Example 1: Finding the hypotenuse
\(a = 3, b = 4 \rightarrow c^2 = 9 + 16 = 25 \rightarrow c = 5\)
Pythagorean Theorem
Find missing side lengths in right triangles using the Pythagorean theorem
Explore the right triangle
Drag the sliders to change the legs and watch the hypotenuse update live. Pick a preset to jump to a famous Pythagorean triple.
Quick presets
3.00
1.0020.00
4.00
1.0020.00
Predict what will happen
If you double leg a while keeping leg b unchanged, what happens to the hypotenuse?
Try moving the a-slider from 3 to 6 and watching c.
Notice
The hypotenuse is always the longest side. Whichever leg you grow, c grows with it — but more slowly because of the square root.
Common mistake
Squaring the sum is not the sum of squares: (a + b)² ≠ a² + b². The Pythagorean theorem adds the squares first, then takes the root.
Why it works
Picture a square built on each side of the right triangle. The areas of the two leg-squares add up exactly to the area of the hypotenuse-square — that is what a² + b² = c² is saying.
Concept check
In a right triangle, which side is always the longest?
Results
Final Answer
The hypotenuse is \(c = 5.00\)
Step-by-step Solution
- The Pythagorean theorem states: \(a^2 + b^2 = c^2\)
- Given: \(a = 3.00\) and \(b = 4.00\)
- Substitute into the formula: \(3.00^2 + 4.00^2 = c^2\)
- Calculate: \(9.00 + 16.00 = c^2\)
- Therefore: \(c^2 = 25.00\)
- Taking the square root: \(c = \sqrt{25.00} = 5.00\)
Theory & Formula
The Pythagorean theorem is a fundamental principle in geometry that relates the sides of a right triangle. It states that the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.
This theorem only applies to right triangles (triangles with one 90-degree angle). The hypotenuse is always the longest side and is opposite the right angle.
Common Pythagorean Triples
- 3, 4, 5
- 5, 12, 13
- 8, 15, 17
- 7, 24, 25
- 9, 40, 41
\(a^2 + b^2 = c^2\)
Worked Examples
Example 2: Finding a leg
\(b = 8, c = 10 \rightarrow a^2 = 100 - 64 = 36 \rightarrow a = 6\)
External educational resource
Construct your own in GeoGebra
Open the GeoGebra geometry tool to drag points, build right triangles, and verify the theorem visually.
GeoGebra (geogebra.org)