Key Idea: Topic 3.1 builds the 3D spatial toolkit you need for all geometry problems: measuring distance between points in 2D and 3D, finding midpoints, and calculating volumes and surface areas of common 3D solids. These appear in every paper and connect to trigonometry in topics 3.2 and 3.3.
✅ Distance and midpoint
📐 Volume and surface area formulas (from IB booklet)
Example: 3D diagonal of a cuboid with dimensions 3 × 4 × 12: d = √(3² + 4² + 12²) = √(9 + 16 + 144) = √169 = 13 Distance from A(2, −1, 5) to B(5, 3, 1): d = √((5−2)² + (3−(−1))² + (1−5)²) = √(9 + 16 + 16) = √41 ≈ 6.40 (3 s.f.) Volume of cone with r = 4 cm, h = 9 cm: V = (1/3)π(4²)(9) = (1/3)π(16)(9) = 48π ≈ 151 cm³
All volume formulas are given in the IB formula booklet. Do not memorise them — but do know what each variable means (r, h, l) so you can substitute correctly. Slant height l ≠ vertical height h. For a cone, l = √(r² + h²). Draw a right-angled triangle to see this.
Paper 1 (GDC allowed): Leave answers in exact form (e.g., 48π or √41) unless told to evaluate. Show the formula substitution clearly. Paper 2 (GDC allowed): Use the calculator to evaluate. Check whether the question asks for volume or surface area — misreading this is a common error.
IB-style question [6 marks]
A garden plan is drawn on a coordinate grid where each unit is 1 m. A water tank stands at A(0, 0) and a tap is fixed at B(8, 6). (a) Find the length of pipe needed to run in a straight line from A to B. (b) The tank is a cylinder of radius 1.2 m and height 3 m. Find the capacity of the tank, in litres, correct to 3 significant figures. (1 m³ = 1000 litres.) (c) Water is pumped into the empty tank at a constant rate of 200 litres per minute. Find the time, in minutes, to fill the tank, correct to 3 significant figures.
Step by step:
(a) Write the 2D distance formula, then substitute A(0, 0) and B(8, 6).
(b) Write the cylinder volume formula, substitute r = 1.2 and h = 3, then convert m³ to litres.
Convert to litres by multiplying by 1000, then round to 3 significant figures.
(c) Time = total volume ÷ rate. Use the unrounded capacity to avoid rounding error.
(a) 10 m of pipe. (b) ≈ 13 600 litres. (c) ≈ 67.9 minutes.
[Diagram: math-solid-volume]