Calculate the missing side of a right triangle using the Pythagorean theorem (a² + b² = c²). Find hypotenuse, legs, area, perimeter, angles, inradius, and circumradius instantly.
The Pythagorean theorem is one of the most fundamental relationships in all of mathematics. Stated simply, for any right triangle the square of the hypotenuse equals the sum of the squares of the other two sides: a² + b² = c². This elegant formula, attributed to the ancient Greek mathematician Pythagoras (though known earlier in Babylon and India), underpins geometry, trigonometry, physics, and engineering. Our Pythagorean Theorem Calculator lets you enter any two sides of a right triangle and instantly computes the third, along with area, perimeter, both acute angles, the altitude to the hypotenuse, the inradius of the inscribed circle, and the circumradius. Eight classic Pythagorean triple presets — from the famous 3-4-5 to 28-45-53 — let you explore integer-sided right triangles with a single click. A reference table lists 16 primitive triples with their areas and perimeters, while interactive comparison bars give you an at-a-glance visual of side proportions and angle breakdown. Whether you are a student checking homework, a carpenter cutting rafters, or a developer computing distances, this tool delivers accurate results with configurable decimal precision and six unit options.
Pythagorean Theorem problems often require several dependent steps, and a small arithmetic slip can propagate through every derived value. This calculator is tailored to that workflow: you enter side a (value), side b (value), hypotenuse c (value), and it returns angle a (°), angle b (°) in one consistent pass. It is useful for homework checks, worksheet generation, tutoring walkthroughs, and fast field/design estimates where you need reliable geometry results without rebuilding the full derivation each time.
c = √(a² + b²) | a = √(c² − b²) | b = √(c² − a²) Area = ½ × a × b Perimeter = a + b + c Angle A = arctan(a / b) | Angle B = arctan(b / a) Altitude to c = (a × b) / c Inradius = Area / s (s = perimeter / 2) Circumradius = c / 2
Result: 3 and b = 4: c = √(9 + 16) = √25 = 5 Area = ½ × 3 × 4 = 6 Perimeter = 3 + 4 + 5 = 12 Angle A = arctan(3/4) ≈ 36
Given a = 3 and b = 4: c = √(9 + 16) = √25 = 5 Area = ½ × 3 × 4 = 6 Perimeter = 3 + 4 + 5 = 12 Angle A = arctan(3/4) ≈ 36.87° Angle B ≈ 53.13° Altitude to c = 6 / 5 = 2.4 Inradius = 6 / 6 = 1 Circumradius = 5 / 2 = 2.5
The Pythagorean theorem is the entry point, but a right triangle encodes far more information. Once you know two sides, every other measurement follows — the hypotenuse or missing leg, the two acute angles via arctangent, the area as half the product of the legs, and the perimeter as the sum of all three sides. The **altitude to the hypotenuse** h = a·b / c creates two smaller triangles that are each similar to the original, giving a powerful tool for proofs and for calculating the altitudes, medians, and angle bisectors of the original triangle.
The acute angles satisfy A + B = 90°, so knowing one immediately fixes the other. Angle A = arctan(a/b) and B = arctan(b/a) = 90° − A. These relationships connect trigonometry directly to the Pythagorean theorem: sin A = a/c, cos A = b/c, tan A = a/b, and the identity sin²A + cos²A = 1 is just the Pythagorean theorem rewritten in trigonometric notation.
The calculator supports three solve modes: **find hypotenuse** (given legs a and b), **find leg a** (given leg b and hypotenuse c), and **find leg b** (given a and c). In find-leg mode the formula rearranges to a = √(c² − b²), which requires c > b for a valid triangle. Entering equal values for a leg and the hypotenuse (c = b) would yield a degenerate triangle with zero area — the calculator flags this.
Right triangle calculations appear in every construction project: checking that a wall is plumb, setting out a building's footprint, cutting roof rafters to the correct length and angle. A **3-4-5 triangle** or any Pythagorean triple lets builders verify square corners without a protractor — if the measured diagonal equals c, the angle is exactly 90°. In **navigation**, the straight-line distance between two map coordinates is the hypotenuse of a right triangle whose legs are the north-south and east-west displacement. In **electrical engineering**, the magnitude of a complex impedance Z = √(R² + X²) is a direct application of the theorem.
It states that in a right triangle, the square of the hypotenuse (c) equals the sum of the squares of the two legs (a and b): a² + b² = c². Use this as a practical reminder before finalizing the result.
No. For non-right triangles you need the Law of Cosines: c² = a² + b² − 2ab·cos(C), which generalises the Pythagorean theorem.
A set of three positive integers (a, b, c) such that a² + b² = c². The smallest is (3, 4, 5). Primitive triples have no common factor greater than 1.
Infinitely many. They can be generated with the formula a = m²−n², b = 2mn, c = m²+n² for any integers m > n > 0.
The inradius r = (a + b − c) / 2, which also equals the area divided by the semi-perimeter. It is the radius of the largest circle that fits inside the triangle.
Angles alone don't determine the triangle's size — you need at least one side length. Combine one side with trigonometric ratios (sin, cos, tan) to find the remaining sides.
Absolutely — in construction (checking right angles), navigation (distance between points), physics (vector magnitudes), computer graphics, and much more. Keep this note short and outcome-focused for reuse.