Lessons 20–21 · 2 lessons · M. A. Mirzaahmedov, Sh. N. Ismailov, A. Q. Amanov (algebra and analysis), B. Q. Haydarov (geometry). Mathematics Grade 11, Parts 1 and 2, 1st edition. ZAMIN NASHR, Tashkent, 2018
20
Polyhedral angles and polyhedra
Textbook, Part 1: pp. 146–152
GoalStudy dihedral and trihedral angles, convex polyhedra, Euler’s theorem (F + V − E = 2) and the regular polyhedra.
New words
dihedral angle · ikkiyoqli burchaktrihedral angle · uch yoqli burchakEuler’s theorem · Eyler teoremasiregular polyhedron · muntazam ko‘pyoq
Explanation
A dihedral angle is a figure formed by two half-planes with a common edge; it is measured by its linear angle, the angle between the rays drawn in the faces perpendicular to the edge from one point of the edge. Among the dihedral angles formed by two intersecting planes, adjacent ones add up to 180°. Three rays from one point form a trihedral angle; its plane angles satisfy: each is less than the sum of the other two, and their sum is less than 360°. A polyhedron is a solid bounded by flat polygons; it is convex if it lies on one side of the plane of each of its faces. Euler’s theorem: for a convex polyhedron F + V − E = 2 (faces, vertices, edges). Consequences: the number of plane angles is 2E and their sum is 360°(V − 2). In a regular polyhedron the faces are equal regular polygons and the same number of edges meet at every vertex; there are five: tetrahedron, cube (hexahedron), octahedron, dodecahedron, icosahedron.
Worked examples
Pentagonal prism: F = 7, V = 10, E = 15, and 7 + 10 − 15 = 2. If a convex polyhedron has 8 vertices and 12 edges then F = 2 − V + E = 2 − 8 + 12 = 6. For a point in one face of a 60° dihedral angle at distance 9 from the edge, the distance to the other face is 9 · sin 60° = 9√3/2.
A trihedral angle with plane angles 50°, 60°, 70° exists: 50 + 60 > 70, 50 + 70 > 60, 60 + 70 > 50 and the sum 180° < 360°. With 40°, 50°, 100° it does not exist, because 40 + 50 = 90 < 100. Regular polyhedron: for the cube F = 6, E = 12, V = 8: 6 + 8 − 12 = 2; three edges meet at each vertex.
Class activity
“Platonic solids”: the class draws nets of the five regular polyhedra on paper (cut with scissors with an adult’s help) and glues them. For each model count faces, edges and vertices into a table and check Euler’s formula.
Practice
1
A convex polyhedron has 10 faces and 16 vertices. How many edges does it have?
24
2
One of two adjacent dihedral angles is 40° larger than the other. How many degrees is the larger one?
110
3
Two plane angles of a trihedral angle are 50° and 80°. Within what limits can the third plane angle lie?
80° − 50° < γ < 80° + 50°, i.e. 30° < γ < 130° (the sum condition 50° + 80° + γ < 360° then holds automatically).
4
Why are there only five regular polyhedra? Use the fact that the plane angles at a vertex sum to less than 360°.
Triangles (60°): 3, 4 or 5 at a vertex (180°, 240°, 300°); 6 would give 360°. Squares (90°): only 3. Pentagons (108°): only 3. Hexagons (120°) with 3 already give 360°. In total 3 + 1 + 1 = 5 kinds.