Pythagorean Theorem Calculator
Find the missing side of a right triangle using a² + b² = c².
Enter any two sides. Leave the unknown blank.
All calculations run live in your browser. Angles computed via inverse trig (arcsin, arccos).
What Is the Pythagorean Theorem Calculator?
The Pythagorean Theorem is one of the most fundamental results in mathematics. In any right triangle, the square of the hypotenuse equals the sum of the squares of the other two sides. This calculator handles three tasks: finding a missing side, checking if three lengths form a right triangle, and computing angles using inverse trigonometry.
- ›Enter any two sides and leave the unknown blank, the calculator finds the third
- ›Use "Check Right Triangle" mode to verify three given sides
- ›Use "Find Angles" to compute both non-right angles from leg a and hypotenuse c
- ›Outputs include area, perimeter, and both acute angles in degrees
Formula
a² + b² = c²
Find hypotenuse c
c = √(a² + b²)
Find leg a
a = √(c² − b²)
Find leg b
b = √(c² − a²)
Where:
- ›a, b, the two legs (shorter sides) of the right triangle
- ›c, the hypotenuse (longest side, opposite the 90° angle)
- ›Area = ½ × a × b, triangle area from the two legs
How to Use
- 1Choose a mode: "Find a Side", "Check Right Triangle", or "Find Angles"
- 2In "Find a Side" mode, enter any two of the three sides (a, b, c) and leave the unknown blank
- 3Click Calculate, the missing side appears along with area, perimeter, and angles
- 4In "Check" mode, enter all three sides to determine right / acute / obtuse
- 5In "Find Angles" mode, enter leg a and hypotenuse c to get both angles and missing leg b
- 6Press Enter in any field as a keyboard shortcut
Example Calculation
Classic 3-4-5 right triangle:
c = √(3² + 4²) = √(9 + 16) = √25 = 5
Area = ½ × 3 × 4 = 6 square units
Perimeter = 3 + 4 + 5 = 12 units
Angle A = arcsin(3/5) ≈ 36.87°
Angle B = 90° − 36.87° ≈ 53.13°
5-12-13 Triple
a = 5, b = 12: c = √(25 + 144) = √169 = 13. Every integer multiple of a Pythagorean triple is also a triple, so (10, 24, 26), (15, 36, 39) etc. all work too.
Understanding Pythagorean Theorem
The Pythagorean Theorem: History and Significance
Although attributed to Pythagoras (c. 570–495 BC), the relationship was known centuries earlier by Babylonian and Indian mathematicians. More than 370 distinct proofs have been documented, including one by US President James Garfield in 1876. It is the most-proved theorem in mathematics.
- ›Euclid's proof (Book I, Proposition 47) uses area dissection
- ›Bhaskara's elegant algebraic proof dates to ~1150 AD
- ›The theorem extends to any dimension: d = √(x₁² + x₂² + … + xₙ²)
- ›It underlies the Euclidean distance metric used throughout science
Common Pythagorean Triples
| a | b | c | Type | Check |
|---|---|---|---|---|
| 3 | 4 | 5 | Primitive | 9+16=25 |
| 5 | 12 | 13 | Primitive | 25+144=169 |
| 8 | 15 | 17 | Primitive | 64+225=289 |
| 7 | 24 | 25 | Primitive | 49+576=625 |
| 6 | 8 | 10 | ×2 of 3-4-5 | 36+64=100 |
| 9 | 12 | 15 | ×3 of 3-4-5 | 81+144=225 |
Frequently Asked Questions
What is the hypotenuse?
The hypotenuse is the longest side of a right triangle, always opposite the 90° right angle. It is labeled c in the formula a² + b² = c² and is always longer than either leg.
- ›It cannot be shorter than either leg
- ›For the 3-4-5 triple, c = 5 is the hypotenuse
- ›In a 45-45-90 triangle, c = a√2 (roughly 1.41× a leg)
- ›In a 30-60-90 triangle, c = 2 × shorter leg
What are Pythagorean triples?
Pythagorean triples are sets of three positive integers (a, b, c) satisfying a² + b² = c². They are useful in construction and navigation because they produce exact right angles without irrational numbers.
- ›(3, 4, 5), the most famous triple, used by ancient Egyptians
- ›(5, 12, 13), second-simplest primitive triple
- ›(8, 15, 17), common in geometry problems
- ›(7, 24, 25), generated by Euclid's formula with m=4, n=3
- ›Any multiple of a triple is also a triple: (6, 8, 10), (9, 12, 15)…
Does the theorem work for non-right triangles?
The Pythagorean Theorem applies exclusively to right triangles (those with a 90° angle). For arbitrary triangles, you need the Law of Cosines, which generalises it.
- ›Acute triangle (all angles < 90°): a² + b² > c²
- ›Right triangle (one angle = 90°): a² + b² = c²
- ›Obtuse triangle (one angle > 90°): a² + b² < c²
- ›Law of Cosines: c² = a² + b² − 2ab·cos(C), works for any triangle
Can sides be decimals or fractions?
Absolutely. The theorem holds for any positive real numbers, not just integers. For example, a right triangle with legs 1 and 1 has a hypotenuse of √2 ≈ 1.41421…
- ›Decimal sides: a = 1.5, b = 2.0 → c = 2.5
- ›Irrational sides are fine, e.g. legs √2 and √3 → hypotenuse √5
- ›The calculator accepts any positive decimal input
How do I find the angles of a right triangle?
Given the sides of a right triangle, you can find the angles using inverse trigonometric functions. The calculator's "Find Angles" mode computes them automatically.
- ›Angle A = arcsin(opposite/hypotenuse) = arcsin(a/c)
- ›Angle B = arcsin(b/c), or simply 90° − Angle A
- ›The right angle is always 90°; the three angles sum to 180°
- ›Enter leg a and hypotenuse c to get all angles and the missing leg b
What is a 45-45-90 triangle?
A 45-45-90 (isosceles right) triangle has two equal legs and a hypotenuse that is √2 ≈ 1.414 times longer. It is one of the two "special right triangles" memorised in geometry.
- ›Sides are in the ratio 1 : 1 : √2
- ›If each leg = 5, hypotenuse = 5√2 ≈ 7.071
- ›Produced by cutting a square diagonally
- ›The other special triangle is 30-60-90 with sides 1 : √3 : 2
How is this used in real life?
The theorem has countless practical applications across many fields. It is one of the most-used geometric results in engineering and technology.
- ›Construction: the "3-4-5 rule" ensures corners are perfectly square
- ›GPS and mapping: distance = √(Δx² + Δy²) in 2D; √(Δx² + Δy² + Δz²) in 3D
- ›Computer graphics: pixel distances, collision detection, ray casting
- ›Navigation: shortest path (straight-line distance) between two points
- ›Physics: magnitude of a vector from its components