Pythagoras Equation Solver
Calculate missing sides, angles, exact radicals, area, and altitude with real-time dynamic geometry diagrams and full step-by-step mathematical proofs.
Enter Any Two Known Values
Leave unknown sides empty. The solver calculates everything in real-time.
Optional: Or solve with Angle α or Area
Full Step-by-Step Mathematical Derivation
Clear proofs and exact radical conversions suitable for homework and exams.
1. Identify the Pythagorean Theorem Formula
Formula: c² = a² + b² ⇒ c = √(a² + b²)
Substitution: c = √(3² + 4²)
Computation: c = √(9 + 16) = √(25)
Note: The hypotenuse c is the longest side, directly opposite the 90° right angle.
Calculate Acute Angles α and β
Formula: α = arcsin(a / c), β = 90° - α
Substitution: α = arcsin(3 / 5)
Computation: α = arcsin(0.6)
Calculate Area & Perimeter
Formula: Area = ½ · a · b, Perimeter = a + b + c
Substitution: Area = ½ · 3 · 4, P = 3 + 4 + 5
Computation: Area = 6 cm²
What is the Pythagoras Equation?
The Pythagoras Equation (commonly known as the Pythagorean Theorem) is a cornerstone of Euclidean geometry establishing a fundamental relationship between the three sides of any right-angled triangle. It states that:
Where a and b are the lengths of the two legs adjacent to the 90° right angle, and c is the length of the hypotenuse directly opposite the right angle.
Geometrically, this implies that if you construct squares on each of the three sides of a right triangle, the area of the square built on the hypotenuse is exactly equal to the combined areas of the squares built on the other two legs.
The 3 Primary Pythagoras Rearrangements
Used when both perpendicular legs (a and b) are known. Always results in the longest side of the triangle.
Used when hypotenuse c and base leg b are known. Requires c > b.
Used when hypotenuse c and vertical leg a are known. Requires c > a.
Interactive Geometric Rearrangement Proof
Visual proof showing that hypotenuse square c² equals leg squares a² + b².
Arrangement 1: The Hypotenuse Square (c²)
Total Square Area = (a + b)² = (3 + 4)² = 49
The 4 corner triangles consume 4 × (½ · 3 · 4) = 24. The inner tilted square has area c² = 25.
Conclusion: Unshaded Area 1 = Unshaded Area 2 ⇒ c² = a² + b²
Secondary Geometric Calculations for Right Triangles
Once the three side lengths are established via Pythagoras, our solver instantly computes every associated trigonometric and Euclidean metric:
Half the product of the two perpendicular legs.
The total boundary distance surrounding the triangle.
The perpendicular distance dropped from vertex C to side c.
The radius of the largest circle that fits completely inside.
Special Right Triangles Ratios
Standardized geometry exams frequently leverage special angle relationships that produce exact radical side ratios:
| Triangle Type | Internal Angles | Side Length Ratio | Hypotenuse Formula |
|---|---|---|---|
| Isosceles Right | 45° - 45° - 90° | 1 : 1 : √2 | c = leg · √2 |
| 30-60-90 Right | 30° - 60° - 90° | 1 : √3 : 2 | c = 2 · (short leg) |
Educational Guides & Proofs
View All Guides4 Beautiful Proofs of the Pythagorean Theorem
Discover visual rearrangement, Garfield's presidential trapezoid proof, Einstein's similar triangles, and Euclid's Windmill.
Number TheoryThe Complete Guide to Pythagorean Triples
Master primitive integer right triangles (3-4-5, 5-12-13) and Euclid's generative formula a = m² - n², b = 2mn, c = m² + n².
Frequently Asked Questions About the Pythagorean Theorem
c = √(a² + b²). When solving for a missing leg, the formula is a = √(c² - b²) or b = √(c² - a²).