Elements of geometry and trigonometry . d on a sphere whose diameter is 30, each angle of thepolygon being 140°? Ans. OF THE REGULAR ^OLYEDRONS. In determining the solidities of the regular polyedrons, itbecomes necessary to know, for each of them, the angle con-tained between any two of the adjacent faces. The determi-nation of this ; ngle involves the following property of a regu-lar polygon, viz.— MENSURATION OF SOLIDS. 295 Half tile diagonal xchich joins the extremities of ttvu adjacentsides of a régula) pohjgon, is equal to the side of the polygonmultiplied by the cosine of the an
Elements of geometry and trigonometry . d on a sphere whose diameter is 30, each angle of thepolygon being 140°? Ans. OF THE REGULAR ^OLYEDRONS. In determining the solidities of the regular polyedrons, itbecomes necessary to know, for each of them, the angle con-tained between any two of the adjacent faces. The determi-nation of this ; ngle involves the following property of a regu-lar polygon, viz.— MENSURATION OF SOLIDS. 295 Half tile diagonal xchich joins the extremities of ttvu adjacentsides of a régula) pohjgon, is equal to the side of the polygonmultiplied by the cosine of the angle which is obtained by di-viding ;{t)0^ by twice the number of sides : the radius beingequal to unity. Let ABCDE he any regular poly-gon. Draw the diagunal AC, aiui fromhe centre F draw FG, perpendicularro AB. Draw also AF, FB ; the lat-ter will be perpendicular to the dia<:-onal AC, and will bisect it at II (Bookin. Prop. VL Sch.). Let tlie number of sides of tlie poly-gon be designated by n : then, O^FB = ?^. and AFC ^ CAB n. 360° *27L But in the right-angled triangle ABH, we have A1I = AB cos A = AB cos :mo 2n (Trig. Th. I. Cor.) Remark 1.—When the polygon in question is the equilateraltriangle, the diagonal becomes a side, and consequently halfthe diagonal becomes half a side of the triangle. Remark L Cor.). 3G0 -The perpendicular BH = AB sin (Trig. 271 To determine the angle included between the two adjacentfaces of either of the regular polyedrons, let us suppose a planeto be passed perpendicular to the axis of a solid angle, andthrough the vertices of the solid anghîs which lie plane will intersect the convex surface of the polyedronm a regular poly<ron ; the number of sides of this polygon willbe ccjual U) the ntmiber of planes which meet at the vertex of<ither of the solid angles, and each side will be a diagonal ofone of the equal faces of the polyedron. Lot D be ; X of a solid angle,CD the, intersection of two adjace
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