Pythagoras' Theorem in 2D & 3D

Master Pythagoras' Theorem a² + b² = c² to calculate missing side lengths in right-angled triangles, coordinate geometry, and 3D shapes.

Learning Objectives

Identify the hypotenuse as the longest side opposite the right angle
Apply the formula a² + b² = c² to find the hypotenuse
Rearrange to find a shorter side: a² = c² - b²
Calculate 3D space diagonals in cuboids and pyramids

Concept Explanation

Pythagoras' Theorem is one of the most famous theorems in mathematics. It establishes a fundamental relationship between the three sides of any right-angled triangle.

In any right-angled triangle, the area of the square built on the hypotenuse (c) is equal to the sum of the areas of the squares on the other two sides (a and b).

Hypotenuse (c): The side directly opposite the 90° right angle. It is ALWAYS the longest side.

Finding the Hypotenuse: Square both shorter sides, add them together, and take the square root: c = √(a² + b²).

Finding a Shorter Side: Square the hypotenuse, subtract the square of the known shorter side, and take the square root: a = √(c² - b²).

Key Mathematical Formulas

a^2 + b^2 = c^2

c = \sqrt{a^2 + b^2}

a = \sqrt{c^2 - b^2}

Worked Example

Step-by-Step Solution

Interactive Walkthrough

Problem Statement:

A right-angled triangle has shorter sides of length 6 cm and 8 cm. Calculate the length of the hypotenuse c.

a = 6\text{ cm}, \quad b = 8\text{ cm}, \quad c = ?

Step 1: Write down Pythagoras' formula

Identify that we are finding the hypotenuse c.

c^2 = a^2 + b^2

Final Answer:

c = 10\text{ cm}

Try It Yourself

A right-angled triangle has a hypotenuse of 13 cm and one shorter side of 5 cm. Calculate the length of the other shorter side.

c = 13\text{ cm}, \quad b = 5\text{ cm}

Common Student Mistakes

❌ Common Pitfall: Adding instead of subtracting when finding a shorter side

✅ Correction: Always remember: if you are finding the hypotenuse, ADD. If you are finding a shorter side, SUBTRACT.

❌ Common Pitfall: Forgetting to take the square root at the final step

✅ Correction: Calculated c² = 100 is the area of the square, not the side length! Remember to calculate √100 = 10.

UK Exam Advice & Tips

Look out for common Pythagorean triples in non-calculator papers: (3, 4, 5), (5, 12, 13), and (8, 15, 17).
Always round your final answer to the required number of decimal places or significant figures stated in the question.

Lesson FAQ

Does Pythagoras' Theorem work on non-right-angled triangles?

No, Pythagoras' Theorem only applies to right-angled triangles. For triangles without a 90° angle, you must use the Sine Rule or Cosine Rule.

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