Right-Triangle Length: Hypotenuse or Missing Leg

Decide which side is missing

In a right triangle, the hypotenuse c satisfies c² = a² + b² for perpendicular legs a and b. If c is unknown, add the squared legs and take a square root. If a leg is unknown, subtract the other leg’s square from the hypotenuse square before taking the root. The choice of addition or subtraction follows from the labelled equation. It should not be chosen by whether a result looks larger or smaller after entering the numbers.

Find a hypotenuse from two legs

For legs 9 m and 12 m, c = √(81 + 144) = √225 = 15 m. The result is longer than either leg, as a hypotenuse must be. An answer of 21 m from adding the original lengths would instead be the sum of the two legs, not the diagonal. Squaring and then taking the square root are essential steps; they encode the right-angle relationship rather than a simple linear sum of side lengths.

Find a leg from a hypotenuse and another leg

If c is 13 cm and one leg is 5 cm, the other leg is √(13² − 5²) = √144 = 12 cm. Adding the squares would produce a value larger than 13, which cannot be a leg of this right triangle. If the proposed hypotenuse were only 4 cm while a known leg were 5 cm, the expression would be negative. That indicates inconsistent side labels or impossible data in this model, not a real negative length to be reported.

Handle roots that do not simplify to an integer

For perpendicular legs 1 m and 1 m, the hypotenuse is √2 m, approximately 1.41421356 m. Keeping √2 in an exact calculation avoids premature rounding. If the task requests two decimal places, report 1.41 m at the end. A rounded value squared may not return exactly 2; that is expected. Use the unrounded result for subsequent calculations such as perimeter or angle checks when the available precision matters.

Identify the given measurements before choosing a formula

A triangle has three sides and three interior angles, but a question rarely supplies all of them. Mark the given information on a sketch and identify the requested quantity. Side lengths must be positive, and each pair of sides must sum to more than the third for a nondegenerate triangle. Three arbitrary positive numbers therefore do not always describe a possible triangle. Angles in an ordinary Euclidean plane triangle sum to 180 degrees. These conditions check the geometric setup before a calculator evaluates an area, side length or angle.

Keep a perpendicular height distinct from a side

The area formula is one half of a chosen base multiplied by its perpendicular height. That height is the distance from the opposite vertex to the line containing the base, and it may fall outside the triangle in an obtuse case. An adjacent sloping side is not automatically the height. In a right triangle, the two perpendicular legs can serve as base and height. The hypotenuse is the side opposite the right angle and is the longest side; labelling it correctly determines whether Pythagoras requires adding or subtracting squared lengths.

Preserve units and check the geometric result

Use the same length unit for every side before substituting values. Perimeter and side lengths retain that unit, whereas area uses its square. Keep exact roots or unrounded intermediate numbers until the requested final precision. If measured values are approximate, the apparent agreement of a squared relation also needs an appropriate tolerance; displayed equality after rounding is not proof of exact geometry. A calculator result describes the supplied ideal shape. It does not establish that a measured object is perfectly flat, exactly right-angled or consistent with an unverified sketch.

Do not replace a missing length with zero

A blank field identifies an unknown side for the local solver. Zero would instead claim that a side has no length, which cannot form the nondegenerate right triangle described here. Keep the unknown field blank and enter only the two known positive lengths.

Related guides

Find x with Pythagoras: Side or Algebraic Expression

Solve for x when it is a missing side or part of a side expression, reject nonphysical roots, and verify the result in the original right-triangle relation.

Use Pythagoras to Find Perimeter

Calculate a missing boundary side with Pythagoras, then sum the outside edges, and avoid adding internal diagonals or shared edges to the perimeter.

Identify a Right Triangle from Its Sides

Test the longest side with the converse of Pythagoras, distinguish acute and obtuse cases, and explain why a sketch or rounded measurements need care.