Right Triangle Calculator

Enter any two known sides or angles of a right triangle and this calculator will determine all missing sides and angles using the Pythagorean theorem, SOH CAH TOA trigonometric ratios, and the angle sum of a triangle. You'll get step-by-step solutions, the area and perimeter, and a to-scale diagram of the solved triangle. Free to use, no sign-up required.

Angle unit: Degrees (°) Radians (rad)

Enter Known Values

Fill in any two or more known values. Leave unknown fields blank. At least one side length is required.

Side a (leg)

Side b (leg)

Side c (hypotenuse)

Angle A (opposite a)

Angle B (opposite b)

Angle C (right angle)

Triangle Solution

Side a (leg):

Side b (leg):

Side c (hypotenuse):

Angle A:

Angle B:

Angle C:

Area:

Perimeter:

Triangle type:

Steps:

Worked Right Triangle Example

📝 Example 1: Legs a = 3, b = 4 — find hypotenuse and angles

Step 1 — Use the Pythagorean theorem: c² = a² + b² = 3² + 4² = 9 + 16 = 25 → c = √25 = 5

Step 2 — Find angle A using SOH (opposite / hypotenuse): sin(A) = a/c = 3/5 = 0.6 → A = sin⁻¹(0.6) ≈ 36.87°

Step 3 — Find angle B: B = 90° − A = 90° − 36.87° ≈ 53.13°

Step 4 — Area: Area = ½ × a × b = ½ × 3 × 4 = 6

Step 5 — Perimeter: P = a + b + c = 3 + 4 + 5 = 12

Result: c = 5, A ≈ 36.87°, B ≈ 53.13°, area = 6, perimeter = 12. This is a 3-4-5 right triangle (a right scalene triangle).

📝 Example 2: Hypotenuse c = 10, angle A = 30° — find both legs

Step 1 — Find side a (opposite A) using SOH: sin(30°) = a/10 → a = 10 × 0.5 = 5

Step 2 — Find side b (adjacent to A) using CAH: cos(30°) = b/10 → b = 10 × (√3/2) ≈ 8.660

Step 3 — Find angle B: B = 90° − 30° = 60°

Step 4 — Verify with the Pythagorean theorem: a² + b² = 25 + 75 = 100 = c² ✓

Result: a = 5, b ≈ 8.660, B = 60°. This is a 30-60-90 right triangle.

How to Use the Right Triangle Calculator

Choose your angle unit — degrees or radians — using the toggle at the top.
Enter any two or more known values of your right triangle. You need at least one side length (two angles alone don't define a unique triangle). Leave unknown fields blank.
Click "Solve Right Triangle." The calculator determines all missing sides and angles using the Pythagorean theorem and SOH CAH TOA.
Read the results. You'll see all three sides, both acute angles, the area, the perimeter, the triangle type, step-by-step working, and a to-scale diagram.
Use the Clear button to reset the inputs and try another triangle.

Calculation Method & Data Handling

This calculator solves right triangles by combining three fundamental relationships:

Pythagorean theorem: a² + b² = c² (where c is the hypotenuse)
SOH CAH TOA: sin(θ) = opposite/hypotenuse, cos(θ) = adjacent/hypotenuse, tan(θ) = opposite/adjacent
Angle sum: A + B + 90° = 180°, so B = 90° − A

Depending on which values you enter, the calculator selects the most direct solution path (e.g., both legs → hypotenuse via Pythagoras; hypotenuse + angle → legs via sine and cosine). Every step is shown in the results. All calculations use full floating-point precision and are performed client-side in your browser — no data is sent to a server.

Assumptions and Limits

Right triangles only: This calculator is specifically for right triangles (one angle = 90°). For oblique triangles, use the Law of Sines or Law of Cosines.
At least one side required: Two angles alone do not determine a unique triangle — you must enter at least one side length.
At least two knowns: Enter at least two values (with at least one side) to solve the triangle.
Angle unit: The angle unit toggle applies to all angle inputs and outputs. Make sure to select the correct unit before solving.
Impossible triangles: If the entered values describe an impossible triangle (e.g., a leg ≥ the hypotenuse, or acute angles that don't sum to 90°), the calculator displays an error.
Negative and zero values: All side lengths must be positive. Angles must be strictly between 0° and 90° (exclusive).
Precision: Results are rounded for display. Minor rounding differences may appear when verifying by hand.

Frequently Asked Questions

What is a right triangle?

A right triangle is a triangle in which one angle is exactly 90° (a right angle). The side opposite the right angle is called the hypotenuse, and the other two sides are called legs. Right triangles satisfy the Pythagorean theorem: a² + b² = c².

How do I find the missing side of a right triangle?

If you know the other two sides, use the Pythagorean theorem: c = √(a² + b²) or a = √(c² − b²). If you know one side and one acute angle, use SOH CAH TOA (sine, cosine, or tangent) to solve for the missing side.

How do I find the missing angle of a right triangle?

If you know one acute angle, the other is 90° − known angle. If you only know sides, use inverse trig: A = sin⁻¹(opposite / hypotenuse), or use cos⁻¹ or tan⁻¹ depending on which sides you know.

What is the 3-4-5 triangle?

The 3-4-5 triangle is a special right triangle where the legs are 3 and 4 and the hypotenuse is 5. It satisfies 3² + 4² = 5² (9 + 16 = 25). It is the simplest integer Pythagorean triple and is often used in construction to check for right angles.

What are 30-60-90 and 45-45-90 triangles?

These are special right triangles with fixed side ratios. In a 30-60-90 triangle, sides are in the ratio 1 : √3 : 2. In a 45-45-90 triangle, sides are in the ratio 1 : 1 : √2.

What is the area of a right triangle?

The area of a right triangle is half the product of its two legs: Area = ½ × a × b. This is because the two legs are perpendicular, so one leg acts as the base and the other as the height.

Can the hypotenuse ever be shorter than a leg?

No. In a right triangle, the hypotenuse is always the longest side because it's opposite the largest angle (90°). If your inputs suggest a leg longer than the hypotenuse, the triangle is impossible, and the calculator will display an error.

Does this calculator work in radians?

Yes. Use the angle unit toggle at the top to switch between degrees and radians. All angle inputs and outputs will use the selected unit.