Right Triangle Calculator
Calculate the sides, angles, area, and other properties of a right triangle.
Understanding Right Triangle Calculator
A right triangle is a triangle containing one angle measuring exactly 90°. The side opposite this right angle is called the hypotenuse, while the other two sides are called the legs. The hypotenuse is always the longest side of a right triangle.
This calculator determines the missing side of a right triangle when two sides are known. It can also accept all three sides and verify whether they satisfy the Pythagorean theorem. In addition to the side lengths, it calculates the two acute angles, area, perimeter, semiperimeter, and altitude to the hypotenuse.
Making the inputs meaningful
Right Triangle Calculator is intended to show the mathematical relationship behind the answer, not just return a number. Enter the quantities requested by the selected method—such as Side A, Side B, Hypotenuse C—and keep their units or mathematical meaning consistent. Where this calculator offers different shapes, methods, or forms, choose the one that matches the problem before entering the values.
Why the inputs matter
For a useful check, look at the units and the scale of the answer before accepting it. A formula can be evaluated correctly while an input is still inappropriate for the problem. Repeating the calculation with a simple test value or checking the result against the expected direction of change can help identify an input mistake.
Right Triangle Calculator
A right triangle is a triangle containing one angle measuring exactly 90°. The side opposite this right angle is called the hypotenuse, while the other two sides are called the legs. The hypotenuse is always the longest side of a right triangle.
This calculator determines the missing side of a right triangle when two sides are known. It can also accept all three sides and verify whether they satisfy the Pythagorean theorem. In addition to the side lengths, it calculates the two acute angles, area, perimeter, semiperimeter, and altitude to the hypotenuse.
Pythagorean Theorem
The fundamental relationship between the sides of a right triangle is the Pythagorean theorem. If a and b are the legs and c is the hypotenuse, then:
When the two legs are known, the hypotenuse can be calculated with:
When the hypotenuse and one leg are known, the other leg can be found by rearranging the theorem.
b = √(c² - a²)
Area of a Right Triangle
The two legs of a right triangle are perpendicular to each other, so either leg can be used as the base while the other serves as the corresponding height.
The area is expressed in square units. For example, if the legs measure 3 and 4 units, the area is 6 square units.
Perimeter
The perimeter is the total distance around the triangle and is found by adding all three side lengths.
Angles of a Right Triangle
One angle is always 90°. Since the three interior angles of every triangle add to 180°, the two remaining acute angles must add to 90°.
The acute angles can be determined from the side ratios using inverse trigonometric functions. For example:
B = sin⁻¹(b / c)
Altitude to the Hypotenuse
A right triangle also has an important altitude drawn from the right angle to the hypotenuse. Its length can be calculated from the two legs and the hypotenuse.
This altitude divides the original right triangle into two smaller right triangles and is useful in several geometric relationships.
Trigonometric Ratios
Right triangles are closely connected to the basic trigonometric functions sine, cosine, and tangent. For an acute angle, these ratios relate the opposite side, adjacent side, and hypotenuse.
cos(θ) = adjacent / hypotenuse
tan(θ) = opposite / adjacent
Checking a Right Triangle
When all three side lengths are supplied, they can be checked using the Pythagorean theorem. The two legs must satisfy a² + b² = c², and the hypotenuse must be longer than either leg.
The calculator performs this verification automatically. If the measurements do not satisfy the required relationship, the calculator reports that the supplied dimensions cannot form the specified right triangle.
How to Use the Calculator
Enter any two known side lengths and leave the missing side empty to have the calculator determine it. You can also enter all three sides when you want to verify whether they form a valid right triangle.
The result provides the complete triangle measurements, calculation steps for a missing side when applicable, angle measurements, geometric properties, trigonometric ratios, and a Pythagorean verification.
Disclaimer
This calculator is intended for informational purposes only. While we strive for accuracy, we cannot guarantee that every result is correct for every situation.