Solids: Volume, Sections & Rotations — Drills
Solids: Volume, Sections & Rotations has 12 drill skills. Each one generates fresh questions every time, with a hint and full worked steps, so you can keep practising a single skill until it settles.
What these drills cover
These questions use 16 ideas from the course. Follow any of them for a definition, where it is taught, and what it leads to.
- Area — Space inside, measured in square units.
- π — The circumference of any circle divided by its diameter — about 3.14159.
- Cross section — The 2-D face revealed when a plane cuts through a solid.
- Volume — Space inside, in cubic units.
- Surface area — Total area of all the outside faces, in square units.
- Prism — A solid with two identical parallel bases joined by rectangles.
- Base — The number being multiplied. In 53 the base is 5.
- Factor — A number that divides evenly into another.
- Radius — Centre to edge. Half the diameter.
- Scale — What one grid step is worth — and where the axis starts.
- Cylinder — Two parallel circles joined by a curved surface. V = π r2 h.
- Cone — A circle tapering to a point. V = 13π r2 h.
- Sphere — A ball. V = 43π r3.
- Parallel — Same slope, different intercept.
- Pyramid — A base tapering to a point. V = 13 Bh.
- Height — The perpendicular distance from the base to the opposite corner.
Prism
Unlimited questions of this kind are generated in the app. Two examples:
Problem 1
A prism has base area 24 and height 15. Find its volume.
Worked answer
360
24 × 15 = 360
Problem 2
A prism has base area 33 and height 12. Find its volume.
Worked answer
396
33 × 12 = 396
Cylinder
Unlimited questions of this kind are generated in the app. Two examples:
Problem 1
Find the volume of a cylinder with r = 6 and h = 9, in terms of π.
Worked answer
324pi
π(36)(9)
= 324π
Problem 2
Find the volume of a cylinder with r = 8 and h = 7, in terms of π.
Worked answer
448pi
π(64)(7)
= 448π
Pyramid
Unlimited questions of this kind are generated in the app. Two examples:
Problem 1
A pyramid has base area 42 and height 12. Find its volume.
Worked answer
168
13 (42)(12) = 168
Problem 2
A pyramid has base area 12 and height 10. Find its volume.
Worked answer
40
13 (12)(10) = 40
Sphere
Unlimited questions of this kind are generated in the app. Two examples:
Problem 1
Find the volume of a sphere with r = 3, in terms of π.
Worked answer
36pi
33 = 27
43 π (27) = 36π
Problem 2
Find the volume of a sphere with r = 6, in terms of π.
Worked answer
288pi
63 = 216
43 π (216) = 288π
Section
Unlimited questions of this kind are generated in the app. Two examples:
Problem 1
What shape does a cut parallel to a face of a cube give?
Worked answer
square
A cut parallel to a face of a cube gives a square.
Problem 2
What shape does a cut parallel to the base of a cylinder give?
Worked answer
circle
A cut parallel to the base of a cylinder gives a circle.
Cone
Unlimited questions of this kind are generated in the app. Two examples:
Problem 1
Find the volume of a cone with r = 12 and h = 11, in terms of π.
Worked answer
528pi
13 π(144)(11)
= 528π
Problem 2
Find the volume of a cone with r = 3 and h = 9, in terms of π.
Worked answer
27pi
13 π(9)(9)
= 27π
Slant
Unlimited questions of this kind are generated in the app. Two examples:
Problem 1
A cone has radius 9 and vertical height 12. Find its slant height.
Worked answer
15
√81 + 144 = √225
= 15
Problem 2
A cone has radius 15 and vertical height 20. Find its slant height.
Worked answer
25
√225 + 400 = √625
= 25
Surface
Unlimited questions of this kind are generated in the app. Two examples:
Problem 1
A cone has radius 16 and slant height 34. Find its total surface area in terms of π.
Worked answer
800pi
Lateral = π(16)(34) = 544π
Base = 256π
Total = 800π
Problem 2
A cone has radius 5 and slant height 13. Find its total surface area in terms of π.
Worked answer
90pi
Lateral = π(5)(13) = 65π
Base = 25π
Total = 90π
Revolve
Unlimited questions of this kind are generated in the app. Two examples:
Problem 1
Spin a semicircle about its diameter. What solid is formed?
Worked answer
sphere
Spinning a semicircle about its diameter sweeps out a sphere.
Problem 2
Spin a rectangle about one of its sides. What solid is formed?
Worked answer
cylinder
Spinning a rectangle about one of its sides sweeps out a cylinder.
Working backwards
Unlimited questions of this kind are generated in the app. Two examples:
Problem 1
A cylinder of height 7 has volume 63π. Find its radius.
Worked answer
3
π r2 (7) = 63π
r2 = 9
r = 3
Problem 2
A cylinder of height 6 has volume 150π. Find its radius.
Worked answer
5
π r2 (6) = 150π
r2 = 25
r = 5
Composite figures
Unlimited questions of this kind are generated in the app. Two examples:
Problem 1
A silo is a cylinder of radius 3 and height 9, topped by a cone of the same radius and height 6. Find the total volume in terms of π.
Worked answer
99pi
Cylinder: 81π
Cone: 18π
Total: 99π
Problem 2
A silo is a cylinder of radius 6 and height 8, topped by a cone of the same radius and height 3. Find the total volume in terms of π.
Worked answer
324pi
Cylinder: 288π
Cone: 36π
Total: 324π
Scale
Unlimited questions of this kind are generated in the app. Two examples:
Problem 1
A solid is scaled up by a factor of 4. By what factor does its volume grow?
Worked answer
64
43 = 64
Length × 4, area × 16, volume × 64.
Problem 2
A solid is scaled up by a factor of 5. By what factor does its volume grow?
Worked answer
125
53 = 125
Length × 5, area × 25, volume × 125.