Pythagorean theorem calculator
Scientific CalculatorsInstantly find the hypotenuse, a missing leg, or verify a perfect square corner with exact radicals and step-by-step proofs.
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How to Use the Pythagorean Theorem Calculator (Step-by-Step)
Solve right triangles, calculate hypotenuse or missing legs, simplify radicals, and verify 90° construction corners.
Select Math Student or Construction Framing Mode
Choose 'Math Student' for classic geometric notation (leg a, leg b, hypotenuse c) or 'Construction Framing' for carpentry and building layout (Wall 1, Wall 2, Diagonal).
Input Any Two Known Side Lengths
Type known measurements into any two fields (both legs, or one leg and the hypotenuse). The smart solver automatically locks and solves for the third side.
Instant Bidirectional Auto-Calculation
The engine solves c = √(a² + b²) for the hypotenuse, or a = √(c² - b²) for a missing leg immediately as you type, enforcing the geometric condition that c > a and c > b.
Inspect Exact Radical Form & Decimals
Review both high-precision decimal values and simplified exact radical notation (e.g., √20 simplifies automatically to 2√5) without manual factoring.
Verify Pythagorean Triples & Right Angles
Check the automatic badge confirming whether your sides form an integer Pythagorean Triple (e.g., 3-4-5, 5-12-13) and verify perfect 90° perpendicular alignment.
Copy Step-by-Step Homework Proof
Click 'Copy Homework' to copy formatted step-by-step algebraic substitutions, square root derivations, and exact values directly to your clipboard for school or reports.
Frequently Asked Questions (FAQ)
Right triangle geometry, formulas, integer triples, construction square checks, and radical simplification.
QWhat is the Pythagorean Theorem and what is its algebraic formula?
The Pythagorean Theorem states that in any right-angled triangle, the square of the length of the hypotenuse (the side opposite the 90° right angle) equals the sum of the squares of the lengths of the two legs: a² + b² = c². Consequently, c = √(a² + b²).
QDoes the Pythagorean Theorem work for acute or obtuse triangles?
No. The theorem applies strictly to right triangles containing an exact 90° angle. For non-right (oblique) triangles, you must use the Law of Cosines: c² = a² + b² - 2ab·cos(C), which reduces to the Pythagorean Theorem when angle C = 90° because cos(90°) = 0.
QHow do you calculate a missing leg when given the hypotenuse and one leg?
Rearrange the Pythagorean equation: a² = c² - b², which yields a = √(c² - b²), or b = √(c² - a²). Note that the hypotenuse c must always be strictly greater than either leg (c > a and c > b); otherwise, no real triangle can be formed.
QWhat is a Pythagorean Triple and what are the most common examples?
A Pythagorean Triple consists of three positive integers (a, b, c) that satisfy a² + b² = c². The most famous primitive triple is (3, 4, 5) because 3² + 4² = 9 + 16 = 25 = 5². Other common triples include (5, 12, 13), (8, 15, 17), (7, 24, 25), and (9, 40, 41), as well as any scalar multiples such as (6, 8, 10).
QHow is the 3-4-5 rule used in construction and carpentry to square a corner?
Builders verify 90° corners using the 3-4-5 ratio. Measure 3 units along one wall/string and 4 units along the adjacent wall from the corner. If the diagonal distance between the two marks measures exactly 5 units (or 6-8-10, 9-12-15), the corner is confirmed to be a perfect 90° right angle.
QWhat is the difference between an exact radical and a decimal approximation?
An exact radical expresses roots in simplest mathematical terms without rounding errors (for example, c = √50 simplifies to 5√2). A decimal approximation rounds this value (e.g., 7.0711). Exact radicals are preferred in pure mathematics, while decimals are standard in engineering and trade.
QHow is the Pythagorean Theorem related to the Euclidean Distance Formula?
The Cartesian distance formula between two points (x₁, y₁) and (x₂, y₂) is d = √[(x₂ - x₁)² + (y₂ - y₁)²]. This formula is a direct application of the Pythagorean Theorem, where the horizontal change Δx and vertical change Δy serve as legs, and distance d is the hypotenuse.
QCan the hypotenuse ever be shorter than or equal to a leg in a real triangle?
No. In Euclidean geometry, the hypotenuse is opposite the largest angle (90°) in a right triangle, so it must always be the longest side. If c ≤ a or c ≤ b, the radical √(c² - a²) results in zero or an imaginary number, meaning no valid geometric triangle exists.
- c (Hypotenuse): The longest side (opposite the right angle).
- a, b (Legs): The two shorter sides (a and b).