Pythagoras' Theorem

For any right-angled triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides.

Before You Begin: Make sure you have reviewed:

  • Square numbers (e.g., 1² = 1, 2² = 4, 3² = 9, 4² = 16, etc.)
  • Square roots (finding the number that multiplied by itself gives the square)
  • Types of triangles (right-angled, equilateral, isosceles, scalene)

In any right-angled triangle, the side opposite the right angle is called the hypotenuse and this is always the longest side.

Triangle showing the hypotenuse

Naming the Sides of a Right-Angled Triangle

Side designations shift dynamically based on the observed target angle.

  • Hypotenuse: The side opposite the right angle. Always the longest side.
  • Opposite Side: The side directly facing the reference angle.
  • Adjacent Side: The side neighboring the reference angle that is not the hypotenuse.
Naming sides of a right angled triangle Naming sides of a right angled triangle

Context Comparison

When studying angle A, side BC represents the opposite side. Side AB represents the adjacent side. If switching focus to angle C, side AB becomes the opposite side. Side BC changes into the adjacent side.

Pythagorean Theorem

The Pythagorean Theorem describes the relationship between the three sides of a right-angled triangle. It states that the square of the hypotenuse is equal to the sum of the squares of the other two sides.

Pythagoras theorem

Where:

  • c is the length of the hypotenuse (the side opposite the right angle, always the longest side)
  • a and b are the lengths of the other two sides (the legs)

This formula can be rearranged to find any missing side, depending on which values are known.

Finding the Hypotenuse (c)

When the lengths of both shorter sides (a and b) are known, the hypotenuse can be found using:

c = a ² + b ²

Example:

If a = 3 cm and b = 4 cm, then:

c = 3 ² + 4 ² = 9 + 16 = 25 = 5 cm

Finding a Shorter Side (a)

When the hypotenuse (c) and side b are known, side a can be found using:

a = c ² - b ²

Example:

If c = 13 cm and b = 12 cm, then:

a = 13 ² - 12 ² = 169 - 144 = 25 = 5 cm

Finding a Shorter Side (b)

When the hypotenuse (c) and side a are known, side b can be found using:

b = c ² - a ²

Example:

If c = 10 cm and a = 6 cm, then:

b = 10 ² - 6 ² = 100 - 36 = 64 = 8 cm

Summary of Formulas

Hypotenuse (c)

c = a ² + b ²

Shorter Side (a)

a = c ² - b ²

Shorter Side (b)

b = c ² - a ²

Summary: To find any side, substitute the known values into the appropriate formula and solve.