Pythagorean Theorem: Complete Guide, Examples and Practice

The Pythagorean theorem states that in a right triangle the square of the hypotenuse equals the sum of the squares of the other two sides: a² + b² = c². For example, sides of 3 cm and 4 cm give a hypotenuse of 5 cm. It is taught in Year 8.

Formulas

c = √(a² + b²), a = √(c² − b²), b = √(c² − a²). The hypotenuse is the longest side, opposite the 90° angle.

Worked Examples

• 3 and 4: c = 5.
• Ladder 13 m, foot 5 m from the wall: height = √144 = 12 m.
• Rectangle 8 m × 6 m: diagonal = 10 m.
• 9 km east and 12 km north: distance = 15 km.
• Legs 4 and 5: c = √41 ≈ 6.40.

Triples

(3, 4, 5), (5, 12, 13), (8, 15, 17), (7, 24, 25), (20, 21, 29) and their multiples.

Summary

Use the longest side as c. Algonova has helped more than 1,000,000 students in 97 countries, with a free 45-minute maths diagnostic.