The Pythagorean theorem states that in any right triangle, the square of the hypotenuse equals the sum of the squares of the two legs: a² + b² = c², where c is the longest side opposite the right angle. For legs a=3 and b=4: c = √(9+16) = √25 = 5. It is one of the most useful formulas in geometry, applied daily in construction, navigation, engineering, and architecture to find missing distances and verify right angles.
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What Is the Pythagorean Theorem?
The Pythagorean theorem is a fundamental relationship in geometry that describes the sides of any right triangle. It states that the square of the hypotenuse (the longest side, opposite the right angle) is always equal to the sum of the squares of the other two sides, called legs. Written as a formula: a² + b² = c², where c is the hypotenuse.
This theorem works for every right triangle regardless of its size or proportions. Whether you are working with a tiny triangle on graph paper or calculating distances across a construction site, the relationship a² + b² = c² holds true as long as one angle is exactly 90 degrees.
The theorem is used to find any missing side of a right triangle when two sides are known. If you know both legs, you can find the hypotenuse. If you know the hypotenuse and one leg, you can find the other leg by rearranging the formula: a = √(c² − b²).
Real-world applications are everywhere. Carpenters use the 3-4-5 rule to check whether a corner is perfectly square. Construction workers calculate stair stringer lengths, roof rafter lengths, and diagonal brace measurements. Pilots and navigators use it to compute straight-line distances between two points. Screen sizes are defined by their diagonal measurement, calculated using the theorem from width and height. The distance formula used in coordinate geometry is a direct extension of the Pythagorean theorem.
A Pythagorean theorem calculator automates these computations — enter any two sides and it solves for the third instantly.
The Formula: Three Variations
Find the hypotenuse (c):
c = √(a² + b²)
Find leg a (given b and c):
a = √(c² − b²)
Find leg b (given a and c):
b = √(c² − a²)
Worked Examples
| Problem | Given | Formula | Answer | |---------|-------|---------|--------| | Find hypotenuse | a=5, b=12 | √(25+144) = √169 | c = 13 | | Find hypotenuse | a=8, b=15 | √(64+225) = √289 | c = 17 | | Find leg | b=9, c=15 | √(225−81) = √144 | a = 12 | | Find leg | a=7, c=25 | √(625−49) = √576 | b = 24 | | Find hypotenuse | a=1, b=1 | √(1+1) = √2 | c ≈ 1.414 |
Pythagorean Triples
Pythagorean triples are sets of three positive integers where a² + b² = c² exactly. These produce right triangles with no irrational numbers:
| Triple | Check | |--------|-------| | 3, 4, 5 | 9 + 16 = 25 ✓ | | 5, 12, 13 | 25 + 144 = 169 ✓ | | 8, 15, 17 | 64 + 225 = 289 ✓ | | 7, 24, 25 | 49 + 576 = 625 ✓ | | 9, 40, 41 | 81 + 1600 = 1681 ✓ | | 6, 8, 10 | 36 + 64 = 100 ✓ (multiple of 3-4-5) |
Any multiple of a Pythagorean triple also forms a right triangle. The 3-4-5 triple scales to 6-8-10, 9-12-15, 30-40-50, etc.
Real-World Applications
Construction — checking right angles (squaring a corner): The classic "3-4-5 method": measure 3 feet along one wall and 4 feet along the adjacent wall. If the diagonal between those marks is exactly 5 feet, the corner is perfectly square (90°). Builders use this constantly for foundations, decks, and framing.
Stair construction: A staircase with a 10-foot rise and 15-foot horizontal run: hypotenuse (stringer length) = √(100+225) = √325 = 18.03 feet.
Screen and monitor sizes: Monitor "size" is the diagonal measurement. A monitor 24 inches wide and 13.5 inches tall: diagonal = √(24² + 13.5²) = √(576+182.25) = √758.25 = 27.5 inches (a "27-inch monitor").
Navigation: A ship traveling 120 miles east then 50 miles north is: √(120² + 50²) = √(14400 + 2500) = √16900 = 130 miles from its starting point (straight-line distance).
Distance formula in coordinates: The distance between points (x₁,y₁) and (x₂,y₂) is derived from the Pythagorean theorem: d = √((x₂−x₁)² + (y₂−y₁)²).
Frequently Asked Questions
Who was Pythagoras and did he discover this theorem? Pythagoras of Samos (c. 570–495 BC) was a Greek philosopher and mathematician. While the theorem bears his name, evidence shows the relationship was known in Babylon over 1,000 years before Pythagoras — Babylonian clay tablets from ~1800 BC list Pythagorean triples. Ancient Indian texts (Sulba Sutras, 800–600 BC) also contain the theorem. Pythagoras is credited with providing the first formal proof in the Western tradition.
Can the Pythagorean theorem be used in 3D? Yes — the 3D generalization: the diagonal of a rectangular box with dimensions a, b, c is √(a² + b² + c²). The diagonal of a room 20 feet long × 15 feet wide × 10 feet tall: √(400+225+100) = √725 = 26.9 feet. This extends naturally — in n dimensions, the distance from origin to point (a₁, a₂, ..., aₙ) is √(a₁² + a₂² + ... + aₙ²).
How do I know which side is the hypotenuse? The hypotenuse is always the longest side of a right triangle, and it is always the side opposite the right angle (the 90° angle). If you don't know which angle is the right angle, the hypotenuse is the side that connects the two non-right angles. In the formula a² + b² = c², c is always the hypotenuse.
Does the Pythagorean theorem work for non-right triangles? No — the standard theorem requires a right angle. For non-right triangles, use the Law of Cosines: c² = a² + b² − 2ab·cos(C), where C is the angle between sides a and b. When C = 90°, cos(90°) = 0 and the law of cosines reduces to the Pythagorean theorem.
How many proofs exist for the Pythagorean theorem? Over 370 proofs have been documented, making it arguably the most-proved theorem in mathematics. These include algebraic proofs, geometric proofs (rearrangement, similar triangles), trigonometric proofs, and calculus-based proofs. US President James Garfield published an original proof in 1876 using a trapezoid.
Related Free Tools on RoughTools
- Quadratic Formula Calculator — solve quadratic equations (used in advanced triangle problems)
- Area Calculator — calculate area of triangles and other shapes
- Distance Calculator — find distance between coordinates
Calculate Right Triangle Sides Now
The free Pythagorean Theorem Calculator at RoughTools finds any missing side of a right triangle when two sides are known. Enter any two sides (a, b, or c) to get the third. No account needed, completely free.