Half the diagonal which joins the extremities of two adjaceni sides of a regular polygon, is equal to the side of the polygon multiplied by the cosine of the angle which is obtained by dividing 360° by twice the number of sides: the radius being equal to unity. Let ABCDE be any regular polygon. Draw the diagonal AČ, and from the centre F draw FG, perpendicular to AB. Draw also AF, FB; the latter will be perpendicular to the diagonal AC, and will bisect it at H (Book III. Prop. VI. Sch.). Let the number of sides of the polygon be designated by n: then, and AFG-CAB= 9 360° AFB= Fou H 360° n 2n But in the right-angled triangle ABH, we have 2n Let D be the vertex of a solid angle, CD the intersection of two adjacent faces, and ABC the section made in the convex surface of the polyedron by a plane perpendicular to the axis through D. Through AB let a plane be drawn perpendicular to CD, produced if necessary, and suppose AE, BE, to be the lines in 360° 2n B REMARK 1.-When the polygon in question is the equilateral triangle, the diagonal becomes a side, and consequently half the diagonal becomes half a side of the triangle. REMARK 2.-The perpendicular BH=AB sin (Trig. Th. I. Cor.). " F To determine the angle included between the two adjacent faces of either of the regular polyedrons, let us suppose a plane to be passed perpendicular to the axis of a solid angle, and through the vertices of the solid angles which lie adjacent. This plane will intersect the convex surface of the polyedron in a regular polygon; the number of sides of this polygon will be equal to the number of planes which meet at the vertex of either of the solid angles, and each side will be a diagonal of one of the equal faces of the polyedron. B which this plane intersects the adjacent faces. Then will AEB be the angle included between the adjacent faces, and FEB will be half that angle, which we will represent by A. Then, if we represent by n the number of faces which meet at the vertex of he solid angle, and by m the number of sides of each face. we shall have, from what has already F been shown, But hence, BF=BC cos BF EB 360° =sin FEB=sin A, to the radius of unity; Tetraedron. Octaedron 360o 2n 360° 2m This formula gives, for the plane angle formed by every two adjacent faces of the sin A and EB=BC sin COS Having thus found the angle included between the adjacent faces, we can easily calculate the perpendicular let fall from the centre of the polyedron on one of its faces, when the faces themselves are known. The following table shows the solidities and surfaces of the regular polyedrons, when the edges are equal to 1. A TABLE OF THE REGULAR POLYEDRONS WHOSE EDGES ARE 1. No. of Faces. Solidity. 0.1178513 4 6 . 1.0000000 Names. Tetraedron Hexacdron Octaedron. Dodecaedron Icosaedron 8 0.4714045 12 7.6631189 2.1816950 20 sin 360° 2m Surface. 1.7320508 6.0000000 3.4641016 20.6457288 . B PROBLEM XXI. To find the solidity of a regular polyedron. RULE I.-Multiply the surface by one-third of the perpendicular let fall from the centre on one of the faces, and the product will be the solidity. RULE II.-Multiply the cube of one of the edges by the solidity of a similar polyedron, whose edge is 1. The first rule results from the division of the polyedron into as many equal pyramids as it has faces. The second is proved by considering that two regular polyedrons having the same number of faces may be divided into an equal number of similar pyramids, and that the sum of the pyramids which make up one of the polyedrons will be to the sum of the pyramids which make up the other polyedron, as a pyramid of the first sum to a pyramid of the second (Book II. Prop. X.); that is, as the cubes of their homologous edges (Book VII. Prop. XX.); that is, as the cubes of the edges of the polyedron. 1. What is the solidity of a tetraedron whose edge is 15? Ans. 397.75. 2. What is the solidity of a hexaedron whose edge is 12? Ans. 1728. 3. What is the solidity of a octaedron whose edge is 20? Ans. 3771.236. 4. What is the solidity of a dodecaedron whose edge is 25? Ans. 119736.2328. 5. What is the solidity of an icosaedron whose side is 20? Ans. 17453.56. A TABLE LOGARITHMS OF NUMBERS OF FROM 1 TO 10,000. Log. 1.447158 53 0.778151 31 1.491362 56 Log. N. Log. 1.707570 76 1.880814 1.716003 77 1.886491 1.724276 78 1.892095 1.732394 79 1.897627 1.740363 80 1.903090 1.748188 81 1.908485 1.755875 82 1.913814 1.763428 83 1.919078 1.770852 84 1.924279 1.778151 85 1.929419 1.556303 1.041393 36 1.568202 62 1.146128 39 1.591065 64 15 1.176091 40 1.602060 65 1.204120 41 19 20 1.612784 66 17 1.230449 42 1.623249 67 18 1.255273 43 1.633468 68 1.278754 44 1.643453 69 1.301030 45 1.653213 70 21 1.322219 46 1.662758 71 1.851258 96 1.982271 22 1.342423 47 1.672098 72 1.857333 97 1.986772 23 1.361728 48 1.681241 73 1.863323 98 1.991226 24 1.380211 49 1.690196 74 1.869232 25 1.397940 50 1.698970 75 1.875061 99 1.995635 100 2.000000 1.785330 86 1.934498 94 1.973128 N. B. In the following table, in the last nine columns of each page, where the first or leading figures change from 9's to O's, points or dots are introduced instead of the O's through the rest of the line, to catch the eye, and to indicate that from thence the annexed first two figures of the Logarithm in the second column stand in the next lower line. |