How many inputs do I need?

Provide any two independent values, at least one of which must be a side. The right angle is assumed at C (90°).

Can I enter two angles only?

No. Two angles alone determine only the shape, not the size. Enter at least one side with an angle.

What units does the calculator use?

You may use any length units (e.g., m, cm, in). The outputs use the same units as your inputs.

What accuracy do you use?

Results are shown with up to 6 significant decimals. Calculations use double-precision floating point.

What happens if I enter inconsistent values?

The tool validates inputs (e.g., c must be the largest side; angles must be between 0° and 90° and sum to 90°) and will show constructive error messages if values conflict.

Is the math authoritative?

Yes. It is based on Euclid’s Elements (Pythagorean theorem) and standard trigonometric identities as catalogued by the NIST Digital Library of Mathematical Functions.

Right Triangle Calculator

Compute every property of a right triangle from any two independent values. Ideal for students, educators, and professionals who need fast, accurate, and accessible results grounded in authoritative mathematics.

Enter any two values

Provide exactly two fields. At least one must be a side (a, b, or c). Angles are degrees.

Leg a (opposite angle A)

Leg b (opposite angle B)

Hypotenuse c

Angle A (degrees)

Angle B (degrees)

Fields filled: 0/2. Ready when it reaches 2.

Results

Enter two values and press Compute. Results update instantly when valid.

Leg a

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Leg b

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Hypotenuse c

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Angle A (°)

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Angle B (°)

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Area

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Perimeter

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Altitude to hypotenuse h

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Inradius r

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Circumradius R

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Authoritative Content Ecosystem

Data Source and Methodology

Authoritative Source: NIST Digital Library of Mathematical Functions (DLMF), Chapter 4: Elementary Functions, Release 1.1.10 (2023-12-15). Direct link: https://dlmf.nist.gov/4. Complementary reference: Euclid’s Elements, Book I, Proposition 47 (Pythagorean Theorem) via Perseus Digital Library: I.47.

Tutti i calcoli si basano rigorosamente sulle formule e sui dati forniti da questa fonte.

The Formulas Explained

Right angle at C, sides: a opposite A, b opposite B, c opposite C (hypotenuse).

1) Pythagorean theorem: \\(c = \sqrt{a^2 + b^2}\\)

2) Trigonometry: \\(\sin A = \dfrac{a}{c},\ \cos A = \dfrac{b}{c},\ \tan A = \dfrac{a}{b}\\)

3) Angle sum: \\(A + B = 90^\circ\\)

4) Given any two independent values, solve others by algebraic rearrangement, e.g.:

\\(b = \sqrt{c^2 - a^2}\\), when \\(a\\) and \\(c\\) are known.
\\(a = c\sin A,\quad b = c\cos A\\)
\\(b = a\tan B,\quad c = \dfrac{a}{\cos B}\\)

5) Area and perimeter: \\(\text{Area}=\dfrac{ab}{2},\quad P=a+b+c\\)

6) Other properties: \\(h=\dfrac{ab}{c}\\) (altitude to hypotenuse), \\(r=\dfrac{a+b-c}{2}\\) (inradius), \\(R=\dfrac{c}{2}\\) (circumradius)

Glossary of Variables

a: leg opposite angle A (length > 0)

b: leg opposite angle B (length > 0)

c: hypotenuse opposite the right angle (largest side)

A: acute angle opposite a (0° < A < 90°)

B: acute angle opposite b (0° < B < 90°, A + B = 90°)

Area: triangle area = ab/2

Perimeter: sum of all sides = a + b + c

h: altitude from right angle to hypotenuse = ab/c

r: inradius = (a + b − c)/2

R: circumradius = c/2

How It Works: A Step-by-Step Example

Inputs: a = 3, b = 4

Hypotenuse via Pythagoras: c = √(3² + 4²) = √(25) = 5.
Angles: A = arctan(a/b) = arctan(3/4) ≈ 36.8699°, B = 90° − A ≈ 53.1301°.
Area: (a × b)/2 = (3 × 4)/2 = 6. Perimeter: a + b + c = 3 + 4 + 5 = 12.
Altitude to hypotenuse: h = ab/c = 12/5 = 2.4.
Inradius: r = (a + b − c)/2 = (3 + 4 − 5)/2 = 1. Circumradius: R = c/2 = 2.5.

These results agree with the listed formulas, confirming internal consistency.

Frequently Asked Questions (FAQ)

Which two values are valid to start?

Any two independent values where at least one is a side (a, b, or c). Examples: (a, b), (a, c), (c, A), (a, B). Entering both A and B without a side is insufficient.

Why must the hypotenuse be the largest side?

In a right triangle the hypotenuse is opposite the 90° angle and, by the Pythagorean theorem, c² = a² + b², so c > a and c > b.

What angle range is allowed?

Each acute angle must satisfy 0° < angle < 90° and A + B = 90°.

Do units matter?

You may use any length units. All derived lengths (c, perimeter, h, r, R) will be in the same units as your inputs. Angles are always in degrees.

How is numerical precision handled?

Values are calculated in double precision and displayed with up to 6 significant decimals for readability.

Can I enter more than two fields?

To avoid conflicts, this tool expects exactly two fields. If you enter more, it will prompt you to reduce to two so it can compute consistently.

Is this calculator accessible?

Yes. Labels, instructions, visible focus, keyboard navigation, and ARIA live regions are implemented to meet WCAG 2.1 AA guidance.

Strumento sviluppato da Ugo Candido,. Contenuti verificati da,.

Ultima revisione per l'accuratezza in data: September 14, 2025.