IB Math AI SL — 3D space, volume, angles, and midpoints

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.