Lessons 21 · 1 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
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Practice and applications: polyhedra and solids of revolution
Textbook, Part 1: pp. 119–125
GoalSolve problems on the surfaces of right prisms, regular pyramids, cylinders and cones, using the relations among height, apothem and generator.
The lateral faces of a right prism are rectangles, so its lateral surface equals the base perimeter times the lateral edge: S_lat = P · h; the total surface is S_tot = S_lat + 2S_base. In a regular pyramid the height h falls on the centre O of the base; the distance from O to a base vertex is the circumradius R and the distance from O to the midpoint of a base side is the inradius r. The lateral edge equals √(h² + R²) and the apothem equals √(h² + r²) (each is the hypotenuse of a right triangle). The lateral surface is S_lat = p · a (p — semi-perimeter of the base) and the total surface adds the base area. For a square r = a/2, R = a√2/2; for a regular triangle r = a√3/6, R = a√3/3; for a regular hexagon R = a, r = a√3/2. The lateral surface of a cylinder is the base circumference times the generator; for a cone l = √(h² + r²) and S_lat = πrl. If the generator makes an angle α with the base, then r = l cos α.
Worked examples
Regular quadrilateral pyramid: base side 10 cm, height 12 cm. r = 5 cm, apothem = √(12² + 5²) = 13 cm. Lateral surface = p · a = 20 · 13 = 260 cm² (p = 40/2 = 20), total surface = 260 + 100 = 360 cm².
Right prism: the base is an isosceles trapezoid with bases 4 cm and 10 cm and legs 5 cm; the lateral edge is 6 cm. The trapezoid’s height is √(5² − 3²) = 4 cm and its perimeter 4 + 10 + 5 + 5 = 24 cm. Lateral surface 24 · 6 = 144 cm², base area (4 + 10)/2 · 4 = 28 cm², total surface 144 + 2 · 28 = 200 cm².
Class activity
Groups measure an empty box (for example a cardboard box), compute its total surface, and then unfold it (an adult helps with scissors) to compare with the measured area of the net.
Practice
1
What is the formula for the lateral surface of a right prism?
S_lat = P · h, where P is the base perimeter and h the lateral edge.
2
The base of a right prism is a triangle with sides 6, 8, 10 cm and the lateral edge is 5 cm. What is the total surface in cm²?
168
3
A regular quadrilateral pyramid has base side 16 cm and height 15 cm. What is its lateral surface in cm²? (apothem 17 cm)
544
4
Why is the apothem √(h² + r²) while the lateral edge is √(h² + R²)?
The apothem goes from the apex to the midpoint of a base side, which is at distance r (inradius) from the centre. The lateral edge goes from the apex to a base vertex, at distance R from the centre. Each forms a right triangle with the height.