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Pythagoras: squares on three sides

MathematicsSuggested ages: 12–15

Why does the sum of the squares on the legs equal the square on the hypotenuse?

Change the two legs of a right triangle and compare the areas of squares on all three sides to verify a²+b²=c².

Learning goals

  • Identify c as the hypotenuse in a right triangle.
  • Interpret a squared side length as the area of a square on that side.
  • Use a²+b²=c² to find an unknown side and check whether the result is reasonable.

Before you begin

  • Calculate square areas and positive square roots.
  • Recognise a right angle, its adjacent legs, and the longest side.

Squares on three sides

  1. 1Find hypotenuse
  2. 2Read areas
  3. 3Combine areas
  4. 4Find length
Squares on three sidesDefault example: legs 3 and 4, hypotenuse 5; 9 + 16 = 25. a b c
a = 3 · a² = 9 b = 4 · b² = 16 c = 5 · c² = 25 · hypotenuse
a² + b² = c² · 9 + 16 = 25 · c = 5 (hypotenuse)

Observe, then explain

Work through four steps. Each button moves the interactive model to the matching observation state.

  1. Legs a and b meet at the right angle. Side c lies opposite it and is the hypotenuse; the theorem uses c this way only for a right triangle.

    Your task: Find the right-angle marker and identify the side that does not touch that angle.

  2. A square with side a has area a²; similarly the other areas are b² and c². The diagram is proportional to side lengths, while the readout gives exact calculations.

    Your task: Choose a=3 and b=4; compare 9, 16, and 25.

  3. For every right triangle, the total area of the squares on the legs equals the area of the square on the hypotenuse: a²+b²=c².

    Your task: Change one leg to 6 and check both sides of the equation.

  4. First find c²=a²+b², then take the positive square root to get c=√(a²+b²). A length is positive, so the negative root is not used here.

    Your task: Choose a=5 and b=5, estimate the two integers c lies between, then read the result.

Build an explanation

Square means an operation and an area

a² means a×a and is also the area of a square with side a. Squares on the sides turn the algebraic equation into an area relationship.

The hypotenuse is not just any longest side

Hypotenuse specifically means the side opposite the right angle in a right triangle. Without a right angle, this theorem does not apply directly and no side should be labelled hypotenuse c.

The converse can test for a right angle

If c is the longest side of a triangle and a²+b²=c², then the triangle is right-angled. This is the converse of the Pythagorean theorem.

Check your understanding

1Which side of a right triangle is the hypotenuse?
2If the legs are 3 and 4, what is the hypotenuse length?
3If a²=36 and b²=64, what is c²?
4When can a²+b²=c² be used directly to find the third side?

Printable worksheet

Complete the worksheet first. Answers and explanations are on a separate page.

Mathematics

Pythagoras: squares on three sides

Name: __________________Date: __________________

Why does the sum of the squares on the legs equal the square on the hypotenuse?

  1. Which side of a right triangle is the hypotenuse?

    A. The side opposite the right angle B. Any vertical side C. Always the shortest side

  2. If the legs are 3 and 4, what is the hypotenuse length?

    A. 5 B. 7 C. 25

  3. If a²=36 and b²=64, what is c²?

    A. 28 B. 100 C. 10

  4. When can a²+b²=c² be used directly to find the third side?

    A. For any triangle B. When the triangle is right-angled and c is the hypotenuse C. Only when all side lengths are integers

Answer page

Pythagoras: squares on three sides

  1. The side opposite the right angle

    The hypotenuse is opposite the right angle and is the longest side of a right triangle.

  2. 5

    c=√(3²+4²)=√25=5. The value 25 is c², not the side length.

  3. 100

    The hypotenuse square has the combined area of the two leg squares: 36+64=100.

  4. When the triangle is right-angled and c is the hypotenuse

    The theorem applies to right triangles; their side lengths may be integers or irrational numbers.