Right Triangle Calculator

Valid Known Input Combinations

Because every right triangle inherently includes one fixed 90° right angle, you only need any two independent measurements (at least one of which must be a side length) to completely solve the triangle:

• Two legs ($a$ and $b$): Solves hypotenuse $c$ via Pythagorean theorem and acute angles via arctan.
• One leg and hypotenuse ($a$ and $c$ or $b$ and $c$): Solves missing leg via subtraction of squares and angles via arcsin/arccos.
• One leg and an acute angle: Solves remaining sides using tangent, sine, or cosine.
• Hypotenuse and an acute angle: Solves both legs using sine ($a = c \sin A$) and cosine ($b = c \cos A$).

A right triangle (also referred to as a right-angled triangle) is a fundamental polygon in Euclidean geometry that contains exactly one 90-degree ($90^\circ$) right angle. This perpendicular corner forms the cornerstone of trigonometry, architecture, navigation, and engineering.

The two sides that intersect to form the $90^\circ$ right angle are called the legs (traditionally denoted as leg $a$ and leg $b$). The third side, situated directly across from the right angle, is the hypotenuse (denoted as $c$). Because the hypotenuse stretches across the widest opening of the triangle, it is mathematically guaranteed to always be the longest side.

Since the sum of interior angles in any Euclidean triangle must equal $180^\circ$, and the right angle accounts for $90^\circ$, the two remaining interior angles ($A$ and $B$) must be acute angles (less than $90^\circ$) that together sum to exactly $90^\circ$ ($A + B = 90^\circ$). This complementary angle relationship allows missing angles to be calculated effortlessly once one acute angle is identified.

Diagram Key: $a$ = opposite leg, $b$ = adjacent leg, $c$ = hypotenuse, $C = 90^\circ$ right angle, $A, B$ = complementary acute angles.

Understanding Reference Angles: While the hypotenuse $c$ is permanently fixed as the side opposite the $90^\circ$ corner, the designations of opposite and adjacent depend strictly on which acute angle is chosen as the reference point. If you select Angle $A$, leg $a$ is opposite and leg $b$ is adjacent. If you select Angle $B$, leg $b$ becomes opposite and leg $a$ becomes adjacent.

The Pythagorean theorem is one of the most famous equations in geometry, establishing that in any right triangle, the square of the hypotenuse length equals the sum of the squares of the two leg lengths:

This theorem allows us to easily find the hypotenuse, calculate a missing leg, check if a triangle is a true right triangle, and explore integer combinations called Pythagorean triples.

Need a dedicated tool focused solely on radical simplifications and exact square roots? Visit our specialized Pythagorean Theorem Calculator.

How to Find the Hypotenuse

To find the hypotenuse of a right triangle when both legs are known, follow these steps:

1. Identify the two perpendicular leg lengths ($a$ and $b$).
2. Square both values ($a^2$ and $b^2$).
3. Add the two squared numbers together ($a^2 + b^2$).
4. Take the square root of the sum to find the hypotenuse ($c = \sqrt{a^2 + b^2}$).

Finding a missing side of a right triangle depends on which two sides are already known:

For focused side-only calculations, check out our dedicated Right Triangle Side Calculator and Pythagorean Theorem Calculator.

Every right triangle has three interior angles: one fixed $90^\circ$ right angle and two acute angles ($A$ and $B$).

• Right Angle ($C$): Always exactly $90^\circ$ (or $\pi/2$ radians).
• Acute Angles ($A$ and $B$): Always strictly between $0^\circ$ and $90^\circ$.
• Complementary Relationship: $A + B = 90^\circ$. Therefore, if Angle $A = 30^\circ$, then Angle $B = 60^\circ$.

If no acute angles are given, you can solve an unknown angle using inverse trigonometric functions from any two side lengths:

Explore more angle properties with our dedicated Right Triangle Angle Calculator.

Inverse Trigonometric Functions: When side lengths are known but angles are unknown, inverse trigonometric functions ($\sin^{-1}, \cos^{-1}, \tan^{-1}$) reverse the operation to output the angle magnitude in degrees or radians.

Try our dedicated SOHCAHTOA Calculator or Right Triangle Ratios Guide for complete ratio tables.

How to Choose the Right Ratio: Identify which two sides are involved in your problem (e.g., opposite and hypotenuse) and choose the corresponding ratio from SOHCAHTOA. For interactive practice, visit our SOHCAHTOA Calculator.

Because the two legs ($a$ and $b$) of a right triangle are mutually perpendicular ($90^\circ$ angle), one leg serves perfectly as the geometric base and the other as the height.

Calculate surface area directly with our Right Triangle Area Calculator.

The perimeter of a right triangle is the total continuous distance around its outer boundary, calculated by adding all three side lengths:

Area vs. Perimeter: Area measures the 2D space inside the boundary (in square units), while perimeter measures the total length around the outside (in linear units).

Need to solve perimeter from angles or single sides? Try our Right Triangle Perimeter Calculator.

In right triangle geometry, it is crucial to distinguish between the perpendicular legs and the altitude to the hypotenuse ($h$).

The altitude to the hypotenuse is the perpendicular line segment drawn from the $90^\circ$ right-angle vertex directly to hypotenuse $c$. It splits the main right triangle into two smaller, similar right triangles.

Explore altitude geometric mean theorems on our Right Triangle Altitude Calculator page.

"Solving" a right triangle means finding the lengths of all three sides and the measures of all three angles. Depending on what is known, use the following methods:

Our right triangle calculator supports both metric ($\text{mm}, \text{cm}, \text{m}, \text{km}$) and imperial ($\text{in}, \text{ft}, \text{yd}$) units.

Linear vs. Area Conversions: Remember that while linear side conversions multiply by a scale factor $k$, surface area conversions multiply by $k^2$:

Always ensure all inputs are entered in consistent length units before performing manual calculations to avoid conversion errors.

To verify whether any triangle is a true right triangle, test its three side lengths against the converse of the Pythagorean theorem:

If the lengths of the two shorter sides ($a$ and $b$) and the longest side ($c$) satisfy:

then the triangle is guaranteed to be a right triangle with a $90^\circ$ angle opposite side $c$.

• Longest Side: The hypotenuse $c$ is always the longest side in any right triangle.
• Angle Opposition: The larger acute angle is always directly opposite the longer leg, while the smaller acute angle is opposite the shorter leg.
• Complementary Acute Angles: The two acute interior angles always add up to $90^\circ$ ($A + B = 90^\circ$).
• Pythagorean Constraint: Side lengths are linked quadratically via $a^2 + b^2 = c^2$.
• Trigonometric Ratios: The ratios of side lengths depend solely on the acute angle measures.

1. Identifying the wrong hypotenuse: Always verify that $c$ is opposite the $90^\circ$ corner before squaring.
2. Using the wrong reference angle: Re-check whether your target leg is opposite or adjacent to angle $\theta$.
3. Confusing opposite and adjacent: Remember that opposite is across the triangle, adjacent is next to the angle.
4. Forgetting to square values: Adding $a + b$ instead of $a^2 + b^2$ is a frequent algebra mistake.
5. Forgetting the square root: Failing to take $\sqrt{c^2}$ at the final step yields $c^2$ instead of $c$.
6. Using degrees vs. radians incorrectly: Ensure your calculator or function matches the expected unit.
7. Mixing measurement units: Mixing inches with feet or centimeters with meters leads to wrong answers.
8. Using linear units for area: Writing $\text{cm}$ instead of $\text{cm}^2$ for area measurements.
9. Rounding too early: Truncating intermediate values during multi-step calculations causes accuracy drift.
10. Entering impossible dimensions: Attempting to set leg $a \ge c$ violates right triangle existence.

Our right triangle calculator maintains 64-bit double-precision floating-point accuracy internally throughout all intermediate operations.

Calculated results display up to 4 decimal places for general clarity, while exact radical forms (such as $\sqrt{2}$ or $3\sqrt{5}$) are presented alongside decimal outputs whenever applicable.