Right Triangle Solver
Find the hypotenuse of a right triangle when you know the adjacent side length and the angle at the base. Useful in construction, navigation, and any geometry problem involving right triangles.
About this calculator
In a right triangle, the cosine of an angle equals the ratio of the adjacent side to the hypotenuse: cos(θ) = adjacent / hypotenuse. Rearranging this gives the hypotenuse directly: hypotenuse = adjacent / cos(θ). This calculator applies that formula with the angle converted from degrees to radians for computation: hypotenuse = adjacent / cos(angle × π / 180). This is a direct application of SOH-CAH-TOA, the mnemonic for sine, cosine, and tangent in right triangles. Knowing the adjacent side and one non-right angle is sufficient to fully determine the triangle because the angle and one side fix the shape and scale. This approach is used constantly in surveying, architecture, carpentry, and physics.
How to use
Suppose a ramp has a horizontal base (adjacent side) of 8 metres and rises at an angle of 20° from the ground. Step 1: Convert the angle: 20 × π / 180 = 0.3491 radians. Step 2: Compute cos(20°) ≈ 0.9397. Step 3: Hypotenuse = 8 / 0.9397 ≈ 8.51 metres. So the ramp surface is approximately 8.51 m long. Enter adjacent = 8 and angle = 20 into the calculator to confirm. This length tells you exactly how much material is needed for the ramp surface.
Frequently asked questions
How do I find the hypotenuse of a right triangle using the adjacent side and angle?
Use the cosine relationship: hypotenuse = adjacent / cos(θ), where θ is the angle between the adjacent side and the hypotenuse (i.e., the base angle). First ensure your angle is the one adjacent to the known side, not the opposite angle. For example, if the adjacent side is 10 units and the angle is 30°, then hypotenuse = 10 / cos(30°) = 10 / 0.866 ≈ 11.55 units. This calculator performs this computation automatically after you enter the adjacent length and angle.
What is the difference between the adjacent, opposite, and hypotenuse sides of a right triangle?
In a right triangle, the hypotenuse is always the longest side, directly opposite the 90° angle. For a given reference angle θ, the adjacent side is the leg that forms the angle with the hypotenuse (it 'touches' the angle along with the hypotenuse), and the opposite side is the leg that does not touch the angle. The mnemonics SOH-CAH-TOA summarises the relationships: Sine = Opposite/Hypotenuse, Cosine = Adjacent/Hypotenuse, Tangent = Opposite/Adjacent. Identifying sides correctly before calculating is crucial to getting the right answer.
When would I use a right triangle solver in real-world applications?
Right triangle solvers are essential whenever distances or lengths cannot be measured directly. Surveyors use them to calculate ground distances from elevation angles. Builders and carpenters use them to find rafter lengths when they know the horizontal span and roof pitch angle. Navigation and aviation use right triangle relationships to compute distances from bearing and altitude. Even in everyday tasks like hanging a picture or building a shelf bracket, knowing one side and an angle lets you calculate the required length of the diagonal support.