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
21
The prism and its surface
Textbook, Part 1: pp. 153–160
GoalKnow the elements of the prism, parallelepiped and cube, diagonal and perpendicular sections; compute the lateral and total surface areas.
New words
right / oblique prism · to‘g‘ri / og‘ma prizmaregular prism · muntazam prizmadiagonal section · diagonal kesimlateral surface · yon sirt
Explanation
A prism has two equal polygons as bases and parallelograms as lateral faces. If the lateral edges are perpendicular to the base the prism is right, otherwise oblique; a right prism whose base is a regular polygon is a regular prism. The height is the perpendicular between the bases. A diagonal section passes through corresponding diagonals of the bases; an n-gonal prism has n(n − 3)/2 diagonal sections and n(n − 3) diagonals. A parallelepiped is a prism with a parallelogram base; for a cuboid with edges a, b, c the diagonal satisfies d² = a² + b² + c². The lateral surface of a right prism is S_lat = P · h (P is the base perimeter); for an oblique prism S_lat = P_⊥ · l (the perimeter of the perpendicular section times the lateral edge). The total surface is S_total = S_lat + 2S_base. For a cube of edge a the diagonal is a√3 and the total surface 6a².
Worked examples
A regular triangular prism with base side 8 and height 5. S_lat = 3 · 8 · 5 = 120. S_base = (√3/4) · 8² = 16√3. S_total = 120 + 2 · 16√3 = 120 + 32√3 ≈ 175.4.
A cuboid 3 × 4 × 12: d² = 9 + 16 + 144 = 169, d = 13. Its total surface is 2(3 · 4 + 3 · 12 + 4 · 12) = 2(12 + 36 + 48) = 192. An oblique prism with perpendicular-section perimeter 14 and lateral edge 5 has S_lat = 14 · 5 = 70.
Class activity
“Surface of a box”: bring a box (a shoebox or book box) and measure its three dimensions. Compute the diagonal with the formula and check it with a piece of string; compute the total surface to decide how much wrapping paper is needed.
Practice
1
Find the diagonal of a cuboid with dimensions 2, 5 and 14.
15
2
A right prism has a rhombus base with side 5 and height 7. Find its lateral surface.
140
3
How many diagonals does a hexagonal prism have?
18
4
Why is the lateral surface of an oblique prism equal to the perimeter of the perpendicular section times the lateral edge?
The perpendicular section cuts the prism into two pieces; moving one so that the bases coincide gives a right prism with that section as base and lateral edge l, whose lateral surface P_⊥ · l equals that of the original prism.