How to use the right triangle calculator
- Enter exactly two values and leave the rest blank. You can use the legs a and b, the hypotenuse c, the angles α and β, or the area.
- Angle α is opposite leg a and angle β is opposite leg b. Both are in degrees and must be less than 90°, because the third angle is the right angle.
- Read the missing sides at the top, then the angles, area, perimeter and height to the hypotenuse, with the working underneath.
Two angles on their own aren't enough: they set the shape but not the size, so the calculator asks for a side or the area too.
The formulas
The legs a and b meet at the right angle, and the hypotenuse c is the long side opposite it. With two sides, Pythagoras gives the third and the inverse tangent gives the angles. With one side and one angle, the sine, cosine and tangent give the other two sides. With the area and a leg, the other leg is 2 × area ÷ leg. With the area and the hypotenuse, the calculator solves a + b and a − b together, since (a + b)² = c² + 4 × area and (a − b)² = c² − 4 × area.
Worked example
The calculator starts with the legs a = 3 and b = 4.
- Hypotenuse: c = √(3² + 4²) = √(9 + 16) = √25, so c = 5.
- Angle α = arctan(3 ÷ 4) ≈ 36.8699°, and β = 90° − 36.8699° ≈ 53.1301°.
- Area = ½ × 3 × 4 = 6, and the perimeter is 3 + 4 + 5 = 12.
- Height to the hypotenuse = 3 × 4 ÷ 5 = 2.4.
The inscribed circle has radius (3 + 4 − 5) ÷ 2 = 1, and the circle through the three corners has radius 5 ÷ 2 = 2.5. The hypotenuse is always a diameter of that circle.
Special right triangles
Two shapes come up so often that their side ratios are worth knowing:
| Triangle | Angles | Sides (short : long : hypotenuse) |
|---|---|---|
| Half a square | 45°, 45°, 90° | 1 : 1 : √2 ≈ 1.4142 |
| Half an equilateral triangle | 30°, 60°, 90° | 1 : √3 ≈ 1.7321 : 2 |
| 3-4-5 | 36.87°, 53.13°, 90° | 3 : 4 : 5 |
So a hypotenuse of 10 with a 30° angle gives legs of 5 and 5√3 ≈ 8.6603. A square with sides of 1 meter has a diagonal of about 1.4142 meters.
Real-world uses
- Ladders. A 4 m ladder with its foot 1 m from the wall reaches √(16 − 1) ≈ 3.873 m up and leans at about 75.5° to the ground.
- Ramps and roofs. The rise and run are the legs, the sloping surface is the hypotenuse, and the angle tells you the pitch. The slope calculator gives the same angle from two points.
- Screens. A TV's size is its diagonal. A 16 : 9 screen with a 55-inch diagonal is about 47.9 by 27 inches.
Frequently asked questions
Which side is the hypotenuse?
It is the side opposite the right angle and always the longest of the three. If the value you enter as c is not longer than both legs, the calculator tells you, because no right triangle can have that shape.
Can I enter the area and an angle?
Yes. With area K and angle α, the legs are b = √(2K ÷ tan α) and a = b × tan α. For an area of 24 and α = 36.8699°, that gives the 6-8-10 triangle.
What if my triangle has no right angle?
Use the triangle calculator, which solves any triangle from three sides or angles with the laws of sines and cosines.
Why are there two answers for the area and hypotenuse?
There aren't really: the two legs simply swap places. A hypotenuse of 10 and an area of 24 give legs of 8 and 6, and it doesn't matter which one you call a. The calculator puts the longer leg first.
Related calculators
- Pythagorean Theorem Calculator — Find the hypotenuse or a missing leg of a right triangle with a² + b² = c², shown step by step.
- Triangle Calculator — Solve any triangle from three sides or angles: missing sides and angles, area, perimeter, heights and radii.
- Slope Calculator — Slope of a line from two points, or its equation from a point and a slope, with intercepts, angle and distance.
- Area Calculator — Area of a square, rectangle, triangle, circle, ellipse, trapezoid, parallelogram, sector or regular polygon.
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