculars from the centre to one of the sides, the side being supposed equal to unity.

To apply this rule generally, multiply the square of the side of the given polygon by the number found in the table of areas for a polygon, having the same number of sides as that whose area is required.