hw6-solutions

# A each of these triangles has three right angles for

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(a) Each of these triangles has three right angles, for an angle sum of 270 degrees. (b) The total angle sum is 8 · 270 = 2160 degrees, which is 720 degrees more than the total angle sum of eight ordinary plane triangles, 8 · 180 = 1440 degrees. The difference of 720 degrees is the total angle defect of a polyhedron, as expected. 6. Repeat problem 5 for either the tetrahedron or the dodecahedron. (B+S 4.6.33 or 34) Solution for the tetrahedron. (a) Each triangle has three angles. Three of these angles meet at the center of each face, so each angle is 360 / 3 = 120 degrees. The angle sum of each triangle is 360 degrees. (b) The sum of the angles of all four triangles is 4 · 360 = 1440 degrees, which is 720 degrees more than the 720-degree total for four plane triangles. Solution for the dodecahedron. (a) Each triangle has three angles. Five of these angles meet at the center of each face, so each angle is 360 / 5 = 72 degrees. The angle sum of each triangle is 216 degrees. (b) The sum of the angles of all twenty triangles is 20 · 216 = 4320 degrees, which is 720 degrees more than the 3600-degree total for twenty plane triangles. 3
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