Right-Angled Triangle Calculator

Find all side lengths, angles, perimeter, and surface area of a right triangle from any two inputs.

Preparing Right Triangle Calculator

Please wait ...

Input

Enter any two known values (such as two side lengths, or one side length and one acute angle) below to calculate the complete triangle properties.

Input summary

Your calculator summary shows here.

Step by Step Calculation

⊿

Step by step calculation shows here

Calculate first and the full working will appear below automatically.

A Guide to Right Triangle Calculations and Trigonometry

A right triangle is a three-sided flat shape characterized by one ninety-degree angle. Because of its rigid structure, a right triangle possesses special mathematical properties that allow you to calculate all of its features if you know only two pieces of information (as long as at least one of those inputs is a side length). The math of right triangles forms the foundation for trigonometry, architectural blueprints, physics vector resolutions, and spatial navigation.

Solving Triangle Sides and Angles

To solve a right triangle, we combine the Pythagorean theorem with trigonometric ratios. The side lengths are labeled relative to the angle you are analyzing: the opposite side (directly across), the adjacent side (next to the angle), and the hypotenuse (the longest diagonal).

The primary trigonometric functions relate the sides to the angles. Sine is the ratio of opposite to hypotenuse, cosine is adjacent to hypotenuse, and tangent is opposite to adjacent. Because the angles of any triangle always add up to 180 degrees, and one angle is always 90 degrees, the other two acute angles must always add up to exactly 90 degrees.

For calculating diagonal lengths directly without angle considerations, you can use our hypotenuse calculation solver. To find the flat space enclosed inside, check out our flat shapes area solver.

Common Applications

Architectural Slopes: Roofers use right triangle calculations to determine the exact pitch and rafter lengths needed for buildings.
Accessibility Ramps: Engineers design wheelchair ramps by calculating the required length to achieve safe slope percentages, which you can evaluate using our percentage rates converter.
Land Surveying: Surveyors determine height changes across sloped property boundaries, which you can map using our coordinate distance mapping tool.
Daily Engineering Checks: Mechanical engineers split forces into vertical and horizontal components using simple daily math tools like our standard daily math helper.

Trigonometric Relations

When solving a triangle with one side and one angle, you select the ratio that connects your known values. For example, if you know the hypotenuse and want to find the height (opposite side), you multiply the hypotenuse by the sine of the angle.

If you know the two legs, you can find the acute angles by using inverse trigonometric functions (like arctangent). Our online solver performs all these checks and displays the intermediate steps for sides, angles, area, and perimeter.

Example of Tree Height Estimation

Suppose you want to estimate the height of a tall pine tree. You stand 20 feet away from its base and measure the angle of elevation to the top of the tree, which is 30 degrees.

Here, the ground distance of 20 feet is the adjacent leg, and the tree height is the opposite leg (opposite to the 30-degree angle). Using the tangent ratio: tangent(30°) = opposite / 20. Tangent of 30 degrees is approximately 0.577. Multiplying by 20 reveals the tree height: 0.577 × 20 = 11.54 feet. The hypotenuse (straight-line distance to the treetop) is 20 / cosine(30°) = 20 / 0.866 = 23.09 feet. This practical example shows how simple angle measurements reveal heights without physical climbing.