Right Triangle Calculator
- Updated
- Formula shown
- Free, no sign-up
Results
Right triangle with legs 3 and 4, hypotenuse 5, angles 36.87° and 53.13°, area 6 units²
How to use the right triangle calculator
- Type exactly two known values into any of the six boxes. Leave the others blank, or press Clear values to start again.
- At least one value must be a length or the area. Two angles alone fix the shape but not the size.
- Angles are in degrees. Angle α sits opposite leg a, and angle β opposite leg b; together they always make 90°.
- Pick a length unit if you want the results labeled. The numbers themselves do not change.
Right triangle formulas
sin α = a ÷ c · cos α = b ÷ c · tan α = a ÷ b
α + β = 90°
Area = a × b ÷ 2 · Altitude to hypotenuse h = a × b ÷ c
Inradius r = (a + b − c) ÷ 2 · Circumradius R = c ÷ 2
In a right triangle the two legs (a and b) meet at the 90° corner and the hypotenuse (c) is the longest side, opposite that corner. The Pythagorean theorem links the three sides, and the trigonometric ratios link a side to an angle: sine is opposite over hypotenuse, cosine is adjacent over hypotenuse and tangent is opposite over adjacent (often remembered as SOH-CAH-TOA). Because the legs are perpendicular, one leg is the height on the other, so the area is simply half their product. The midpoint of the hypotenuse is the same distance from all three corners, which is why the circumradius is exactly half of c.
How each pair of inputs is solved
| You know | First step |
|---|---|
| Both legs a and b | c = √(a² + b²), α = arctan(a ÷ b) |
| A leg and the hypotenuse | Other leg = √(c² − a²) |
| Hypotenuse and an angle | a = c × sin α, b = c × cos α |
| A leg and an angle | b = a ÷ tan α, or a = b × tan α |
| Area and a leg | Other leg = 2 × Area ÷ leg |
| Area and an angle | b = √(2 × Area ÷ tan α), a = b × tan α |
| Area and the hypotenuse | a, b = (√(c² + 4·Area) ± √(c² − 4·Area)) ÷ 2 |
Worked example
A right triangle has a hypotenuse of 10 and an angle α of 30°. Leg a, opposite α, is 10 × sin 30° = 5, and leg b is 10 × cos 30° = 8.6603. The other acute angle is β = 90° − 30° = 60°. The area is 5 × 8.6603 ÷ 2 = 21.6506, the perimeter is 23.6603, and the altitude from the right angle to the hypotenuse is 5 × 8.6603 ÷ 10 = 4.3301.
A practical version: a 6 m ladder standing at 75° to level ground reaches 5.8 m up the wall, with its foot 1.55 m out from the wall (6 × sin 75° and 6 × cos 75°). Ladder safety guidance commonly recommends about 75°, the familiar "1 out for every 4 up" rule of thumb.
If you know the area and the hypotenuse instead, say an area of 6 and c = 5, the legs come out as 4 and 3: the 3-4-5 triangle. The largest area any right triangle with hypotenuse c can have is c² ÷ 4, when both legs are equal.
Special right triangles
These shapes come up so often that their side ratios are worth knowing. Angles and areas below are computed with the calculator's engine.
| Triangle | Sides a : b : c | Angles α, β | Area | Altitude to c |
|---|---|---|---|---|
| 45-45-90 | 1 : 1 : √2 | 45°, 45° | 0.5 | 0.7071 |
| 30-60-90 | 1 : √3 : 2 | 30°, 60° | 0.866 | 0.866 |
| 3-4-5 | 3 : 4 : 5 | 36.87°, 53.13° | 6 | 2.4 |
| 5-12-13 | 5 : 12 : 13 | 22.62°, 67.38° | 30 | 4.6154 |
| 8-15-17 | 8 : 15 : 17 | 28.07°, 61.93° | 60 | 7.0588 |
| 7-24-25 | 7 : 24 : 25 | 16.26°, 73.74° | 84 | 6.72 |
Tips for right triangle problems
- The hypotenuse is always the longest side. If a leg you entered is longer than c, the values cannot form a right triangle.
- Check your calculator mode. Hand calculations in radians instead of degrees are the most common source of wrong trig answers. This calculator works in degrees.
- Slopes and pitches are right triangles. A rise and a run are the two legs, so the slope angle is arctan(rise ÷ run). The slope calculator does this from two points.
- Not a right angle? Use the general triangle calculator, which handles any triangle from three values. For just the a² + b² = c² working, see the Pythagorean theorem calculator.
Frequently asked questions
How do I find the hypotenuse of a right triangle?
Square both legs, add them and take the square root: c = √(a² + b²). With legs of 3 and 4, c = √(9 + 16) = √25 = 5.
How do I find a missing leg of a right triangle?
Subtract the square of the known leg from the square of the hypotenuse and take the square root: b = √(c² − a²). With a hypotenuse of 13 and a leg of 5, the other leg is √(169 − 25) = 12.
How do I find the angles of a right triangle from two sides?
Use an inverse trig function. With legs a and b, α = arctan(a ÷ b) and β = 90° − α. For legs 3 and 4, α ≈ 36.87° and β ≈ 53.13°.
How do I find the sides from the hypotenuse and one angle?
The leg opposite the angle is c × sin α and the adjacent leg is c × cos α. For a hypotenuse of 10 and α = 30°, the legs are 5 and about 8.6603, and the area is about 21.65.
Can I solve a right triangle from its area and hypotenuse?
Yes, as long as the area is no more than c² ÷ 4. An area of 6 with a hypotenuse of 5 gives legs of 4 and 3. The two legs can be swapped, so there are two mirror-image answers with the same measurements.