3d Pythagoras And Trigonometry

3D Pythagoras Theorem

• **3D Pythagoras theorem**: $d^{2} = x^{2} + y^{2} + z^{2}$, where d is the distance between two points and x, y, z are distances in three perpendicular directions.
• You can always break a 3D problem into **two 2D right-angled triangles** instead of using the 3D formula.
• The 3D formula is **not given** in the exam; you must derive it or use 2D steps.
• Example: In a cuboid of dimensions 3 $cm \times 4$ $cm \times 6$ cm, the longest diagonal $is \sqrt{3^{2}+4^{2}+6^{2}} = \sqrt{61}$ ≈ 7.81 cm.

Using Pythagoras in 3D Shapes

• Identify **right-angled triangles** within the 3D shape (e.g., on faces or cross-sections).
• Redraw the 2D triangle **flat** on the page to apply Pythagoras or trigonometry.
• Find missing lengths step by step: e.g., first find a face diagonal, then the space diagonal.
• Common shapes: cuboids, prisms, pyramids with square bases.

Trigonometry (SOHCAHTOA) in 3D

• Use **SOHCAHTOA** in right-angled triangles formed within the 3D shape.
• Look for triangles that include the required angle or side; you may need to combine multiple triangles.
• Always draw the relevant 2D triangle **flat** to avoid confusion.
• Example: In a cuboid, to find the angle between a diagonal and a face, form a triangle with the height perpendicular to that face.

Angle Between a Line and a Plane

• The angle between a line and a plane is the angle between the line and its **projection** onto the plane.
• Construct a right-angled triangle where the height is **perpendicular** to the plane.
• Use SOHCAHTOA to find the angle; the perpendicular side is often a vertical edge.
• Tip: Imagine lowering a vertical fishing line from the line to the plane to visualise the right angle.

Angle Between Two Lines or Planes

• The angle between two lines in 3D can be found by placing them in a common triangle.
• The angle between two planes is found by considering the angle between lines perpendicular to their line of intersection.
• In pyramids, the angle between a sloping edge and the base is found using the vertical height and half the base diagonal.
• Example: In a square-based pyramid with side 8 cm and vertical height 10 cm, the angle between VA and base is tan⁻¹(10 $/ (8\sqrt{2}/2))$.

Solving Multi-Step Problems

• Break the problem into **2D right-angled triangles**; solve each step sequentially.
• Redraw each triangle **flat** on paper with all known lengths.
• Use Pythagoras to find missing sides, then trigonometry to find angles.
• If stuck, start finding any lengths or angles – they may lead to the answer and earn partial marks.

Common Question Types

• Finding the length of a space diagonal (e.g., longest object that fits in a box).
• Calculating the angle between a diagonal and a face (e.g., angle between AG and base ABCD).
• Finding the angle between a sloping edge and the base in a pyramid.
• Using given ratios or volumes to find missing dimensions and then angles.

Examiner Tips

• The 3D Pythagoras formula is **not provided** – you must derive it or use 2D steps.
• Always **redraw 2D triangles** flat to avoid misinterpreting the 3D diagram.
• If unsure, start calculating any lengths or angles – you may score method marks.
• Label all known lengths on the diagram before starting calculations.

Cuboid with diagonal and face diagonal

Right-angled triangle from 3D shape

Angle between line and plane

Square-based pyramid with vertical height