The cone, the frustum and chapter review
Answers are for parents and teachers.
1
A cone has base radius 9 and slant height 15. Find its height, total surface and volume.
h = √(225 − 81) = 12; S_total = π · 9 · 24 = 216π; V = ⅓ π · 81 · 12 = 324π.
2
A sector of radius 12 and central angle 150° is rolled into a cone. Find the base radius and the lateral surface.
2πr = (150/360) · 2π · 12, so r = 5; S_lat = π · 5 · 12 = 60π (the same as the sector’s area).
3
A frustum has base radii 8 and 4 and height 3. Find the slant height, lateral surface and volume.
l = √(9 + 16) = 5; S_lat = π · 5 · 12 = 60π; V = (3π/3)(64 + 32 + 16) = 112π.
4
A regular triangular pyramid is inscribed in a cone whose base radius is 6 and height 8. Find the volumes of the cone and the pyramid.
V_cone = ⅓ π · 36 · 8 = 96π. The base side is a = R√3 = 6√3, its area is (√3/4) · 108 = 27√3; V_pyramid = ⅓ · 27√3 · 8 = 72√3.
5
Why is the lateral surface of a cone equal to πrl?
The net is a sector of radius l with arc 2πr. A sector’s area is ½ · arc · radius = ½ · 2πr · l = πrl.