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MathidiaMathematics Academy
Pythagoras and right triangles
Lesson 56 of 75
Introduction

Squares on triangle sides

Objective. Interpret Pythagoras through areas of squares.

  1. 1. Learn
  2. 2. Worked example
  3. 3. Practice
  4. 4. Feedback
  5. 5. Continue

Learn: the key idea

Pythagoras' theorem can be understood by drawing a square on each side of a right-angled triangle and comparing their areas. The theorem states that the area of the square on the hypotenuse equals the sum of the areas of the squares on the other two sides. Checking this with a 3, 4, 5 triangle: the squares have areas 9, 16 and 25, and indeed 9 + 16 = 25. This area model explains why the algebraic rule a² + b² = c² works, rather than just stating it as a formula to memorise.

Worked example

A right-angled triangle has squares of area 36 cm² and 64 cm² drawn on its two shorter sides. Find the area of the square on the hypotenuse.

Diagram of a triangle
  1. 1Pythagoras' theorem states the hypotenuse square area equals the sum of the other two square areas.
  2. 2Add the two given areas: 36 + 64.
  3. 3This gives 36 + 64 = 100.
  4. 4The area of the square on the hypotenuse is 100 cm².

Answer: 100 cm²

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Practice

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Common mistake. Pupils sometimes assume the areas of all three squares should be equal rather than the hypotenuse square equalling the sum of the other two; always check that the two smaller squares' areas add to the largest one.

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