Right Triangle Calculator
Based on Pythagorean Theorem ($a^2 + b^2 = c^2$) to calculate the hypotenuse.
What is a Right Triangle?
A triangle where one angle is 90°. The longest side opposite to 90° is called the hypotenuse (b), and the other two sides are the base (a) and height (h).
Pythagorean Theorem
In a right triangle, the square of the hypotenuse equals the sum of the squares of the other two sides.
When base (a) and height (h) are known
When base (a) and hypotenuse (b) are known
When base (a) and angle (θ) are known
When hypotenuse (b) and height (h) are known
Congruence Conditions for Right Triangles
RHA: Hypotenuse and one acute angle are equal
RHS: Hypotenuse and another side are equal
Special Angle Ratios
30°-60°-90° → side ratio: 1 : √3 : 2
45°-45°-90° → side ratio: 1 : 1 : √2
Worked Examples
With legs 3 and 4 the hypotenuse is √(9 + 16) = 5, the acute angles are 36.87° and 53.13°, and the area is 6.
With hypotenuse 10 and one leg 6, the other leg is √(100 − 36) = 8 and the angle opposite it is arcsin(0.8) = 53.13°.
What This Does Not Cover
This page solves right triangles only, the ones with a 90° angle. Oblique triangles need the law of sines or cosines, and two sides plus a non-included angle can describe two different triangles (the ambiguous case), which this tool does not explore. Results are rounded to four decimals and angles are in degrees.
Frequently Asked Questions
- What is the Pythagorean theorem?
- In a right triangle the square of the hypotenuse equals the sum of the squares of the two legs: c² = a² + b². It holds in Euclidean geometry only, and whole-number solutions such as 3-4-5 or 5-12-13 are called Pythagorean triples.
- How do I find an angle from two sides?
- Use the inverse trigonometric functions: sin θ = opposite / hypotenuse, cos θ = adjacent / hypotenuse, tan θ = opposite / adjacent. One acute angle is enough, because the other is 90° − θ.
- Why can a triangle have only one right angle?
- The angles of a triangle add up to 180°. Two right angles would use all of them, so a right triangle always has one 90° angle and two acute angles that are complementary.