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O Level · Mathematics 0580

3D Pythagoras & Trigonometry

Revise 3D Pythagoras & Trigonometry for Mathematics 0580 (O Level) — revision notes, 24 practice questions and instant AI marking.

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3D Pythagoras & Trigonometry

Cambridge IGCSE Maths: Extended

The Big Idea: Extend your 2D geometry skills into three dimensions by breaking 3D problems into 2D right-angled triangles, then applying Pythagoras' theorem and SOHCAHTOA.

What You'll Learn

Apply Pythagoras' theorem in 3D shapes
Use the 3D Pythagoras formula
Break 3D problems into 2D problems
Find angles between lines and planes
Apply SOHCAHTOA in 3D spaces
Solve complex multi-step 3D problems

1. Pythagoras' Theorem in 3D Shapes

How to identify and use right-angled triangles

The principle is simple: Inside any 3D shape, you can often find right-angled triangles. Once you spot one, you can use Pythagoras' theorem exactly as you would in 2D.

For example, in a cone with base radius 5 cm and perpendicular height 12 cm, to find the slant height, you form a right-angled triangle where:

One leg = the radius (5 cm) — measured along the base
Other leg = the perpendicular height (12 cm) — straight up
Hypotenuse = the slant height we're finding

Using Pythagoras: l² = 5² + 12² = 25 + 144 = 169, so l = 13 cm.

Key insight: The perpendicular height is crucial. It forms the 90° angle of the right-angled triangle. If you don't have a perpendicular line, you can't use Pythagoras directly.

Worked Example: Cone Slant Height

Problem: A cone has base radius 5 cm and perpendicular height 12 cm. Find the slant height l.

Step 1: Identify the right-angled triangle. The radius, height, and slant height form a right angle at the base centre.

Step 2: Apply Pythagoras: l² = 5² + 12²

Step 3: Calculate: l² = 25 + 144 = 169

Step 4: Take the square root: l = √169 = 13 cm

Cone Practice

A cone has a base radius of 6 cm and a perpendicular height of 8 cm. Find the slant height.

2. The 3D Pythagoras Formula

The distance between two points in space

If you want to find the straight-line distance between two points in 3D space, there's a direct formula:

3D Distance Formula

d² = x² + y² + z²

What this means:

d = the straight-line distance between the two points
x, y, z = the horizontal and vertical separations in each of the three perpendicular directions

This is like extending the 2D Pythagoras formula (a² + b² = c²) by adding a third dimension.

Worked Example: Distance in a Cuboid

Problem: Find the distance from point A to point B in a cuboid with dimensions 2 cm × 6 cm × 2 cm (width × length × height).

Step 1: Identify the three perpendicular distances:
• x = 2 cm (width)
• y = 6 cm (length)
• z = 2 cm (height)

Step 2: Apply the 3D formula: d² = 2² + 6² + 2²

Step 3: Calculate: d² = 4 + 36 + 4 = 44

Step 4: Take the square root: d = √44 ≈ 6.63 cm

Exam tip:

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Also in the full note

3. Breaking 3D Problems into 2D Triangles
4. Using SOHCAHTOA in 3D
5. The Angle Between a Line and a Plane
6. Tackling Complex 3D Problems
Key Terms & Formulas to Memorise
Concepts Checklist
Exam Tips & Common Mistakes

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📖 Revision notes ◉ 24 practice questions 🎯 Learn mode ✦ AI flashcards ✓ Instant AI marking 🧊 3D explorers 🧪 Experiments & simulations 📈 Progress tracking

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3D Pythagoras & Trigonometry

3D Pythagoras & Trigonometry

What You'll Learn

1. Pythagoras' Theorem in 3D Shapes

Worked Example: Cone Slant Height

2. The 3D Pythagoras Formula

Worked Example: Distance in a Cuboid

Past papers that test 3D Pythagoras & Trigonometry

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