Pythagoras' Theorem and Trigonometry in 3D

2026 Syllabus Objectives

By the end of this topic, you should be able to:

  1. Carry out calculations and solve problems in three dimensions using Pythagoras' theorem and trigonometry, including calculating the angle between a line and a plane.

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What is 3D Trigonometry?

3D trigonometry means using Pythagoras' theorem and trigonometry (SOHCAHTOA) to solve problems involving three-dimensional shapes like cubes, cuboids, pyramids, cones, and prisms.

In 2D (flat shapes), you work with triangles on a flat surface. In 3D (solid shapes), you work with triangles that exist in space, involving height, width, and depth.

The key skill is finding and extracting right-angled triangles from 3D shapes. Once you spot these triangles, you can use the same methods you already know from 2D geometry.

The Basic Strategy: Extract Right-Angled Triangles

When faced with a 3D problem, follow this strategy:

  1. Look carefully at the 3D shape and identify where right-angled triangles exist.
  2. Extract (draw out) the triangle as a flat 2D triangle on your page.
  3. Label the sides with the measurements you know.
  4. Use Pythagoras' theorem or SOHCAHTOA to find the missing length or angle.

This strategy works for all 3D problems. You break down the complicated 3D shape into simpler 2D triangles.

Using Pythagoras' Theorem in 3D

Pythagoras' theorem states that in a right-angled triangle: a² + b² = c², where c is the hypotenuse (the longest side opposite the right angle).

Method 1: Breaking Into Two 2D Triangles (Step-by-Step Method)

This is the most reliable method and works for every 3D problem.

Example 1: Finding a Space Diagonal in a Cuboid

Imagine a box (cuboid) with dimensions: length = 10 cm, width = 2 cm, height = 20 cm. You want to find the length of the diagonal line that goes from one corner at the bottom to the opposite corner at the top (called a space diagonal).

Step 1: First, find the diagonal across the base (the face diagonal).

Extract the right-angled triangle from the base of the box. If the base has sides 10 cm and 2 cm, the triangle looks like this:

  • One side = 10 cm
  • Other side = 2 cm
  • Hypotenuse (face diagonal) = unknown

Using Pythagoras:

  • AC² = 10² + 2²
  • AC² = 100 + 4 = 104
  • AC = √104 ≈ 10.20 cm

Step 2: Now find the space diagonal using the face diagonal you just found.

Extract a new right-angled triangle that cuts through the inside of the box:

  • Base = face diagonal = √104 cm (from Step 1)
  • Height = 20 cm
  • Hypotenuse (space diagonal) = unknown

Using Pythagoras:

  • AH² = (√104)² + 20²
  • AH² = 104 + 400 = 504
  • AH = √504 ≈ 22.45 cm

Important tip: Keep your answer in exact form (like √104) until the final step to avoid rounding errors.

Method 2: The 3D Pythagoras Formula (Direct Method)

There is a formula that lets you find the space diagonal in one step:

d² = x² + y² + z²

Where:

  • d = the distance you want to find (the space diagonal)
  • x, y, z = the three perpendicular distances (length, width, height) between two points

Using the same cuboid example:

  • x = 10 cm
  • y = 2 cm
  • z = 20 cm

d² = 10² + 2² + 20²
d² = 100 + 4 + 400 = 504
d = √504 ≈ 22.45 cm

This gives the same answer as Method 1, but in fewer steps.

Which method should you use?

Both methods are correct. Method 1 (breaking into two triangles) is more reliable because you can always see exactly what you're doing. Method 2 (the formula) is faster but only works when you have three perpendicular distances. In exams, you can use either method, but if you get stuck, always go back to Method 1.

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