The pyramid and the frustum
Answers are for parents and teachers.
1
A regular quadrangular pyramid has base side 16 and height 6. Find its lateral surface and volume.
The apothem is √(36 + 64) = 10; S_lat = ½ · 64 · 10 = 320; V = ⅓ · 256 · 6 = 512.
2
A pyramid has a 12 × 16 rectangular base and every lateral edge is 26. Find its volume.
The diagonal is 20 and half of it is 10 (the height meets the centre of the base). H = √(676 − 100) = 24; V = ⅓ · 192 · 24 = 1536.
3
A section parallel to the base is made at distance 6 from the apex of a pyramid with height 9 and base area 81. Find the section area and what fraction of the whole volume the small pyramid is.
k = 6/9 = 2/3. The section area is 81 · 4/9 = 36; the volume ratio is k³ = 8/27.
4
A regular quadrangular frustum has base sides 3 and 9 and height 4. Find its volume.
Q₁ = 81, Q₂ = 9, √(Q₁Q₂) = 27; V = (4/3)(81 + 27 + 9) = (4/3) · 117 = 156.
5
Why is a pyramid’s volume one third of that of the prism with the same base and height?
A triangular prism can be cut into three pyramids; they have bases of equal area and the same height, so their volumes are equal. Each is 1/3 of the prism. Any pyramid can be split into triangular pyramids.