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.
Cylinder
- base × height
- 2 circles + a wrap
Cone
- ⅓ of the cylinder
- l = slant height
Sphere
- all from r
- no height
Pyramid / prism: A prism is base-area × length; a pyramid (or cone) is ⅓ × base × height.
Pick the formula, substitute, evaluate: Identify the solid, read off the radius and height, then substitute. Keep π exact unless told to round.
IB-style question — volume of a cone
Find the volume of a cone with base radius 3 and height 4.
Step by step
- Use V = ⅓πr²h.
Final answer
V = 12π ≈ 37.7.
Square the radius, not the height: In V = ⅓πr²h, only r is squared. A common slip is squaring h too.
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Add up every face: Total surface area adds all the surfaces. A cone's slant uses l = √(r² + h²); a closed cylinder is 2 circles + the curved wrap (2πr² + 2πrh).
IB-style question — cone surface area
A cone has base radius 6 and height 8. Find its total surface area.
Step by step
- Slant height first.
- Total = base + curved.
Final answer
A = 96π ≈ 302.
Open or closed?: Read whether a surface is included — an open cylinder (no lid) drops one circle; a hemisphere's flat face is one extra circle.
Add the pieces (or subtract a hole): For a solid made of parts (e.g. a cylinder topped by a hemisphere), add the volumes. For surface area, add only the exposed faces (a shared join is not counted twice).
IB-style question — cylinder + hemisphere
A solid is a cylinder (r = 3, h = 10) with a hemisphere (r = 3) on top. Find its volume.
Step by step
- Cylinder volume.
- Hemisphere = half a sphere.
Final answer
V = 90π + 18π = 108π ≈ 339.
Don't double-count the join: When adding surface areas, the circle where two pieces meet is internal — leave it out.
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Given the volume, solve for r or h: Set the formula equal to the given volume (or area) and solve for the unknown dimension — often a cube root or a square root.
IB-style question — find the radius
A sphere has volume 36π. Find its radius.
Step by step
- Set the formula equal.
- Solve.
Final answer
r = 3.
Cancel the π: If π appears on both sides, divide it out first — the numbers get much friendlier.