Know the booklet formulas: Volume and surface-area formulas for the standard solids are in the formula booklet — but you must know which to use and read the right radius / height.
[Diagram: math-solid-volume] - Available in full study mode
What's given vs what you memorise: ✓ in the booklet — you can look it up in the exam.
★ not given — memorise it, or build it from the given parts (add a base circle, halve a sphere, sum the faces…).
| Solid | Volume | Surface area |
|---|---|---|
| Cuboid | ✓ | ★ |
| Cylinder | ✓ | curved ✓; closed adds ★ |
| Cone | ✓ | curved ✓; total adds base ★ |
| Sphere | ✓ | ✓ |
| Hemisphere | ★ | ★ (½ sphere + flat base) |
| Pyramid | ✓ (A = base area) | ★ |
| Prism | ✓ | sum of all faces ★ |
[Diagram: math-base-areas] - Available in full study mode
| Shape | Area — also given | Where you use it |
|---|---|---|
| Parallelogram | cross-section of a slanted prism | |
| Triangle | the base A of a triangular prism / a pyramid | |
| Trapezoid | cross-section of a trough / trapezoidal prism | |
| Circle | a cylinder or cone base, or a hemisphere's flat face | |
| Circle — rim | the distance around the edge (e.g. how far a wheel rolls in one turn) |
[Diagram: math-prism-cross-section] - Available in full study mode
Worked example — volume of a cylinder
A can has radius 4 cm and height 10 cm.
Find its volume.
Step by step
- Write the cylinder volume formula, where r is the radius and h is the height.
- Substitute r = 4 and h = 10.
- Evaluate and round to 3 s.f.
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 |
| Hemisphere (solid) | SA = 3πr² | Curved 2πr² + flat base πr² |
| Cone (closed) | SA = πr² + πrl | Base circle + curved side; l = slant height |
[Diagram: math-solid-volume] - Available in full study mode
Worked example — surface area of a sphere
Find the surface area of a sphere with radius 5 cm.
Step by step
- Write the sphere surface area formula, where r is the radius.
- Substitute r = 5.
- Evaluate and round to 3 s.f.
Final answer
Surface area = 100π ≈ 314 cm².
Worked example — surface area of a capsule
A capsule is a cylinder of radius r and height h with a hemisphere on each end.
Find a formula for its total surface area S.
Step by step
- The curved side of the cylinder is 2πrh. The two hemispheres are domes only — their flat faces sit against the cylinder, so they aren't exposed.
- Two hemisphere domes together make one whole sphere's surface.
- Add only the exposed surfaces (no flat circles — they are internal joins).
Final answer
S = 2πrh + 4πr².
[Diagram: math-capsule] - Available in full study mode
Open vs closed containers: IB questions sometimes ask for an open cylinder (no lid).
In that case, remove one circular end: SA = πr² + 2πrh.
Surface area — decide which faces actually count: Before you add areas, picture the real object and count only the surfaces that exist:
• Open top / no lid → drop one circle (an open cylinder is ).
• Solid hemisphere → the curved dome plus its flat circular base: . (A hollow dome with no base is just .)
• Composite solid (e.g. a cylinder topped by a hemisphere) → count only the exposed faces; the circle where the pieces join is internal, so it is not painted on either side.
Know your predicted grade
Take timed mock exams and get detailed feedback on every answer. See exactly where you're losing marks.
Slant height vs vertical height: For a 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 Pythagoras: l² = r² + h².
Pyramids work the same way: A right pyramid has the same slant-height idea. Its slant height is the height of a triangular face — from the apex down to the midpoint of a base edge.
For a square base of side b, the vertical height h, the slant height l and half the base (b ÷ 2) form a right triangle:
.
Its volume is V = ⅓ × base area × height.
[Diagram: math-solid-volume] - Available in full study mode
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. The slant height l is the diagonal from tip to base edge, found by Pythagoras on the radius r and vertical height h.
- Substitute r = 3 and h = 4.
- (b) Volume. Write the cone volume formula (one third of a cylinder), using the vertical height h.
- Substitute r = 3 and h = 4.
- Evaluate and round to 3 s.f.
- (c) Total surface area. Write the closed-cone formula — base circle plus curved side, where l is the slant height.
- Substitute r = 3 and l = 5.
- Evaluate and round to 3 s.f.
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. Write the cylinder volume formula, where r is the radius and h is the height.
- Substitute r = 4 and h = 12.
- Evaluate and round to 3 s.f.
- Total surface area. Write the closed-cylinder formula — two circular ends plus the curved side.
- Substitute r = 4 and h = 12.
- Evaluate and round to 3 s.f.
Final answer
Volume ≈ 603 cm³; metal area needed ≈ 402 cm².
[Diagram: math-solid-volume] - Available in full study mode
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.
IB-style question — a chocolate dome (surface area → mass)
A chocolate is a solid hemisphere of radius 15 mm.
(a) Find the total surface area of one chocolate.
(b) The whole surface is coated in an edible glaze. 1 gram of glaze covers 200 mm². Find the mass of glaze needed for one chocolate, correct to 3 significant figures.
Step by step
- (a) Write the solid-hemisphere surface area formula — a curved dome 2πr² plus a flat circular base πr², so 3πr².
- Substitute r = 15.
- (b) Mass of glaze is proportional to the area covered: write it as the area divided by the coverage rate c (the area covered per gram).
- Substitute A = 2120.6 and c = 200.
- Evaluate and round to 3 s.f.
Final answer
(a) A = 675π ≈ 2120 mm². (b) ≈ 10.6 g of glaze.
[Diagram: math-solid-volume] - Available in full study mode
Hollow cylinder (a pipe): A pipe is a cylinder with a smaller cylinder removed (the hole). Picture the ring-shaped end: outer radius R, inner radius r.
Volume of metal = big cylinder − hole = .
IB-style question — volume of metal in a pipe
A metal pipe is a hollow cylinder with outer radius 5 cm, inner radius 3 cm and length 20 cm.
Find the volume of metal in the pipe.
Step by step
- The metal is the outer cylinder with the inner cylinder (the hole) taken out — so subtract the two volumes. Both share the same length h, so factorise.
- Substitute R = 5, r = 3 and h = 20.
Final answer
V = 320π ≈ 1005 cm³ of metal.
[Diagram: math-hollow-cylinder] - Available in full study mode