Right Triangle Calculator: Sides, Angles, Area
Enter any two values of a right triangle (two sides, or one side and one acute angle) and get the missing sides, both angles, area, perimeter, height to the hypotenuse, inradius and circumradius, with the working shown.
Right triangle calculator
Fill in exactly two boxes. Side a is opposite angle A, side b is opposite angle B, and c is the hypotenuse.
How the calculator works
A right triangle has one 90° angle (C). The side opposite it is the hypotenuse (c), and the other two sides are the legs (a and b). Knowing any two independent values fixes the whole triangle.
| You enter | Method used |
|---|---|
| a and b | c = √(a² + b²) |
| a and c (or b and c) | b = √(c² − a²); c must be greater than a |
| A side and angle A | a = c · sin A, b = c · cos A, a = b · tan A |
| A side and angle B | A = 90° − B, then the same ratios |
Formulas for the extra results
- Area = ½ × a × b
- Perimeter = a + b + c
- Height to the hypotenuse = (a × b) ÷ c
- Inradius = (a + b − c) ÷ 2
- Circumradius = c ÷ 2
- Angle A = tan⁻¹(a ÷ b), and angle B = 90° − A
Worked examples
Example 1: two legs (3 and 4)
c = √(3² + 4²) = √25 = 5. Angle A = tan⁻¹(3 ÷ 4) = 36.87°, so B = 53.13°. Area = ½ × 3 × 4 = 6, perimeter = 12, height to the hypotenuse = 12 ÷ 5 = 2.4, inradius = (3 + 4 − 5) ÷ 2 = 1 and circumradius = 2.5. Because 3, 4 and 5 are whole numbers, this is a Pythagorean triple.
Example 2: one side and an angle (a = 7, A = 30°)
c = 7 ÷ sin 30° = 14 and b = 7 ÷ tan 30° ≈ 12.124. Angle B = 60°. This is a 30-60-90 triangle, where the hypotenuse is twice the shortest side.
Special right triangles
| Type | Angles | Side ratio a : b : c |
|---|---|---|
| Isosceles right | 45°, 45°, 90° | 1 : 1 : √2 |
| 30-60-90 | 30°, 60°, 90° | 1 : √3 : 2 (a opposite the 30° angle) |
| 3-4-5 (Pythagorean triple) | about 36.87°, 53.13°, 90° | 3 : 4 : 5 |
Common mistakes
- Mixing up the sides. Side a is opposite angle A, not next to it.
- Calculator set to the wrong angle mode when you check results by hand. Check degrees versus radians.
- Entering three values. They can contradict each other, so this tool asks for exactly two.
- Rounding too early. Keep extra decimals in your working and round at the end.
For real-world jobs such as framing, roofing or cutting material, measure twice and check critical results independently.
Frequently asked questions
- Which two values can I enter?
- Any two values as long as at least one is a side: two legs, a leg and the hypotenuse, or a side and one acute angle. Two angles alone are not enough, because they only fix the shape, not the size.
- Why does it say the hypotenuse must be longer than a leg?
- In a right triangle the hypotenuse is always the longest side. If c is not greater than a (or b), no right triangle exists with those lengths.
- What are the inradius and circumradius?
- The inradius is the radius of the largest circle that fits inside the triangle: (a + b − c) ÷ 2 for a right triangle. The circumradius is the radius of the circle through all three corners, and it equals half the hypotenuse.
- Are the results exact?
- Calculations use standard double-precision arithmetic and are rounded only for display. Choose more decimal places if you need them, and round at the final step of your own work.
- Can I enter angles in radians?
- Yes. Switch the angle unit to radians, then enter angle A or B in radians. Results show angles in both degrees and radians.