Enter any two known sides (two legs, or one leg and the hypotenuse) to find the missing side, both acute angles, and the area.
Fill in exactly two values, leave the unknown one blank.
A Right Triangle Calculator solves for unknown sides and angles of a right triangle given a minimum set of known values, such as two sides, or one side and one angle. This tool applies the Pythagorean theorem and basic trigonometric ratios to quickly solve problems that would otherwise require multiple manual calculation steps, useful in construction, navigation, physics, and geometry education.
The Pythagorean theorem states a² + b² = c², where c is the hypotenuse (the side opposite the right angle) and a and b are the other two sides, allowing calculation of any missing side if the other two are known. For angle calculations, trigonometric ratios apply: sine = opposite/hypotenuse, cosine = adjacent/hypotenuse, and tangent = opposite/adjacent, allowing calculation of angles or additional sides when at least one angle and one side are known.
Enter any two known values from the triangle (two sides, or one side and one non-right angle), and the calculator solves for all remaining sides and angles. This eliminates the need to manually select and apply the correct trigonometric formula for each specific scenario.
Example 1: A right triangle with legs of 3 and 4 units has a hypotenuse of the square root of (3² + 4²) = square root of 25 = 5 units, the classic 3-4-5 right triangle.
Example 2: A right triangle with a hypotenuse of 10 units and one angle of 30 degrees has an opposite side of 10 × sin(30°) = 10 × 0.5 = 5 units.
The theorem's specific relationship between the squares of the sides relies on the 90-degree angle's unique geometric properties, meaning triangles without a right angle require different formulas (like the law of cosines) to relate their side lengths.
The hypotenuse is always the longest side, located directly opposite the right angle, while the two legs form the right angle itself and are always shorter than the hypotenuse.
This mnemonic helps recall that Sine equals Opposite over Hypotenuse, Cosine equals Adjacent over Hypotenuse, and Tangent equals Opposite over Adjacent, providing a memorable shortcut for applying the correct trigonometric ratio to a given problem.
Right angles are structurally fundamental to buildings and construction, and the well-defined mathematical relationships in right triangles make it straightforward to calculate unknown measurements, verify square corners, and determine diagonal bracing lengths accurately.
Depending on which combination of sides and angles are provided, the calculator automatically selects between the Pythagorean theorem (for two known sides) or the appropriate trigonometric ratio (for a known side and angle) to solve for the remaining unknowns.
These are right triangles with specific fixed angle combinations that produce predictable, consistent side length ratios (such as 1:1:√2 for a 45-45-90 triangle), which are frequently used as shortcuts in geometry problems since their proportions are well-established and don't require recalculating trigonometric ratios each time.
Since a ladder leaning against a wall, or a ramp rising to a platform, naturally forms a right triangle with the ground and the vertical surface, this calculator can determine the necessary ladder length, ramp length, or safe angle based on the known height and horizontal distance involved.
Yes, since a right triangle's area is simply half the product of its two legs (Area = ½ × base × height), once the two legs are known or calculated, the area follows directly using this straightforward formula.
Surveyors and navigators frequently use right triangle relationships to calculate distances that can't be measured directly, such as the height of a tall structure or the width of a river, using a known baseline distance and measured angle.
If the provided measurements don't satisfy the Pythagorean relationship, the calculator will indicate an inconsistency, signaling that the input values don't correspond to a genuine right triangle and should be rechecked.
Screen sizes are measured diagonally, and knowing the aspect ratio (essentially the two legs of a right triangle) allows calculating the actual width and height of a screen from just its diagonal measurement and aspect ratio.