Right Triangle Formulas Explained
I built this calculator to solve right triangles completely — give it any two values (sides or angles) and it finds everything else. Right triangles are the foundation of trigonometry and appear in navigation, construction, engineering, and physics. The right angle (90°) is what makes these triangles special and solvable with a compact set of formulas.
The Key Formulas
- Pythagorean theorem:
a² + b² = c²(where c is the hypotenuse) - Sine:
sin(θ) = opposite / hypotenuse - Cosine:
cos(θ) = adjacent / hypotenuse - Tangent:
tan(θ) = opposite / adjacent - Area:
A = (1/2) × a × b(half base times height) - Perimeter:
P = a + b + c - Angles sum:
A + B + 90° = 180°, so the two acute angles always add to 90°
The mnemonic SOH-CAH-TOA helps remember the trig ratios: Sine = Opposite/Hypotenuse, Cosine = Adjacent/Hypotenuse, Tangent = Opposite/Adjacent.
Real-World Applications of Right Triangle Geometry
Right triangle trigonometry is one of the most applied branches of mathematics. Any time you need to find a height you cannot directly measure, or a distance along a slope, a right triangle is usually hiding in the problem.
- Roof pitch: the rise and run of a roof form the two legs; the rafter length is the hypotenuse
- Surveying: trigonometry calculates heights of buildings, trees, and cliffs from ground measurements and angles
- Ramp design: given a rise height and maximum angle, calculate the required ramp length (hypotenuse)
- Navigation: if you travel due north then due east, the direct return distance is the hypotenuse
Frequently Asked Questions
What information do I need to fully solve a right triangle?
You need at least two pieces of information, and at least one must be a side length. Two angles (without any side) leave the triangle's size completely undefined. Valid combinations include: two sides; one side and one acute angle; or (in practice) the hypotenuse and one other side. Given any of these, the calculator finds all remaining sides and angles.
What is the hypotenuse and why is it always the longest side?
The hypotenuse is the side opposite the 90° angle. It must be the longest side because the largest angle in any triangle is always opposite the longest side — and 90° is the largest possible angle in a right triangle (the other two must sum to 90°, so both are less than 90°).
What is a 45-45-90 triangle?
A 45-45-90 triangle is an isosceles right triangle where both acute angles are 45°. Its sides are in the ratio 1 : 1 : √2. It is one of the two "special triangles" (along with 30-60-90) that appear constantly in geometry problems and architecture. If the legs have length 1, the hypotenuse is exactly √2 ≈ 1.414.