Pythagorean Theorem
Relates the sides of a right triangle.
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Example
3² + 4² = 5²
How it works
In a right triangle, the two shorter sides (the legs) and the longest side (the hypotenuse) are linked by a precise square relationship: the sum of the squares of the legs equals the square of the hypotenuse. This works because the geometry of a right angle creates a special distance relationship, and it can be confirmed by comparing the areas of squares built on each side. It is a way to calculate an unknown side when the other two are known.
When to use it
Use this when you need to find a missing side of a right triangle, such as measuring the diagonal of a rectangle or checking whether three lengths can form a right triangle. It also appears in navigation, construction, and physics whenever perpendicular distances are involved.
Worked example, step by step
Suppose a right triangle has legs of 6 cm and 8 cm. Use the theorem: a^2 + b^2 = c^2 6^2 + 8^2 = c^2 36 + 64 = c^2 100 = c^2 c = √100 c = 10 So, the hypotenuse is 10 cm.
Common mistakes to avoid
- Putting the longest side into a or b instead of c
- Forgetting to square both legs before adding them
- Taking the square root too early or not at the end
Did You Know?
There are over 350 published proofs of the Pythagorean theorem — including one by US President James Garfield.
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