Right Triangle Solver: Find Missing Sides and Angles Easily
ByMuhammad Ali•Founder of KruskalCode
16:05
6 min read
Right-angled triangles are fundamental building blocks in geometry and trigonometry, appearing everywhere from construction to navigation. Whether you're tackling homework or a real-world problem, knowing how to find missing sides and angles is a crucial skill. This guide will walk you through the principles of solving right triangles and show you how to use our online Right Triangle Solver to make calculations quick and accurate.
Explanation
A right triangle is defined by having one angle that measures exactly 90 degrees. The side opposite this right angle is called the hypotenuse, which is always the longest side. The other two sides are often referred to as legs. To solve a right triangle means to find the lengths of all its sides and the measures of all its angles, given some initial information. The key tools for solving right triangles are the Pythagorean theorem and the basic trigonometric ratios (SOH CAH TOA). The Pythagorean theorem relates the lengths of the three sides, while SOH CAH TOA connects angles to the ratios of side lengths. By combining these, you can determine any unknown value as long as you have at least two pieces of information (excluding the 90-degree angle itself).
Formula
For a right-angled triangle with angles A, B, C (where C = 90°) and opposite sides a, b, c (where c is the hypotenuse): 1. **Pythagorean Theorem:** a² + b² = c² 2. **Angle Sum Property:** A + B = 90° 3. **Trigonometric Ratios (SOH CAH TOA):** * **SOH:** Sine(Angle) = Opposite / Hypotenuse sin(A) = a / c sin(B) = b / c * **CAH:** Cosine(Angle) = Adjacent / Hypotenuse cos(A) = b / c cos(B) = a / c * **TOA:** Tangent(Angle) = Opposite / Adjacent tan(A) = a / b tan(B) = b / a
Example
Let's work through an example. Suppose you have a right triangle where side 'a' is 6 units long and Angle A is 30 degrees. We need to find side 'b', side 'c' (hypotenuse), and Angle B. 1. **Find Angle B:** Since A + B = 90°, B = 90° - 30° = 60°. 2. **Find Hypotenuse 'c' (using sin A):** sin(A) = a / c sin(30°) = 6 / c 0.5 = 6 / c c = 6 / 0.5 = 12 units. 3. **Find Side 'b' (using tan A):** tan(A) = a / b tan(30°) = 6 / b 0.5774 ≈ 6 / b b ≈ 6 / 0.5774 ≈ 10.39 units. So, the missing values are side b ≈ 10.39, side c = 12, and Angle B = 60°.
How to use the related calculator
Using the ProMathTools Right Triangle Solver is straightforward. Simply input the two known values into the corresponding fields. For example, if you know two sides, enter their lengths into 'Side a', 'Side b', or 'Side c'. If you know one side and one angle, enter those values. The calculator will automatically determine the remaining sides and angles. Remember to leave the fields for unknown values blank. The results will be displayed with high precision, giving you all the properties of your right triangle.
Try the related calculator
Open toolFAQ
What is a right triangle?
A right triangle is a triangle that has one angle measuring exactly 90 degrees (a right angle). The side opposite the right angle is called the hypotenuse, and it's always the longest side of the triangle.
What information do I need to solve a right triangle?
To solve a right triangle, you typically need at least two pieces of information, besides the inherent 90-degree angle. This could be two sides, or one side and one acute angle (an angle less than 90 degrees). Our calculator will then find all the other missing sides and angles.
How does the solver work?
The solver uses fundamental geometric principles and trigonometry. It applies the Pythagorean theorem (a² + b² = c²) for finding sides and trigonometric ratios (SOH CAH TOA: Sine = Opposite/Hypotenuse, Cosine = Adjacent/Hypotenuse, Tangent = Opposite/Adjacent) to determine angles and sides.
Can I use this for any triangle?
No, this specific solver is designed only for right-angled triangles. For other types of triangles (like acute or obtuse triangles), you would typically use the Law of Sines or the Law of Cosines, which are different formulas.
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About the author
Muhammad Ali. Muhammad Ali is a full-stack developer and founder of KruskalCode. He builds SaaS platforms and automation systems with React and Laravel, and helps teams ship fast, scalable tools.
Need a custom calculator, dashboard, or automation workflow? Reach out to KruskalCode.
