Pythagorean Theorem Calculator

Euclidean Geometry & Right Triangle Trigonometry

Attributed to the ancient Greek philosopher Pythagoras of Samos (~570–495 BC), the Pythagorean Theorem establishes that in any right-angled triangle, the area of the square whose side is the hypotenuse equals the sum of the areas of the squares on the other two sides.

Core Geometric & Trigonometric Equations

• Pythagorean Theorem: a^2 + b^2 = c^2

• Hypotenuse: c = √(a^2 + b^2)

• Missing Leg: a = √(c^2 - b^2) or b = √(c^2 - a^2)

• Triangle Area: Area = (1) / (2) × a × b

• Triangle Perimeter: P = a + b + c

• Acute Angles: α = ((a) / (c)) = ((a) / (b)), β = 90^ - α

• Altitude to Hypotenuse: h = (a × b) / (c)

• Inradius: r_in = (a + b - c) / (2)

• Circumradius: R = (c) / (2)

Famous Primitive Pythagorean Triples

Integer sets (a, b, c) satisfying a^2 + b^2 = c^2 generated via Euclid's formula:

• (3, 4, 5): 3^2 + 4^2 = 9 + 16 = 25 = 5^2

• (5, 12, 13): 25 + 144 = 169 = 13^2

• (8, 15, 17): 64 + 225 = 289 = 17^2

• (7, 24, 25): 49 + 576 = 625 = 25^2

• (9, 40, 41): 81 + 1600 = 1681 = 41^

2.

How to Use This Calculator

Enter your Solve For — choose from Hypotenuse c: c = √(a² + b²), Leg a: a = √(c² - b²), Leg b: b = √(c² - a²), Leg a Length (e.g. 3.0), Leg b Length (e.g. 4.0), and Hypotenuse c Length (e.g. 5.0). Results update instantly as you type — adjust any value to compare scenarios.

The Formula

Altitude h = (a * b) / c

Worked Example

Framing a right-angle deck corner with legs 3 feet and 4 feet: The diagonal hypotenuse measures exactly 5 feet (the classic 3-4-5 rule), verifying a true 90-degree square corner.

Understanding Your Result

Evaluate your result in the context of your specific use case and relevant industry standards or reference values. A single calculated result is a starting point — test two to three scenarios to understand how sensitive the outcome is to your key input assumptions. For high-stakes decisions, verify your result against a secondary source or subject-matter expert before acting.