Lessons 38 · 1 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
38
The pyramid and the truncated pyramid
Textbook, Part 2: pp. 113–126
GoalLearn the elements of a pyramid and a frustum, how to find their lateral and total surfaces and volumes, and how to use the ratio of volumes of similar solids.
New words
pyramid · piramidaslant height (apothem) · apofemafrustum of a pyramid · kesik piramidasimilarity ratio · o‘xshashlik koeffitsiyenti
Explanation
A pyramid has one polygonal face (the base) and triangular lateral faces with a common vertex; the perpendicular from the apex to the base plane is the height H. If the base is a regular polygon and the height meets its centre, the pyramid is regular; the altitude of a lateral face from the apex is the slant height (apothem) l. For a regular pyramid Slat = ½ P l (P is the base perimeter) and Stotal = Slat + Sbase, while for any pyramid V = ⅓ Sbase H. A plane parallel to the base cuts off a smaller similar pyramid and leaves a frustum; if the similarity ratio is k, lengths scale by k, areas by k² and volumes by k³. For a frustum with base areas Q₁ and Q₂ and height H the volume is V = (H/3)(Q₁ + √(Q₁Q₂) + Q₂); for a regular frustum Slat = ½(P₁ + P₂) l. In problems use the right triangle formed by the height, the apothem and half the base side (or the inradius).
Worked examples
A regular quadrangular pyramid has base side 10 and height 12. Half the base side is 5 and the apothem is l = √(144 + 25) = 13. Slat = ½ · 40 · 13 = 260, Sbase = 100, Stotal = 360, V = ⅓ · 100 · 12 = 400. The lateral edge is √(144 + 50) = √194.
A regular quadrangular frustum has base sides 12 and 6 and height 4. Q₁ = 144, Q₂ = 36, √(Q₁Q₂) = 72; V = (4/3)(144 + 72 + 36) = (4/3) · 252 = 336. Apothem: the difference of the half-sides is 6 − 3 = 3, l = √(16 + 9) = 5; Slat = ½ (48 + 24) · 5 = 180.
Class activity
“Paper pyramid”: with an adult’s help use scissors to cut a 6 cm square and four isosceles triangles (base 6 cm, height 5 cm) and build a pyramid. The apothem is 5 and the half-side 3, so H = 4 cm. Compute Slat = ½ · 24 · 5 = 60 cm² and V = ⅓ · 36 · 4 = 48 cm³ and check by measuring the height.
Practice
1
A regular quadrangular pyramid has base side 6 and height 4. Find its volume.
48
2
A regular quadrangular pyramid has base side 8 and apothem 5. Find its lateral surface.
80
3
A frustum has base areas 36 and 9 and volume 84. Find its height.
4
4
Why is the ratio of areas k² but the ratio of volumes k³ for a section parallel to the base?
Area is two-dimensional: both linear dimensions change by k, so k · k = k². Volume is three-dimensional: length, width and height all change by k, so k³.
