Right Triangle Calculator

Free Online Right Triangle Calculator — Solve Every Side and Angle

This right triangle calculator (also called a right angle triangle calculator or right triangle solver) instantly solves a right triangle for every missing side and angle. Enter any two known values — two side lengths, or one side plus one acute angle — and the calculator applies the Pythagorean theorem and the SOH-CAH-TOA trigonometric ratios to find the remaining leg, the hypotenuse, both acute angles, the area, the perimeter, and more, all shown on a labeled, to-scale diagram with the full step-by-step solution underneath.

Whether you're checking geometry or trigonometry homework, working through a construction or surveying problem, calculating a roof pitch, or simply need to solve right triangle equation quickly, this hypotenuse calculator and right triangle solver handles every valid combination of inputs — no matter which two quantities you already know.

What Is a Right Triangle?

A right triangle (or right-angled triangle) is a triangle in which one interior angle measures exactly 90°. The side opposite this right angle is called the hypotenuse — always the longest side of the triangle — while the other two sides are called the legs. In the standard labeling used by this calculator, side c is the hypotenuse, sides a and b are the two legs, angle A is opposite side a, and angle B is opposite side b, with the right angle sitting at vertex C.

Because one angle is fixed at 90°, and all three interior angles of any triangle sum to 180°, the two remaining acute angles A and B in a right triangle always add up to exactly 90° — they are complementary angles. This single fact, combined with the Pythagorean theorem, is what makes a right triangle fully solvable from just two known measurements.

Right Triangle Formulas Used by This Calculator

Two families of formulas power every calculation in this tool: the Pythagorean theorem, which relates the three sides, and the trigonometric ratios, which relate a side to an angle. Together they form a right triangle formula toolkit that can solve for anything from any two starting values.

• Pythagorean theorem: a² + b² = c² — relates the two legs to the hypotenuse.
• SOH — sin(A) = Opposite ÷ Hypotenuse = a ÷ c
• CAH — cos(A) = Adjacent ÷ Hypotenuse = b ÷ c
• TOA — tan(A) = Opposite ÷ Adjacent = a ÷ b
• Angle sum rule: A + B = 90° (the two acute angles are always complementary)
• Area = (a × b) ÷ 2
• Perimeter = a + b + c
• Altitude to the hypotenuse: h = (a × b) ÷ c

How to Use This Right Triangle Calculator — Step by Step

Solving a right triangle is straightforward with this tool: just tell it which two quantities you already know.

• Select your first known value from the dropdown — a leg, the hypotenuse, or an angle — and type in its measurement.
• Select a second, different known value and enter its measurement. You need at least one side; two angles alone are not enough because they only fix the triangle's shape, not its size.
• Instantly read off every side, every angle, the area, perimeter, altitude, inradius, and circumradius in the results panel.
• Study the labeled diagram above the solution — every side length and angle value is printed directly on the triangle itself, drawn to scale, so you can see exactly how the numbers fit together.
• Scroll down to the step-by-step solution to see precisely which formula was applied at each stage, in the correct order.

Worked Example 1 — Solving a Right Triangle From Two Legs

Suppose leg a = 3 and leg b = 4. Using the Pythagorean theorem, c = √(a² + b²) = √(3² + 4²) = √(9 + 16) = √25 = 5. This is the famous 3-4-5 right triangle. To find angle A, apply the inverse tangent: A = tan⁻¹(a ÷ b) = tan⁻¹(3 ÷ 4) = tan⁻¹(0.75) ≈ 36.87°. Since the acute angles are complementary, B = 90° - 36.87° = 53.13°. The area is (3 × 4) ÷ 2 = 6, and the perimeter is 3 + 4 + 5 = 12.

Worked Example 2 — Solving From a Side and an Angle

Now suppose you know the hypotenuse c = 10 and angle A = 30°. Using SOH, side a = c × sin(A) = 10 × sin(30°) = 10 × 0.5 = 5. Using CAH, side b = c × cos(A) = 10 × cos(30°) ≈ 10 × 0.866 ≈ 8.66. Angle B is simply 90° - 30° = 60°. This is the well-known 30°-60°-90° right triangle, where the side opposite the 30° angle is always exactly half of the hypotenuse.

How to Solve Any Right Triangle Equation

Every right triangle problem reduces to one of two setups: two sides given, or one side and one angle given. If you have two sides, apply the Pythagorean theorem to find the third side, then use an inverse trigonometric function (sin⁻¹, cos⁻¹, or tan⁻¹) to find one angle, and subtract from 90° for the other. If you have one side and one angle, rearrange SOH, CAH, or TOA to isolate the unknown side, using the known angle and the known side as inputs — this calculator automatically detects your combination and applies the correct rearranged formula, so you can enter virtually any equation involving a right triangle's sides and angles and have it solved instantly.

Common Right Triangles and Special Ratios

Certain right triangles come up so often in geometry, trigonometry, and standardized tests that it pays to recognize their exact ratios by sight:

• 3-4-5 triangle (a=3, b=4, c=5) — The smallest whole-number (Pythagorean) right triangle.
• 5-12-13 triangle (a=5, b=12, c=13) — Another common integer right triangle.
• 8-15-17 triangle (a=8, b=15, c=17) — A larger Pythagorean triple used in construction.
• 7-24-25 triangle (a=7, b=24, c=25) — A Pythagorean triple frequently seen in textbooks.
• 45°-45°-90° triangle (legs equal, c = leg × √2) — An isosceles right triangle — both acute angles are 45°.
• 30°-60°-90° triangle (sides in ratio 1 : √3 : 2) — The short leg is half the hypotenuse; the long leg is √3 times the short leg.

Understanding the Diagram: Reading the Values Directly

The diagram drawn above the step-by-step solution is scaled proportionally to your actual triangle, so a triangle with a much longer leg b than leg a will visibly look that way. Every side is labeled with its exact computed length, and small arcs mark each acute angle with its degree value printed right beside it, along with a small square marking the 90° right angle at vertex C. This means you don't need to cross-reference the results table at all — every number needed to understand the solved triangle is visible directly on the shape itself.

Area, Perimeter, and Other Right Triangle Properties

Beyond the basic sides and angles, this calculator also reports several derived properties. The area, computed as half the product of the two legs, tells you how much surface the triangle covers. The perimeter is simply the sum of all three sides. The altitude to the hypotenuse is the shortest distance from the right-angle vertex to the hypotenuse, computed as (a × b) ÷ c — a value frequently needed in construction and design. The inradius (radius of the largest circle that fits inside the triangle) equals (a + b - c) ÷ 2, while the circumradius (radius of the circle passing through all three vertices) is always exactly half the hypotenuse, since the hypotenuse is a diameter of the triangle's circumscribed circle.

Real-World Applications of Right Triangle Calculations

Right triangle math is one of the most practically useful branches of geometry, appearing constantly outside the classroom:

• Construction and carpentry: calculating roof pitch, staircase rise-and-run, and rafter lengths.
• Navigation and surveying: finding distances and elevations using angle-of-elevation measurements.
• Physics and engineering: resolving forces, velocities, and vectors into perpendicular components.
• Architecture: determining diagonal bracing lengths and structural support angles.
• Everyday problem solving: finding the shortest distance across a park, or the height of a tree or building using its shadow and the sun's angle of elevation.

Tips for Getting Accurate Results

Always double-check which quantity you're entering as which — mixing up a leg with the hypotenuse is the most common mistake when solving right triangles by hand. Remember that the hypotenuse must always be the longest side; if you enter a leg longer than the hypotenuse, the calculator will flag the input as invalid because no such right triangle can exist. When entering an angle, make sure it's the acute angle you intend (A or B, not the fixed 90° angle), and enter it in degrees. Finally, remember that knowing only the two acute angles is never enough on its own — angles alone describe a triangle's shape, not its actual size, so at least one side length is always required.