Lessons 22–23 · 2 lessons · M. A. Mirzaahmedov, Sh. N. Ismailov, A. Q. Amanov (algebra and analysis), B. Q. Haydarov (geometry). Mathematics Grade 10, Parts 1 and 2, 1st edition. EXTREMUM PRESS, Tashkent, 2017
23
Polyhedra and constructing their simple sections
Textbook, Part 1: pp. 131–134
GoalKnow how to draw a prism and a pyramid and construct the section of a polyhedron by a plane using traces.
To draw a prism we first draw one base as a polygon, then from each vertex draw equal parallel segments and join their ends; hidden edges are dashed. For a pyramid we draw the base, mark the apex and join it to every base vertex. The section of a polyhedron is the figure formed by the points of the polyhedron lying in the cutting plane; it is a polygon whose sides are segments in the faces. To construct it we find two common points of the cutting plane with each face: joining them gives the side of the section in that face (a trace). If two such points are not on the same face, we extend a line lying in the face until it meets a line of the cutting plane and use the meeting point. The number of sides of a section does not exceed the number of faces of the polyhedron.
Worked examples
Triangular pyramid QABC with points K ∈ QA, L ∈ QB, M ∈ QC. Draw KL, LM and KM: each pair lies in one face (QAB, QBC, QAC), so the section is triangle KLM.
The plane through vertices A, C and B₁ of a cube: AC lies in the bottom face, CB₁ in a side face and AB₁ in the front face. The three sides lie in three faces, so the section is triangle ACB₁; its sides are face diagonals, hence equal (a√2).
Class activity
Draw a cube in your notebook, mark points on three edges and construct the section. Compare your drawing, with hidden edges dashed, with your neighbour’s.
Practice
1
How does the second base appear when drawing a prism?
Equal parallel segments are drawn from the vertices of the first base and their ends are joined.
2
How many faces does a cube have? What is the maximum number of sides of a section of a cube?
A cube has 6 faces; a section has at most 6 sides (a hexagon).
3
Find the number of faces of a pentagonal prism.
7
4
Why does the number of sides of a section not exceed the number of faces?
Each side is a segment in one face, and in a convex polyhedron the plane gives only one segment in each face.