math

Right Triangle Trigonometry Calculator

Solve a right triangle given two sides. Enter any two sides (legs a and b) to compute the hypotenuse, all angles, area, and trigonometric ratios using the Pythagorean theorem and inverse trigonometric functions for physics and engineering problems.

Reviewed by Chase FloiedUpdated April 16, 2026

This free online right triangle trigonometry calculator provides instant results with no signup required. All calculations run directly in your browser — your data is never sent to a server. Enter your values below and see results update in real time as you type. Perfect for everyday calculations, homework, or professional use.

Leg a

Length of the first leg (adjacent to angle B)

Leg b

Length of the second leg (adjacent to angle A)

Results

Hypotenuse (c)

How to Use This Calculator

Enter your input values

Fill in all required input fields for the Right Triangle Trigonometry Calculator. Most fields include unit selectors so you can work in your preferred unit system — metric or imperial, whichever matches your problem.

Review your inputs

Double-check that all values are correct and that you have selected the right units for each field. Incorrect units are the most common source of calculation errors and can produce results that are off by factors of 2, 10, or more.

Read the results

The Right Triangle Trigonometry Calculator instantly computes the output and displays results with units clearly labeled. All calculations happen in your browser — no loading time and no data sent to a server.

Explore parameter sensitivity

Try adjusting individual input values to see how the output changes. This is a quick and effective way to develop intuition about how different parameters influence the result and to identify which inputs have the largest effect.

Formula Reference

Right Triangle Trigonometry Calculator Formula

See calculator inputs for the governing equation

Variables: All variables and their units are labeled in the calculator interface above. Input fields accept values in multiple unit systems — select your preferred unit from the dropdown next to each field.

When to Use This Calculator

•Use the Right Triangle Trigonometry Calculator when you need a quick mathematical result without writing out all the steps manually, saving time on repetitive calculations.
•Use it to verify hand calculations on tests or assignments and catch arithmetic mistakes.
•Use it when teaching or explaining mathematical concepts to others, demonstrating how changing inputs affects the result.
•Use it to explore the behavior of mathematical functions across a range of inputs.

About This Calculator

The Right Triangle Trigonometry Calculator is a free mathematical calculation tool for students, educators, and professionals who need quick, reliable results. Solve a right triangle given two sides. Enter any two sides (legs a and b) to compute the hypotenuse, all angles, area, and trigonometric ratios using the Pythagorean theorem and inverse trigonometric functions for physics and engineering problems. The underlying algorithms implement well-established mathematical formulas and numerical methods. Results are computed instantly in the browser. This tool is useful for learning, verification of hand calculations, and rapid exploration of mathematical relationships. All computation happens locally — no data is sent to a server.

About Right Triangle Trigonometry Calculator

The Right Triangle Trigonometry Calculator solves a right triangle given two legs. It computes the hypotenuse using the Pythagorean theorem, both acute angles using inverse tangent, and the area. Right triangle trigonometry is the foundation of all trigonometry, connecting angle measures to side length ratios. The six trigonometric functions (sine, cosine, tangent, and their reciprocals) are all defined in terms of right triangle side ratios. Applications span virtually every technical field: physics (resolving forces into components), engineering (structural analysis), navigation (bearing calculations), surveying (distance measurement), and computer graphics (rotation and projection calculations).

The Math Behind It

The Pythagorean theorem, a^2 + b^2 = c^2, is perhaps the most famous theorem in mathematics. It states that in a right triangle, the square of the hypotenuse equals the sum of the squares of the two legs. There are over 370 known proofs, including one by a young James Garfield (later President of the United States). The trigonometric ratios for a right triangle are: sin(A) = opposite/hypotenuse = a/c, cos(A) = adjacent/hypotenuse = b/c, tan(A) = opposite/adjacent = a/b. These definitions extend to all angles via the unit circle. The inverse functions recover angles from ratios: A = arctan(a/b). The two acute angles in a right triangle are complementary (sum to 90 degrees), so A + B = 90. This means sin(A) = cos(B) and vice versa, which is why the 'co' prefix exists in cosine (complementary sine). The Pythagorean theorem generalizes to the law of cosines: c^2 = a^2 + b^2 - 2ab*cos(C), which reduces to the Pythagorean theorem when C = 90 degrees (since cos(90) = 0). In higher dimensions, the Pythagorean theorem extends to: |v|^2 = v1^2 + v2^2 + ... + vn^2 for a vector in n-dimensional Euclidean space. Practical applications include: computing the distance a ladder reaches up a wall, determining the viewing angle of a screen, calculating the resultant of perpendicular forces, and converting between rectangular and polar coordinates. In surveying, right triangles are used with theodolites to measure heights and distances indirectly.

Formula Reference

Pythagorean Theorem

c = sqrt(a^2 + b^2)

Variables: a, b = legs, c = hypotenuse

Angle from Tangent

A = arctan(a/b), B = arctan(b/a)

Variables: A = angle opposite leg a, B = angle opposite leg b

Worked Examples

Example 1: Classic 3-4-5 Triangle

A right triangle has legs a = 3 and b = 4.

Step 1:Hypotenuse: c = sqrt(9 + 16) = sqrt(25) = 5

Step 2:Angle A = arctan(3/4) = 36.87 degrees

Step 3:Angle B = arctan(4/3) = 53.13 degrees

Step 4:Area = (3 * 4) / 2 = 6

c = 5, A = 36.87 deg, B = 53.13 deg, Area = 6 square units.

Example 2: Equal Legs

A right isosceles triangle has legs a = b = 10.

Step 1:c = sqrt(100 + 100) = sqrt(200) = 14.142

Step 2:A = B = arctan(1) = 45 degrees

Step 3:Area = 10 * 10 / 2 = 50

c = 14.142, both angles = 45 deg, Area = 50 square units.

Common Mistakes & Tips

!Applying the Pythagorean theorem to non-right triangles. The formula a^2 + b^2 = c^2 ONLY works for right triangles. For others, use the law of cosines.
!Confusing which side is the hypotenuse. The hypotenuse is always the longest side and is opposite the right angle.
!Using the wrong inverse trig function. Use arctan(opposite/adjacent) for angles, not arcsin or arccos of the same ratio.

Related Concepts

30-60-90 Triangle

A special right triangle with angles 30, 60, and 90 degrees, having side ratios 1 : sqrt(3) : 2.

45-45-90 Triangle

A special right isosceles triangle with side ratios 1 : 1 : sqrt(2).

Used in These Calculators

Calculators that build on or apply the concepts from this page:

30-60-90 Triangle Calculator

45-45-90 Triangle Calculator

Complementary Angles Calculator

Scalene Triangle Area Calculator

Frequently Asked Questions

What are Pythagorean triples?

Pythagorean triples are sets of three positive integers (a, b, c) that satisfy a^2 + b^2 = c^2. The most common are (3,4,5), (5,12,13), (8,15,17), and (7,24,25). Any triple can be scaled: (6,8,10) is also Pythagorean. All primitive triples can be generated by the formula a = m^2-n^2, b = 2mn, c = m^2+n^2.

Why is SOH-CAH-TOA useful?

SOH-CAH-TOA is a mnemonic: Sine = Opposite/Hypotenuse, Cosine = Adjacent/Hypotenuse, Tangent = Opposite/Adjacent. It helps students remember which ratio corresponds to which trigonometric function in a right triangle.

Can I use this calculator if I know the hypotenuse and one leg?

This calculator takes two legs. If you know the hypotenuse c and leg a, first compute the other leg: b = sqrt(c^2 - a^2), then use those two legs as input.