Angle of Right Triangle Calculator
Enter any two known measurements of a right triangle and this calculator will find both acute angles, all three side lengths, the altitude to the hypotenuse, and the area and perimeter. Choose your solve mode: two legs, one leg plus the hypotenuse, one side plus area, or one side plus a known angle. Results update instantly with a labeled triangle diagram and a full show-your-work panel.
How to find the angles of a right triangle
A right triangle has exactly one 90-degree angle. The other two angles are called acute angles and they always add up to 90 degrees, making them complementary. Once you know one acute angle you can find the other by subtracting from 90. To calculate angle alpha from two sides, pick the appropriate inverse trig function: alpha = arctan(a/b) when you have both legs, alpha = arcsin(a/c) when you have leg a and the hypotenuse, and alpha = arccos(b/c) when you have leg b and the hypotenuse. This calculator handles all three cases automatically, plus two additional modes using area.
The four solve modes explained
Mode 1 - Both legs: enter a and b. The calculator uses alpha = arctan(a/b) and the Pythagorean theorem (c = sqrt(a^2 + b^2)) for all remaining values. Mode 2 - One leg plus hypotenuse: the missing leg is recovered first (b = sqrt(c^2 - a^2)), then the angle follows. Mode 3 - One side plus area: the area formula A = (1/2) x a x b rearranges to b = 2A/a, giving the second leg, and angle calculation proceeds as normal. Note that only hypotenuse plus area is an unsolvable combination, because many different right triangles can share the same hypotenuse and area. Mode 4 - One side plus a known angle: b = a / tan(alpha) or a = b x tan(alpha) recovers the missing leg.
Altitude, area, and perimeter of a right triangle
The altitude h drawn from the right-angle vertex to the hypotenuse equals (a x b) / c. It splits the hypotenuse into two segments, each of which forms a smaller right triangle similar to the original. The area is simply half the product of the two legs: A = (1/2) x a x b. This works because the two legs themselves are the base and height. The perimeter is the sum of all three sides: P = a + b + c. These quantities are all computed automatically so you have everything needed for a geometry problem in one place.
Special right triangles and Pythagorean triples
Two angle families appear constantly in geometry. The 45-45-90 triangle is isosceles: both legs are equal and the hypotenuse is leg x sqrt(2). It arises whenever you cut a square along its diagonal. The 30-60-90 triangle has legs in the ratio 1 : sqrt(3) and a hypotenuse twice the shorter leg; it appears when you bisect an equilateral triangle. Pythagorean triples are integer-sided right triangles. The smallest and most common is 3-4-5 (angles approximately 36.87 and 53.13 degrees). Multiples such as 6-8-10 and 9-12-15 also satisfy the Pythagorean theorem. The reference table above lists the most frequently tested triples.
Common special right triangles
| Type | Angle alpha | Angle beta | Leg a : Leg b : Hypotenuse c |
|---|---|---|---|
| 45-45-90 (isosceles) | 45 deg | 45 deg | 1 : 1 : sqrt(2) approx 1.414 |
| 30-60-90 | 30 deg | 60 deg | 1 : sqrt(3) : 2 (approx 1 : 1.732 : 2) |
| 3-4-5 Pythagorean triple | 36.87 deg | 53.13 deg | 3 : 4 : 5 |
| 5-12-13 Pythagorean triple | 22.62 deg | 67.38 deg | 5 : 12 : 13 |
| 8-15-17 Pythagorean triple | 28.07 deg | 61.93 deg | 8 : 15 : 17 |
| 7-24-25 Pythagorean triple | 16.26 deg | 73.74 deg | 7 : 24 : 25 |
These ratios appear frequently in geometry and trigonometry. Memorising them saves time on exams and in practical work.
Frequently asked questions
What formula finds the angle of a right triangle?
It depends which two values you know. With both legs a and b, use alpha = arctan(a/b). With leg a and hypotenuse c, use alpha = arcsin(a/c). With leg b and hypotenuse c, use alpha = arccos(b/c). Once you have one acute angle, the other is 90 minus that value because the two acute angles in a right triangle always sum to 90 degrees.
Can I find the angles knowing only one side?
No. Knowing one side tells you the shape's scale but not its angles. You need at least two pieces of information, either two side lengths or one side length combined with a second measurement such as the other leg, the hypotenuse, the area, or one of the angles.
Why can't I solve the triangle from just the hypotenuse and the area?
Infinitely many right triangles can have the same hypotenuse length and the same area while differing in leg lengths and angles. The two constraints are not independent enough to pin down a unique triangle, so the system is underdetermined for that combination.
How do I convert degrees to radians?
Multiply the angle in degrees by pi/180. For example, 45 degrees = 45 x (pi/180) = pi/4 approximately 0.7854 radians. To go the other way, multiply radians by 180/pi. This calculator displays angles in degrees, which is the most common format in everyday geometry and construction work.
What are the angles of a 3-4-5 right triangle?
The 3-4-5 right triangle has angles of 90 degrees (the right angle), approximately 36.87 degrees (opposite the side of length 3), and approximately 53.13 degrees (opposite the side of length 4). You can verify this with alpha = arctan(3/4) approximately 36.87 and beta = 90 - 36.87 approximately 53.13.
What is the altitude to the hypotenuse?
The altitude h is the perpendicular line segment drawn from the right-angle vertex to the hypotenuse. Its length is h = (a x b) / c, where a and b are the legs and c is the hypotenuse. It divides the original triangle into two smaller right triangles, each similar to the original and to each other. This altitude is useful for computing areas and for proving geometric relationships.