Right Triangle Calculator
Use this right triangle calculator to solve a right triangle from either two known sides or one side plus one acute angle. The page calculates all sides, both acute angles, area, perimeter, and a step-by-step explanation using the Pythagorean theorem and basic right triangle trigonometry.
All calculations use standard published formulas. Results are for informational use only.
Solving mode
Step-by-step solution
Calculate a valid right triangle to generate the full working.
Formula section
| Method | Formula | When to use it |
|---|---|---|
| Pythagorean theorem | c^2 = a^2 + b^2 | Use when two sides are known and c is the hypotenuse. |
| Missing leg form | a = sqrt(c^2 - b^2), b = sqrt(c^2 - a^2) | Use when the hypotenuse and one leg are known. |
| Sine | sin(A) = a / c, sin(B) = b / c | Use when the opposite side and hypotenuse relationship is most direct. |
| Cosine | cos(A) = b / c, cos(B) = a / c | Use when the adjacent side and hypotenuse relationship is most direct. |
| Tangent | tan(A) = a / b, tan(B) = b / a | Use when the opposite and adjacent sides are known or easy to solve. |
Mode example
Example 1 - Two legs known
Given a = 9 and b = 12
Method: Pythagorean theorem
c = 15, A = 36.86989765 deg, B = 53.13010235 deg
Square both legs, add them, take sqrt(), then use tan(A) = a / b for the acute angle.
How to use this right triangle calculator
- Choose whether you know two sides or one side plus one acute angle.
- Enter only the values requested for that mode. Use c for the hypotenuse and keep acute angles between 0 and 90.
- Press Calculate to solve all sides, both acute angles, area, and perimeter.
- Use the step-by-step section to verify how the formula was chosen and how each result was computed.
What is a right triangle?
A right triangle is a triangle with one angle equal to 90 degrees. The side opposite that right angle is called the hypotenuse, and it is always the longest side. The other two sides are the legs, usually labeled a and b. Because one angle is fixed at 90 degrees, right triangles are easier to solve than general triangles and are the foundation of both geometry and trigonometry.
That is why a good right triangle calculator is so useful. Once you know enough information about a right triangle, the remaining sides and angles are tightly constrained. This page is built to take the two most practical inputs people actually have, solve the full triangle, and explain the method in plain language.
How to solve right triangles
There are two main solving patterns. If two sides are known, the Pythagorean theorem gives the missing side. If one side and one acute angle are known, basic trigonometry does the rest. In either case, once all three sides are known, area and perimeter are easy to calculate. This is why the page is organized into two clear modes instead of mixing every case into one confusing form.
A strong right triangle solver should also explain why the method changes. The Pythagorean theorem uses only side lengths and applies only to right triangles. Trigonometry uses the relationship between angles and side ratios, which is why one acute angle plus one side is enough to recover the entire triangle. The calculator mirrors that logic so the UI matches the math.
Pythagorean theorem explained
The Pythagorean theorem is written as c^2 = a^2 + b^2. It says the square of the hypotenuse equals the sum of the squares of the two legs. If the hypotenuse is missing, the process is direct: square a, square b, add them, and take sqrt(). If one leg is missing, rearrange the same relationship so the missing leg becomes sqrt(c^2 - known_leg^2).
This is the heart of every hypotenuse calculator. The page also checks whether your inputs make sense before calculating. If the supposed hypotenuse is not actually the longest side, or if the subtraction would create a negative value inside sqrt(), the triangle is impossible and the page explains that clearly.
Trigonometry basics for right triangles
Right triangle trigonometry uses three core ratios: sin(), cos(), and tan(). In plain text, sin(A) = a / c, cos(A) = b / c, and tan(A) = a / b. These formulas work because the side lengths scale in predictable ways relative to the acute angles. Once one acute angle is known, the other is automatically 90 - angle, so the full triangle is locked in.
That is why one side plus one angle is enough for this calculator. Depending on the known side, the page chooses the most direct trig formula, substitutes your values, solves the remaining sides, and then calculates the second acute angle. This keeps the workflow beginner-friendly while still being mathematically correct.
How to use this calculator
Choose the mode that matches your inputs. If you know two sides, use the Two sides known mode. If you know one side and one acute angle, use the One side + one angle mode. Then fill in the requested fields only. This focused layout helps prevent the most common data-entry mistakes and keeps the page easy to use on mobile as well as desktop.
After entering your values, click Calculate to solve the triangle. The result section shows all sides, both acute angles, the 90 degree angle, area, perimeter, and a step-by-step explanation. The Example button loads a ready-made problem if you want to study the method first, and Reset clears the current mode quickly.
Worked examples
Worked examples make right triangle solving much easier to learn. A two-leg example shows the classic Pythagorean hypotenuse workflow. A hypotenuse-plus-leg example shows how to rearrange the formula safely. A side-and-angle example shows how trig recreates the entire triangle from minimal information. Reviewing those patterns side by side helps you recognize which method belongs to your own problem.
This page includes both a mode-specific example and a worked examples section lower on the page. That makes it useful not only as a fast calculator, but also as a revision tool when you want to understand the reasoning behind the answer.
Common mistakes
One common mistake is forgetting that side c is the hypotenuse. In a right triangle, c must always be the longest side. Another mistake is entering an acute angle that is 0, 90, or larger than 90. Those values do not work in this page's one-side-plus-one-angle mode because the second acute angle would stop being valid. A third mistake is mixing up which side is opposite which angle, especially when using sin(), cos(), or tan().
Another practical issue is rounding too early. If you round the missing side too aggressively before calculating the angles or the perimeter, your later values can drift slightly. This calculator keeps the internal math precise and only formats the output at the end, which gives cleaner results while still keeping the display easy to read.
Why this right triangle calculator is useful
A good triangle calculator with steps should not feel like a black box. This page explains the Pythagorean theorem, uses plain-text trig formulas, validates impossible inputs, and keeps the result visually easy to scan. That makes it useful for students, teachers, engineers, surveyors, carpenters, and anyone else who needs quick right triangle math without sacrificing understanding.
Right triangles show up everywhere: ramps, ladders, roofs, navigation, vectors, coordinate geometry, and measurement problems. A polished right triangle calculator belongs on a serious calculator site because it handles one of the most universal geometry problems in a way that is both practical and educational.
Worked examples
Example 1 - Two legs known
Given a = 9 and b = 12
Method: Pythagorean theorem
c = 15, A = 36.86989765 deg, B = 53.13010235 deg
Square both legs, add them, take sqrt(), then use tan(A) = a / b for the acute angle.
Example 2 - Hypotenuse and one leg known
Given b = 8 and c = 17
Method: Rearranged Pythagorean theorem
a = 15, A = 61.92751306 deg, B = 28.07248694 deg
Subtract the known leg square from the hypotenuse square, take sqrt(), and then solve the angles.
Example 3 - Hypotenuse and one acute angle known
Given c = 10 and angle A = 35 deg
Method: Right triangle trigonometry
a = 5.73576436, b = 8.19152044, B = 55 deg
Use sin(A) for the opposite leg, cos(A) for the adjacent leg, and then subtract from 90 to get the second acute angle.
Frequently Asked Questions
How does this right triangle calculator solve a triangle?
This right triangle calculator handles the two most common right-triangle setups: two sides known, or one side plus one acute angle known. It then uses the Pythagorean theorem or basic trigonometry to solve the missing sides, calculates the second acute angle, and finishes with area and perimeter. The page also explains each step so you can verify the method instead of just copying the final answer.
When should I use a hypotenuse calculator?
Use a hypotenuse calculator whenever side c is the missing value in a right triangle. Side c is the hypotenuse and must always be the longest side because it sits opposite the 90 degree angle. If the two legs are known, c = sqrt(a^2 + b^2) gives the result directly.
When do I use the pythagorean theorem calculator?
Use the pythagorean theorem calculator when two sides are known. If the missing side is c, use c^2 = a^2 + b^2. If the missing side is a or b, rearrange the same relationship so the missing leg comes from sqrt(c^2 - known_leg^2). This method only applies to right triangles.
How does the triangle calculator with steps work with one side and one angle?
When one side and one acute angle are known, the calculator uses sin(), cos(), or tan() depending on whether the known side is opposite, adjacent, or the hypotenuse. Once one acute angle is known, the other is just 90 - angle, so the full right triangle can be rebuilt from one trig relationship and one angle subtraction.
Can this right triangle solver solve any triangle?
No. This page is specifically for right triangles, where one angle is exactly 90 degrees. If your triangle is not a right triangle, use the general triangle calculator or a missing-side triangle calculator that supports the Law of Cosines and Law of Sines.
Why is the right triangle formula so useful?
The right triangle formula set links every side to every acute angle. sin() connects the opposite side to the hypotenuse, cos() connects the adjacent side to the hypotenuse, and tan() connects the opposite side to the adjacent side. Together they make right triangles one of the most practical geometry topics in math, engineering, navigation, and construction.