JENECONKMaths Universe

Finding the Hypotenuse

Ages 11–15

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Finding the Hypotenuse

In a right-angled triangle, the longest side (opposite the right angle) is called the hypotenuse. The Pythagorean theorem says: a² + b² = c², where c is the hypotenuse.

Your goals

  • ✓You can use a² + b² = c² to find the longest side of a right triangle.
  • ✓You can use hypotenuse, Pythagorean theorem correctly.
  • ✓You can label the hypotenuse before substituting values.

Words to know

hypotenuse:
The side opposite the right angle; it is the longest side of a right-angled triangle.
Pythagorean theorem:
For a right-angled triangle, a² + b² = c², where c is the hypotenuse.

Examples

  • 1If a = 3 and b = 4: c² = 9 + 16 = 25, so c = 5.
  • 2If a = 5 and b = 12: c² = 25 + 144 = 169, so c = 13.

More than one way to solve it

Try each method, then choose the one you can explain and check most clearly.

Method 1

Pythagorean equation

Best for: Exact calculation
  1. 1.Label the shorter sides a and b.
  2. 2.Calculate a² + b².
  3. 3.Take the square root to find c.

Example: For sides 6 and 8: c = √(36 + 64) = √100 = 10.

Method 2

Known triples and scaling

Best for: Fast mental recognition
  1. 1.Look for a known triple such as 3-4-5.
  2. 2.Find the scale factor.
  3. 3.Scale all three sides equally.

Example: 6-8-10 is the 3-4-5 triple multiplied by 2.

Check these

  • • Do not use the theorem on a triangle that is not right-angled.
  • • Keep every number, symbol, label, and unit in its correct place.
  • • Choose the method that makes each step easiest to explain and check.

Helpful habits

  • • Read the whole question before choosing what to do.
  • • Show one clear step at a time and check each step.
  • • If you are stuck, draw a model, try smaller numbers, or revisit the example.

Now try your own

Create your own finding the hypotenuse example, solve it, and explain how you checked it.

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