📐 Right Triangle Calculator
Pythagorean theorem solver: find any missing side, angles, area, perimeter.
Enter any TWO sides; leave the unknown blank.
Missing side
How to Use the Right Triangle Calculator
Enter any two sides of a right triangle and leave the third blank. The calculator finds the missing side with the Pythagorean theorem and also reports both acute angles, area, and perimeter.
Enter two known sides. Leave exactly one field blank: that is the side to find.
Calculate. The missing side, angles, area, and perimeter appear with the working shown.
Verify the answer satisfies a² + b² = c² using the check line.
The Pythagorean Theorem
In a right triangle the square of the hypotenuse equals the sum of the squares of the legs: a² + b² = c². Finding the hypotenuse means c = √(a² + b²). Finding a leg means rearranging: a = √(c² − b²). The theorem applies only to right triangles; for general triangles use the Law of Cosines (see our triangle solver). One of the oldest theorems in mathematics, it appears in construction (checking square corners with 3-4-5), navigation, and computer graphics distance calculations.
Angles, Area, and Perimeter
Once all three sides are known, the acute angles come from inverse trigonometry: angle A = arctan(a/b), and angle B = 90° − A. Area is simply (a × b) / 2 since the legs form the base and height. Perimeter is a + b + c. These follow automatically, so one calculation gives you the complete triangle.
Common Right Triangles
Memorize the classic triples and many problems solve themselves: 3-4-5 (and multiples like 6-8-10), 5-12-13, 8-15-17, 7-24-25, 9-40-41. Special right triangles: the 45-45-90 isosceles right triangle has legs in ratio 1:1:√2, and the 30-60-90 triangle has sides in ratio 1:√3:2. If your numbers are close to one of these, check whether the problem was designed around it.
Frequently Asked Questions
What is the Pythagorean theorem?
For any right triangle, a² + b² = c², where c is the hypotenuse (the side opposite the right angle). If you know any two sides, you can find the third: c = √(a² + b²), a = √(c² − b²). It only works when one angle is exactly 90°.
How do I find the hypotenuse?
Square both legs, add them, and take the square root. For legs 3 and 4: 3² + 4² = 9 + 16 = 25, √25 = 5. The famous 3-4-5 triangle is the simplest integer example. Enter the two legs above and choose "Hypotenuse" as the unknown.
How do I find a missing leg?
Rearrange the theorem: leg = √(hypotenuse² − known leg²). For hypotenuse 13 and leg 5: √(169 − 25) = √144 = 12. Note the hypotenuse must be the longest side; if your numbers make the radicand negative, the triangle is impossible.
What are Pythagorean triples?
Integer sets satisfying a² + b² = c², like (3,4,5), (5,12,13), (8,15,17), and (7,24,25). Multiples work too: (6,8,10) is just 3-4-5 doubled. Recognizing triples lets you solve many textbook problems without a calculator.
How do I find the angles of a right triangle?
Use inverse trig: if you know the legs, angle A = arctan(opposite/adjacent). The two acute angles always sum to 90°. This calculator computes both acute angles automatically from whichever sides you enter.