Find the missing side of any right triangle using a² + b² = c²
The Pythagorean theorem states that in a right triangle, the square of the hypotenuse (the side opposite the right angle) equals the sum of squares of the other two sides. Expressed as a² + b² = c², it's one of the most fundamental relationships in mathematics.
Named after the ancient Greek mathematician Pythagoras (c. 570-495 BC), though the relationship was known to Babylonians and Indians centuries earlier. The theorem only applies to right triangles - triangles with one 90° angle.
This theorem is the foundation of distance calculations, from measuring diagonals in construction to calculating distances in coordinate geometry. It's essential in trigonometry, physics, engineering, navigation, and computer graphics.
The longest side, opposite the right angle. Always the side you're finding with √(a² + b²).
The two sides that form the right angle. Interchangeable in the formula.
The 90° angle formed between the two legs. Marked with a small square.
Area = ½ × a × b for a right triangle.
Pythagorean triples are sets of three integers that satisfy a² + b² = c²:
| a | b | c | Multiples |
|---|---|---|---|
| 3 | 4 | 5 | 6-8-10, 9-12-15, 12-16-20 |
| 5 | 12 | 13 | 10-24-26, 15-36-39 |
| 8 | 15 | 17 | 16-30-34 |
| 7 | 24 | 25 | 14-48-50 |
| 20 | 21 | 29 | 40-42-58 |
| 9 | 40 | 41 | 18-80-82 |
Check if corners are square using the 3-4-5 rule. Measure 3 feet on one side, 4 feet on the other - the diagonal should be exactly 5 feet for a right angle.
Calculate straight-line distance between two points. If you go 3 miles east and 4 miles north, you're 5 miles from your starting point.
Determine safe ladder placement. A 25-foot ladder 7 feet from the wall reaches 24 feet high (7² + 24² = 25²).
TV and monitor sizes are measured diagonally. A 16:9 display that's 40" wide and 22.5" tall is about 46" diagonal.
A right triangle has legs of 3 and 4 units. Find the length of the hypotenuse.
Resultc = 5
Apply a² + b² = c². Compute 3² + 4² = 9 + 16 = 25. Take the square root: c = √25 = 5. This is the smallest primitive Pythagorean triple. The same proportion scales to any wall, screen, or layout where one side is 8 ft and the other is 6 ft - the diagonal will be 10 ft.
You can always find the third side. If you know the two legs, use c = √(a² + b²). If you know the hypotenuse and one leg, use a = √(c² - b²). The hypotenuse must always be the longest side.
No, the Pythagorean theorem only works for right triangles. For other triangles, use the Law of Cosines: c² = a² + b² - 2ab·cos(C).
Use d = √[(x₂-x₁)² + (y₂-y₁)²]. This is the Pythagorean theorem applied to the horizontal and vertical distances between points.
In 3D, the diagonal of a box is d = √(a² + b² + c²), where a, b, c are the three dimensions. It's an extension of the 2D theorem. For a unit cube where a = b = c = 1, the space diagonal is √3 ≈ 1.732.
The four most useful primitive triples are 3-4-5, 5-12-13, 8-15-17, and 7-24-25. Any multiple of a triple is also a triple (so 6-8-10, 9-12-15, and 12-16-20 all work). Carpenters use 3-4-5 to square corners on site without a protractor.
The converse states that if a² + b² = c² holds for a triangle's three sides (with c the longest), then the triangle has a right angle opposite c. Plug the three side lengths in and check equality - this is how you confirm a corner is truly 90° from measurements alone.
In any right triangle, sin(θ) = opposite/hypotenuse and cos(θ) = adjacent/hypotenuse. Squaring and adding gives sin²(θ) + cos²(θ) = (a² + b²)/c² = 1, the fundamental Pythagorean identity in trigonometry. The theorem underlies every unit-circle relationship.
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