Right Triangle Trigonometry Calculator
Find any missing side of a right triangle from one side and one angle using sine, cosine, or tangent. Ideal for students, architects, and anyone working with slopes, heights, or distances.
About this calculator
In a right triangle, the three primary trigonometric ratios relate the sides to the non-right angles. For a given angle θ: sin(θ) = opposite/hypotenuse, cos(θ) = adjacent/hypotenuse, and tan(θ) = opposite/adjacent. Rearranging these gives six solving formulas. For instance, if you know the adjacent side and want the hypotenuse: hypotenuse = adjacent / cos(θ). If you know the opposite side and want the adjacent: adjacent = opposite / tan(θ). These relationships follow directly from the definitions of the trig functions and hold for any right triangle regardless of size. The calculator applies the correct formula automatically based on which side is known and which is sought, covering all six possible combinations.
How to use
Suppose a ladder leans against a wall. The angle at the base is 65° and the base (adjacent side) measures 4 m. To find the ladder length (hypotenuse): hypotenuse = adjacent / cos(65°) = 4 / cos(65°) = 4 / 0.4226 ≈ 9.47 m. Select 'adjacent' as the known side type, enter 4 for the known side length, 65 for the known angle, and 'hypotenuse' as what to calculate. The result is approximately 9.47 units — the required ladder length.
Frequently asked questions
How do I know which trigonometric ratio to use in a right triangle problem?
The choice depends on which two sides are involved relative to your known angle. If you have the opposite and hypotenuse, use sine (SOH). If you have the adjacent and hypotenuse, use cosine (CAH). If you have the opposite and adjacent, use tangent (TOA). The mnemonic SOH-CAH-TOA helps remember these relationships. Identify the sides you have and the side you want, then pick the ratio that connects those two sides.
What is the difference between the opposite side, adjacent side, and hypotenuse in a right triangle?
The hypotenuse is always the longest side, directly opposite the 90° angle. The opposite and adjacent sides are defined relative to a specific non-right angle. The opposite side faces that angle directly, while the adjacent side runs from that angle to the right angle. Note that if you switch to the other non-right angle, the opposite and adjacent sides swap roles — but the hypotenuse stays the same. This labeling is crucial for correctly applying SOH-CAH-TOA.
Can I use this calculator to find the angle if I know two sides of a right triangle?
To find an angle from two sides, you need the inverse trigonometric functions: arcsin, arccos, or arctan. For example, if the opposite side is 3 and the hypotenuse is 5, then the angle θ = arcsin(3/5) = arcsin(0.6) ≈ 36.87°. This calculator focuses on finding sides from a known side and angle. For the reverse operation — finding the angle from two known sides — use an inverse trigonometry calculator, which applies the arcsin, arccos, or arctan function as appropriate.