Elements of geometry and trigonometry . he following is the manner of determiningthe perpendicular when only one side and the number of sidesof the regular polygon are known :— First, divide 3G0 degrees by the number of sides of the poly-gon, and the quotient will be the angle at the centre ; that is,the angle subtended by one of the equal sides. Divide thisangle by 2, and half the angle at the centre will then be known. Now, the line drawn from the centre to an angle of thepolygon, the perpendicular let fall on one of the equal sides,and half this side, form a right-angled triangle, in which
Elements of geometry and trigonometry . he following is the manner of determiningthe perpendicular when only one side and the number of sidesof the regular polygon are known :— First, divide 3G0 degrees by the number of sides of the poly-gon, and the quotient will be the angle at the centre ; that is,the angle subtended by one of the equal sides. Divide thisangle by 2, and half the angle at the centre will then be known. Now, the line drawn from the centre to an angle of thepolygon, the perpendicular let fall on one of the equal sides,and half this side, form a right-angled triangle, in which thereare known, the base, which is half the equal side of the poly-gon, and the angle at the vertex. Hence, the perpendicularcan be determined. Î. To find the area of a regular hexa-gon, whose sides are 20 feet each. 6)360° 60°=ACB,the angle at the centre. 30°—ACD, half the angle at the centre Also, CAD=90°—ACD = 60°; and AD = 10. Then, as sin ACD . . 30°, ar. comp : sin CAD ... 60° •: AD 10 CD Perimeter =120, and half the perimeter =, 60 x , the area. 2. What is the area of an octagon whose side is 20 ? Ans. Remark II.—The area of a regular polygon of any numberof sides is easily calculated by the above rule. Let the areasof the regular polygons whose sides are unity, or 1, be calcu-lated and arranged in the following IVIENSURATION OF SURFACES, ^ 281 TABLE. Namcs. Sides. Areas. Triangle . ... 3 . . Square ... 4 . . Pcfltagon . ... 5 . . Hexagon ... G . Heptagon . ... 7 . . Octagon . . 8 . . Nonagon . . 9 . . Decagon . . 10 . . Unci ec ago r I ... 11 . Dodeca^on . . 12 . . Now, since the areas of similar polygons are to each otherts the squares of their homologous sides (Book IV. ;^XV1I.), we shall have 1- : tabular area : : any side squared : area. Or, to find the area of an
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