Triangle Calculator – Solve Sides, Angles, Area & Hypotenuse

A triangle calculator starts with the measurements you actually know and chooses a relationship that is valid for that information. For example, two perpendicular legs define a right triangle and allow the Pythagorean theorem; three sides use SSS relationships and Heron’s formula; two sides with the included angle use the Law of Cosines and the sine-area formula.

The important point is that there is no single formula that solves every triangle. A good solver first checks whether the given data uniquely determines a triangle, validates the inputs, then calculates the missing sides, angles, area, perimeter and useful heights. Learn Mode explains why that particular theorem applies rather than simply displaying numbers.

A right triangle has one 90° angle. If two side lengths are known, the Pythagorean theorem a² + b² = c² can find the missing side. If one acute angle and one side are known, sine, cosine or tangent can relate the angle to the opposite, adjacent and hypotenuse sides.

After the sides are known, the area is ½ × leg × leg because the two legs are perpendicular. The two acute angles must add to 90°. A common mistake is to call a side the hypotenuse when it is not opposite the 90° angle; the hypotenuse is always the longest side.

If the two perpendicular legs are a and b, use c = √(a² + b²). For a 3–4–5 triangle, c = √(3² + 4²) = √25 = 5. This works only for right triangles.

The hypotenuse is the side opposite the right angle, so it must be longer than either leg. If your computed value is shorter than a leg, the inputs or the chosen relationship are wrong. The calculator also shows the angles and area so you can check the geometry from more than one result.

The method depends on what else is known. If two angles are known, use the fact that the interior angles of a Euclidean triangle total 180°. If the sides are known, the Law of Cosines can be rearranged to find an angle. In a right triangle, trigonometric ratios such as sin, cos and tan can find an acute angle from side ratios.

Always make sure your calculator is using degrees if the problem is written in degrees. A useful check is that all three final angles are positive and add to 180°.

With three valid side lengths, the triangle is fixed by the SSS condition. First verify the triangle inequality: each pair of sides must add to more than the remaining side. Then use the Law of Cosines to find the angles and Heron’s formula to find the area.

Heron’s formula uses the semiperimeter s = (a+b+c)/2 and A = √[s(s−a)(s−b)(s−c)]. Learn Mode shows both the angle calculation and the area derivation so the answer is not a black box.

This is the SAS case. If sides a and b and the angle C between them are known, the Law of Cosines gives the third side: c² = a² + b² − 2ab cos C. The area can be found directly from A = ½ab sin C.

Once c is known, the remaining angles can be calculated and checked so that A+B+C = 180°. The word included matters: C must be the angle between the two known sides. Using a non-included angle with the SAS formula can produce an invalid result.

Two angles immediately determine the third because A+B+C = 180°. Once all angles are known and one side is known, the Law of Sines relates every side to its opposite angle: a/sin A = b/sin B = c/sin C.

This is an ASA or AAS situation and normally gives one unique triangle. A common mistake is pairing a side with the wrong opposite angle. Learn Mode explicitly identifies which side is opposite which angle before performing the ratio calculation.

The most familiar formula is A = ½bh, where h is the perpendicular height to the chosen base. But a triangle solver can use other equivalent methods when height is not directly known. With two sides and their included angle, A = ½ab sin C; with three sides, Heron’s formula can be used.

The perpendicular requirement is important. A sloping side is not automatically the height. The calculator selects an area method that matches the information actually available.

If the area A and a base b are known, rearrange A = ½bh to obtain h = 2A/b. If all three sides are known, first compute the area with Heron’s formula and then use h = 2A/b for the chosen base.

A triangle has three possible altitudes—one to each side—so always state which side is being used as the base. The calculator reports height to a defined side when the solved geometry permits it.

In ordinary Euclidean geometry, draw a line through one vertex parallel to the opposite side. The other two interior angles appear as alternate interior angles along that straight line. Together with the angle at the vertex, they form a straight angle of 180°.

This fact is one of the most useful checks in triangle solving. If three calculated angles do not total approximately 180° after allowing for rounding, something in the calculation or inputs needs review.

The triangle inequality says that the sum of any two side lengths must be greater than the third side. For example, sides 2, 3 and 6 cannot form a triangle because 2+3 is not greater than 6.

This is not merely a formula rule; it is a geometric necessity. If two shorter sides cannot reach each other when joined at their endpoints, the triangle cannot close. The calculator checks this before solving SSS data.