Triangle Calculator
Solve for unknown sides, angles, and the area of a triangle from known values using trigonometric rules and the law of cosines.
Solve for unknown sides, angles, and the area of a triangle from known values using trigonometric rules and the law of cosines.
To fully solve a triangle you generally need three pieces of information, at least one of which is a side length. Common valid combinations include three sides (SSS), two sides and the included angle (SAS), two angles and a side (ASA or AAS), and in some cases two sides and a non-included angle (SSA), which can have more than one solution. With three angles alone you can determine the shape but not the size. The calculator detects which case you have entered and applies the appropriate method.
The most familiar formula is one-half times the base times the height. When you do not know the height, other methods apply: Heron's formula computes area from the three side lengths, and the formula one-half times two sides times the sine of the included angle works when you know two sides and the angle between them. This calculator automatically selects the correct area formula based on the values you provide, so you do not need to remember which one to use.
These are the two key relationships for non-right triangles. The law of sines states that the ratio of each side to the sine of its opposite angle is constant, which is useful when you know certain angle-side pairs. The law of cosines generalises the Pythagorean theorem and relates all three sides to one angle, which is ideal when you know two sides and the included angle or all three sides. The calculator uses whichever law fits your inputs.
Yes. Right triangles are simply a special case where one angle is 90 degrees, so all the general triangle methods apply. For right triangles you can also use the Pythagorean theorem and basic trigonometric ratios directly. If you enter values consistent with a right triangle, the calculator will return the remaining sides, angles, and area just as it does for any other triangle.
In flat, Euclidean geometry, the three interior angles of any triangle always sum to exactly 180 degrees. This is a fundamental property that the calculator uses to find a missing angle once two are known. It holds for every triangle regardless of size or shape. The rule changes only in non-Euclidean geometries, such as on the curved surface of a sphere, which are outside the scope of standard triangle calculations.