Right Triangle Hypotenuse Calculator

The hypotenuse of a right triangle is defined as the side opposite the $90^\circ$ right angle. It possesses several fundamental geometric characteristics:

• Longest Side: It is always strictly longer than either individual leg ($c > a$ and $c > b$).
• Opposite the Right Angle: It faces the perpendicular corner where leg $a$ and leg $b$ meet at a right angle.
• Connects Non-Right Vertices: It links the two acute angle vertices of the right triangle.
• Standard Notation: In mathematical literature and formulas, the hypotenuse length is conventionally represented by the variable $c$.
• Perpendicular Base Legs: The remaining two sides forming the $90^\circ$ corner are called legs ($a$ and $b$).

The primary mathematical formula used to calculate the hypotenuse of a right triangle when both leg lengths are known is:

Where:

• $c$ = length of the hypotenuse
• $a$ = length of the first perpendicular leg
• $b$ = length of the second perpendicular leg

This equation is derived directly by taking the principal square root of both sides of the standard Pythagorean relationship:

When both leg values $a$ and $b$ are known positive real numbers, squaring each leg, adding the squares together, and taking the square root yields the exact length of hypotenuse $c$. For a broader exploration of right-triangle side relationships, visit our Pythagorean theorem calculator.

To calculate the hypotenuse of a right triangle from two known legs by hand, follow these seven step-by-step instructions:

1. Identify the two leg lengths: Label the two perpendicular side lengths as $a$ and $b$.
2. Confirm the right angle: Verify that the triangle contains a $90^\circ$ right angle between legs $a$ and $b$.
3. Square the first leg: Multiply leg $a$ by itself to find $a^2$.
4. Square the second leg: Multiply leg $b$ by itself to find $b^2$.
5. Sum the squared values: Add $a^2$ and $b^2$ together ($a^2 + b^2$).
6. Take the square root: Calculate the principal square root of the sum ($c = \sqrt{a^2 + b^2}$).
7. Report with linear units: State the final hypotenuse length using the same linear measurement unit as the legs.

Because the two legs of a right triangle meet at a fixed $90^\circ$ angle, knowing both leg lengths provides complete geometric information required to construct the triangle.

Depending on your measurement inputs, hypotenuse calculations behave as follows:

• Equal Legs (Isosceles Right Triangle): When $a = b$, the formula simplifies to $c = \sqrt{2a^2} = a\sqrt{2}$.
• Unequal Legs: When leg lengths differ, calculate $c = \sqrt{a^2 + b^2}$ using full precision.
• Integer Measurements: Legs given as whole integers may produce whole integer hypotenuse results (Pythagorean triples) or radical expressions.
• Decimal & Fractional Measurements: Leg inputs containing decimals or fractions follow identical squared operations ($c = \sqrt{a^2 + b^2}$).
• Compatible Units: Ensure both legs are measured in identical units before performing addition under the square root.

When calculating the hypotenuse of a right triangle, results can be expressed in exact radical notation or rounded decimal format:

• Exact Radical Result: Unrounded mathematical form preserving full precision using square roots, such as $\sqrt{50}$.
• Simplified Radical Result: Factoring perfect squares from under the radical, simplifying $\sqrt{50}$ to $5\sqrt{2}$.
• Decimal Approximation: Converting the radical into a floating-point number, such as $7.0711$.
• Integer Results: Occur exclusively when the leg inputs form a Pythagorean triple (e.g. $3^2 + 4^2 = 25 \implies \sqrt{25} = 5$).

In practical applications like woodworking or engineering, decimal approximations rounded to 3 or 4 decimal places provide actionable real-world measurements, whereas academic geometry prefers exact simplified radicals.

For any physically valid right triangle with positive leg lengths ($a > 0$ and $b > 0$), the calculated hypotenuse $c$ must satisfy two fundamental inequalities:

Furthermore, the triangle inequality theorem requires that the sum of any two side lengths must strictly exceed the third side length ($a + b > c$).

Example Check: If $a = 3$ and $b = 4$, then $c = 5$. Notice that $5 > 3$, $5 > 4$, and $3 + 4 = 7 > 5$. If a calculation yields a hypotenuse equal to or shorter than either leg, an error has occurred.

To verify whether a calculated hypotenuse value is mathematically correct, substitute all three side lengths back into the Pythagorean equation:

Because both sides of the equation balance perfectly, the hypotenuse calculation of $13$ is verified as 100% accurate.

The calculated hypotenuse inherits the exact same linear unit of measurement as the legs used in the calculation. Supported standard linear units include:

• Metric Units: Millimeters ($\text{mm}$), Centimeters ($\text{cm}$), Meters ($\text{m}$), Kilometers ($\text{km}$)
• Imperial Units: Inches ($\text{in}$), Feet ($\text{ft}$), Yards ($\text{yd}$)

Unit Compatibility Rule: Both leg inputs must be converted into the same unit before performing squaring or square root operations. Never add squared quantities from different units without converting first.

No. Knowing the length of only one leg is not sufficient to calculate a unique hypotenuse. Infinitely many right triangles can be constructed sharing a single leg length.

To solve for the hypotenuse, you require at least one additional piece of geometric information:

• The length of the second leg ($b$) — use this hypotenuse calculator.
• One acute interior angle ($\theta$) — use our SOHCAHTOA calculator or angle calculator.
• The triangle's area or perimeter — use our right triangle side calculator.

• Using the formula on non-right triangles: Applying $c = \sqrt{a^2 + b^2}$ to triangles without a $90^\circ$ angle.
• Misidentifying legs and hypotenuse: Treating the hypotenuse as a leg input in the formula.
• Adding before squaring: Adding $a + b$ first and then squaring, i.e., $(a + b)^2 \neq a^2 + b^2$.
• Forgetting the square root: Leaving the answer as $c^2 = a^2 + b^2$ without taking the final square root $\sqrt{a^2 + b^2}$.
• Mixing measurement units: Adding inches squared to feet squared without unit normalization.
• Premature rounding: Rounding intermediate squared values prior to taking the square root.
• Entering negative or zero leg lengths: Inputting non-physical lengths ($a \le 0$ or $b \le 0$).

The hypotenuse formula $c = \sqrt{a^2 + b^2}$ can be applied when:

• The geometric figure is strictly a $90^\circ$ right triangle.
• Both perpendicular leg lengths ($a$ and $b$) are known positive real numbers.
• All inputs are measured in compatible linear units.

When NOT to use this specific calculator: Do not use this tool if your triangle is oblique (acute or obtuse), if you are solving for a missing leg given hypotenuse $c$, or if you need to calculate acute angles using trigonometric ratios.