Pythagorean Theorem Calculator

Frequently Asked Questions

What is the Pythagorean theorem?

For a right triangle, a² + b² = c², where c is the hypotenuse (the side opposite the right angle) and a and b are the legs. Rearranged: c = √(a²+b²), or a leg = √(c²−a²).

Can I find a side if I only know one side?

No - you need any two of the three sides. With two sides this tool finds the third, then derives the perimeter, area (½·a·b), and both non-right angles via inverse tangent.

How do I know which side is the hypotenuse?

The hypotenuse is always the longest side and is opposite the 90° angle. If you enter two legs, c is computed as the largest value; if you enter a leg and the hypotenuse, the hypotenuse must exceed the leg or the triangle is impossible.

What is a Pythagorean triple?

A set of three whole numbers satisfying a²+b²=c², such as 3-4-5, 5-12-13, and 8-15-17. They appear constantly in construction and geometry because all three sides are exact integers.

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How This Calculator Works

For any right triangle with legs a and b and hypotenuse c (the side opposite the 90° angle), the Pythagorean theorem states a² + b² = c². The geometric proof: build a square on each side; the two leg-squares together tile exactly into the hypotenuse-square - area a² plus area b² equals area c². Rearranging gives three solving formulas: c = √(a² + b²) for the hypotenuse, and a = √(c² − b²) or b = √(c² − a²) for a missing leg. Enter exactly two of the three sides and leave the unknown blank; the calculator solves for the third, then derives the perimeter (a + b + c), the area ½ab (the legs are perpendicular, so they serve as base and height), and the two acute angles via the arctangent of the opposite-over-adjacent ratio.

Worked Example: the 3-4-5 Triangle

Enter a = 3, b = 4, leave c blank. Then c = √(3² + 4²) = √(9 + 16) = √25 = 5 exactly - a rare clean integer. Perimeter = 3 + 4 + 5 = 12. Area = ½ × 3 × 4 = 6 square units. The angle opposite leg a is arctan(3 ÷ 4) ≈ 36.87°; the angle opposite leg b is arctan(4 ÷ 3) ≈ 53.13°. Check: 36.87 + 53.13 + 90 = 180°, the angle sum every triangle must obey. Now reverse it: enter a = 3 and c = 5, blank b. The calculator computes b = √(5² − 3²) = √(25 − 9) = √16 = 4, recovering the same triangle from the other direction.

Where It Is Used

Construction: the 3-4-5 rule squares foundation corners and door frames without a protractor - measure 3 ft along one wall, 4 ft along the other; if the diagonal is exactly 5 ft, the corner is true 90°.

Navigation & GPS: straight-line ("as the crow flies") distance between two grid coordinates is a direct application.

Screens & displays: a 16:9 TV's advertised diagonal is √(width² + height²).

Physics: the magnitude of a vector, or the resultant of two perpendicular forces, is √(Fₓ² + Fₕ²).

School: it is the bridge from geometry to the coordinate distance formula and to trigonometry.

Common Mistakes & Edge Cases

Treating a leg as the hypotenuse. c is always the longest side and must exceed both legs; if you enter c ≤ a leg, no real triangle exists and the calculator reports an error rather than returning √ of a negative.

Forgetting to take the square root after summing the squares - a² + b² gives c², not c.

Applying the theorem to a non-right triangle. It holds only when one angle is exactly 90°; otherwise use the Law of Cosines.

Mixing units - convert feet and inches to a single unit before squaring, or the areas will not match.

Assuming sides must be integers. Most real right triangles are irrational, e.g. legs 1 and 1 give c = √2 ≈ 1.41421.

Related Concepts & Limitations

Integer triples like (3,4,5), (5,12,13), (8,15,17), and (7,24,25) are Pythagorean triples; any multiple of one (6,8,10) is also a triple. The converse is equally useful: if a² + b² = c² for a triangle's sides, that triangle must be right-angled - this is how surveyors confirm a right angle. The general case for any triangle is the Law of Cosines, c² = a² + b² − 2ab·cos(C), which collapses to Pythagoras when C = 90° because cos(90°) = 0. Limitation: this calculator assumes a flat (Euclidean) plane; on a sphere or curved surface the theorem does not hold, and great-circle formulas are required instead.

Pythagorean Theorem Calculator

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