Triangle calculator
Solve any triangle from three values
Enter any three values, including at least one side. Typing into a fourth hands the oldest back to the calculator.
Side a is opposite angle A, and so on. Use any length unit, the same for all three sides.
Result
Your answer appears here as you type.
About this tool
Enter any three sides and angles to solve the rest — including the ambiguous case where two different triangles fit — with area, perimeter, heights and the inscribed and circumscribed circles.
How the triangle is solved
Three values fix a triangle as long as one of them is a side. Three angles alone fix only its shape, not its size. The calculator works out which case you have and solves it with two rules that hold for any triangle, not just right triangles:
- Law of sines: a ÷ sin A = b ÷ sin B = c ÷ sin C.
- Law of cosines: c² = a² + b² − 2ab × cos C, which becomes the Pythagorean theorem when C is 90°.
- The angles always add up to 180°.
The ambiguous case
Two sides and an angle that is not between them (SSA) can fit no triangle, one, or two different ones. Example: with A = 30°, a = 6 and b = 10, the law of sines gives sin B = 10 × sin 30° ÷ 6, and two angles have that sine: B = 56.44° or 123.56°. Both work, so there are two triangles: one with C = 93.56° and c ≈ 11.98, and one with C = 26.44° and c ≈ 5.34.
Many calculators quietly return only the first. This one shows both and lets you switch between them, because a drawing or a site measurement usually tells you which one you have.
What else it gives you
The area (by Heron’s formula from the three sides), the perimeter, the height onto each side, and the radii of the inscribed and circumscribed circles, along with whether the triangle is acute, right or obtuse and whether it is equilateral, isosceles or scalene. To measure the angles of a real triangle first, use the protractor or photo measurement.
Frequently asked questions
How do you find the missing side of a right triangle?
Use the Pythagorean theorem, a² + b² = c², where c is the hypotenuse: the longest side, opposite the right angle. To find the hypotenuse, add the squares of the other two sides and take the square root, so sides of 3 and 4 give √(9 + 16) = 5.
To find one of the shorter sides, subtract instead: b = √(c² − a²). If you know a side and an angle rather than two sides, use sine, cosine or tangent, or enter the three values above and the calculator solves the rest, as with sides 3 and 4 and a right angle.
How do you find the angles of a right triangle?
The three angles of any triangle add up to 180°, so in a right triangle the other two add up to 90°. Find one of them with inverse trigonometry and subtract it from 90° to get the other.
Use whichever two sides you know: angle = arctan(opposite ÷ adjacent), arcsin(opposite ÷ hypotenuse) or arccos(adjacent ÷ hypotenuse). In a 3-4-5 triangle, arctan(3 ÷ 4) ≈ 36.87°, so the other angle is 53.13°.
How do you find the area of a triangle?
Multiply the base by the height and halve it: area = ½ × base × height, with the height measured at right angles to the base. A triangle with a 10 ft base and a 6 ft height covers 30 square feet. Measure in feet to get square feet, or in metres to get square metres.
If you know the three sides but not the height, use Heron’s formula, or enter the sides above: the calculator gives the area, the perimeter and all three heights. Working backwards, a height is twice the area divided by its base.
Guides
About these measurements
Readings are estimates taken from rendered pixels and consumer device sensors. Do not rely on them for structural engineering, surveying, manufacturing inspection, navigation, or medical decisions. Check anything safety-critical with a calibrated instrument.
All processing happens in your browser. No image, camera frame or sensor reading is ever uploaded.
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