Pythagorean Theorem Calculator
Find the missing side of any right triangle using a² + b² = c²
Pythagorean Theorem
What is the Pythagorean Theorem?
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.
Understanding the Triangle Parts
Hypotenuse (c)
The longest side, opposite the right angle. Always the side you're finding with √(a² + b²).
Legs (a and b)
The two sides that form the right angle. Interchangeable in the formula.
Right Angle
The 90° angle formed between the two legs. Marked with a small square.
Area
Area = ½ × a × b for a right triangle.
Common Pythagorean Triples
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 |
Real-World Applications
Construction
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.
Navigation
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.
Ladder Safety
Determine safe ladder placement. A 25-foot ladder 7 feet from the wall reaches 24 feet high (7² + 24² = 25²).
Screen Size
TV and monitor sizes are measured diagonally. A 16:9 display that's 40" wide and 22.5" tall is about 46" diagonal.
Frequently Asked Questions
What if I only know two sides?
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.
Does it work for non-right triangles?
No, the Pythagorean theorem only works for right triangles. For other triangles, use the Law of Cosines: c² = a² + b² - 2ab·cos(C).
How do I find distance between two points?
Use d = √[(x₂-x₁)² + (y₂-y₁)²]. This is the Pythagorean theorem applied to the horizontal and vertical distances between points.
What's the 3D version?
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.
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