Right Triangle Calculator: Sides, Angles and Area
Solve a right triangle from any two values, at least one a side. Get the missing sides and angles, the area, the perimeter and the height to the hypotenuse, with each step, free in your browser.
Solved triangle
- Side a
- 3
- Side b
- 4
- Hypotenuse c
- 5
- Angle A
- 36.8698976458°
- Angle B
- 53.1301023542°
- Area
- 6
- Perimeter
- 12
- Height to the hypotenuse
- 2.4
Steps
- c² = a² + b² = 3² + 4² = 9 + 16 = 25
- c = √25 = 5
- A = atan(a / b) = 36.8698976458°
- B = 90° − A = 53.1301023542°
- Area = a × b / 2 = 6
- Perimeter = a + b + c = 12
Right Triangle Calculator
Enter any two known values of a right triangle, with at least one of them a side, and this calculator finds everything else. It gives the two legs a and b, the hypotenuse c, the angles A and B in degrees, the area, the perimeter and the height from the right angle to the hypotenuse. Each step of the working is shown so you can follow it or copy it into homework. The page opens with the classic 3-4-5 triangle.
Naming
Sides and angles
- c is the hypotenuse, the longest side, opposite the right angle.
- a and b are the legs, the two sides that meet at the right angle.
- Angle A is opposite side a and angle B is opposite side b. Together they make 90°.
Formulas
With two sides the calculator uses the Pythagorean theorem for the third and the inverse tangent or sine for the angles. With a side and an angle it uses sine, cosine and tangent. The area is ab / 2, the perimeter a + b + c and the height to the hypotenuse ab / c.
How to use it
Steps
- Type two values and leave the other three fields empty. Angles are in degrees, between 0° and 90°.
- Read the solved triangle: all three sides, both angles, the area, the perimeter and the height.
- Follow the steps to see which formula found each value.
Worked examples
Two legs: a = 3, b = 4
c² = 9 + 16 = 25, so c = 5. A = atan(3 / 4) = 36.8698976458° and B = 53.1301023542°. The area is 6, the perimeter 12 and the height to the hypotenuse 3 × 4 / 5 = 2.4.
Hypotenuse and angle: c = 10, A = 30°
a = 10 × sin 30° = 5 and b = 10 × cos 30° = 8.66025403784, with B = 60°. The area is 21.6506350946 and the perimeter 23.6602540378.
What the calculator refuses
It needs exactly two values. Two angles fix only the shape, not the size, so at least one value must be a side. Sides must be greater than 0, angles must lie strictly between 0° and 90°, and the hypotenuse must be longer than a leg you enter. Each problem shows a message next to the field at fault.
Privacy
Nothing is uploaded; every calculation runs in your browser and never leaves your device.
Related tools
For the area of a triangle without a right angle, from base and height or from three sides, use the triangle area calculator. The Pythagorean Theorem Calculator works with the three sides only, and the Angle Converter turns degrees into radians.
Frequently asked questions
Which two values do I need?
Any two of side a, side b, hypotenuse c, angle A and angle B, as long as at least one is a side. Two angles alone fix only the shape of the triangle, not its size, so the calculator asks for a side.
Which side is the hypotenuse?
The hypotenuse c is the longest side, opposite the right angle. Angle A is opposite side a and angle B is opposite side b. The diagram on the page shows where each one sits, and A + B is always 90°.
How does it find the angles?
From two sides it uses A = atan(a / b), or asin(a / c) when the hypotenuse is known. From a side and an angle it uses sin, cos and tan: a = c × sin A, b = c × cos A and a = b × tan A. For a = 3 and b = 4, A = 36.8698976458°.
What else does it calculate?
The area a × b / 2, the perimeter a + b + c and the height from the right angle to the hypotenuse, a × b / c. For the 3-4-5 triangle those are 6, 12 and 2.4.
Why is my hypotenuse rejected?
The hypotenuse must be longer than either leg. If you enter a = 13 and c = 5, or a = 5 and c = 5, no right triangle exists, and the message says the hypotenuse c must be longer than side a.
Can I use it for triangles without a right angle?
No. For the area of any triangle from base and height or from three sides with Heron's formula, use the triangle page of the area calculator.