03 // Bidirectional STEM Calculators // MATHc = √(a² + b²)

Pythagorean Theorem Right Triangle Calculator

In Euclidean geometry, the Pythagorean Theorem states that the square of the hypotenuse c opposite the 90° angle equals the sum of the squares of the two orthogonal legs a and b: c = √(a² + b²).

c = √(a² + b²)

Pick which value you want to calculate (or type into any field):Solving for: c — Hypotenuse (Longest Side)

Hypotenuse (Longest Side)Calculated

Side opposite the 90° right angle

0.110,000.00

Orthogonal Leg A (Altitude)

First perpendicular side of the right triangle

0.110,000.00

Orthogonal Leg B (Base)

Second perpendicular side of the right triangle

0.110,000.00

Calculated Result · Hypotenuse (Longest Side) (c)

13m

Converted units:

Step-by-Step Calculation

01Canonical Law: c = √(a² + b²)

02Given Inputs (Base Units): a = 5 m, b = 12 m

03Solve for c (Hypotenuse (Longest Side)): c = 13 m

In Euclidean geometry, the Pythagorean Theorem states that the square of the hypotenuse c opposite the 90° angle equals the sum of the squares of the two orthogonal legs a and b: c = √(a² + b²).

Real-World Reference Values

3-4-5 Carpenter's Triangle3² + 4² = 5² (9 + 16 = 25)

Smallest primitive Pythagorean triple for squaring foundations

5-12-13 & 8-15-17 Triples25 + 144 = 169 / 64 + 225 = 289

Exact integer right triangles

Isosceles Right (45°-45°-90°)c = a × √2 ≈ 1.41421 × a

Square diagonal length

20-Row Comparison Table

Hypotenuse (Longest Side) (c) · Orthogonal Leg A (Altitude) (a)

Compare across:

Pythagorean Theorem Right Triangle Calculator — Orthogonal Leg A (Altitude) → Hypotenuse (Longest Side)
#	Orthogonal Leg A (Altitude) (m)	Hypotenuse (Longest Side) (m)	Δ %
01	5 mCurrent	13 m	0.00%
02	72.7256 m	73.709 m	+466.99%
03	145.3512 m	145.8457 m	+1021.89%
04	217.9768 m	218.3068 m	+1579.28%
05	290.6024 m	290.85 m	+2137.31%
06	363.228 m	363.4261 m	+2695.59%
07	435.8536 m	436.0187 m	+3253.99%
08	508.4791 m	508.6207 m	+3812.47%
09	581.1047 m	581.2286 m	+4370.99%
10	653.7303 m	653.8405 m	+4929.54%
11	726.3559 m	726.455 m	+5488.12%
12	798.9815 m	799.0716 m	+6046.70%
13	871.6071 m	871.6897 m	+6605.31%
14	944.2327 m	944.3089 m	+7163.91%
15	1,016.86 m	1,016.93 m	+7722.53%
16	1,089.48 m	1,089.55 m	+8281.15%
17	1,162.11 m	1,162.17 m	+8839.78%
18	1,234.74 m	1,234.79 m	+9398.41%
19	1,307.36 m	1,307.42 m	+9957.04%
20	1,379.99 m	1,380.04 m	+10515.68%

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Frequently Asked Questions

How do I solve for a missing leg a or b when the hypotenuse c is known?+

Click SOLVE FOR on a (or b) to evaluate the algebraic inverse a = √(c² - b²).

Why must the hypotenuse c always be larger than legs a and b?+

Because c sits opposite the largest angle (90°), c² = a² + b² guarantees c > a and c > b for all non-degenerate right triangles.

What if the triangle does not have a 90° angle?+

Use the Law of Cosines calculator (c² = a² + b² - 2ab cos C), which generalizes Pythagoras to any oblique angle C.