All volume formulas are in the IB booklet: You do not need to memorise these — but you must be fluent in setting them up quickly. The most common solids in IB Paper 1 and 2 are: cuboid, cylinder, cone, sphere, and pyramid.
| Solid | Volume formula | Notes |
|---|---|---|
| Cuboid | V = l × w × h | Length × width × height |
| Cylinder | V = πr²h | π × radius² × height |
| Cone | V = ⅓πr²h | One-third of a cylinder with same r and h |
| Sphere | V = 4/3 πr³ | Depends only on radius |
| Pyramid | V = ⅓ × Abase × h | One-third × base area × perpendicular height |
Worked example — volume of a cylinder
A can has radius 4 cm and height 10 cm. Find its volume.
Step by step
- Use the cylinder volume formula.
- Calculate.
Final answer
Volume = 160π ≈ 502 cm³ (3 s.f.)
Surface area = sum of all faces: Surface area is the total area of the outer surface. For each solid, think about all faces separately and add them up.
| Solid | Surface area formula | Notes |
|---|---|---|
| Cuboid (closed) | SA = 2(lw + lh + wh) | 3 pairs of rectangular faces |
| Cylinder (closed) | SA = 2πr² + 2πrh | 2 circles + curved side |
| Sphere | SA = 4πr² | All one curved surface |
| Cone (closed) | SA = πr² + πrl | Base circle + curved side; l = slant height |
Worked example — surface area of a sphere
Find the surface area of a sphere with radius 5 cm.
Step by step
- Use the sphere surface area formula.
- Round to 3 s.f.
Final answer
Surface area = 100π ≈ 314 cm².
Open vs closed containers: IB questions sometimes ask for an open cylinder (no lid). In that case, remove one circular end: SA = πr² + 2πrh.
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Slant height vs vertical height: For cone, the slant height l is the distance along the curved surface from tip to base edge. The vertical height h goes straight down from the tip. Use Pythagora: l² = r² + h².
Worked example — cone volume and surface area
A cone has base radius 3 cm and vertical height 4 cm. Find its (a) slant height, (b) volume, (c) total surface area.
Step by step
- (a) Slant height.
- (b) Volume.
- (c) Total surface area.
Final answer
l = 5 cm, V ≈ 37.7 cm³, SA ≈ 75.4 cm².
Worked example — material needed for a can
A tin of food is a closed cylinder with radius 4 cm and height 12 cm. Find the volume and the minimum area of metal sheet needed to make it.
Step by step
- Volume.
- Total surface area (closed cylinder).
Final answer
Volume ≈ 603 cm³; metal area needed ≈ 402 cm².
Units matter: Volume is always in cubic units (cm³, m³). Surface area is in square units (cm², m²). Mixing them up in an exam loses marks.