Provide any 3 values below — including at least one side — and click Calculate. The calculator finds the remaining sides and angles along with the area, perimeter, heights, medians, inradius, circumradius, and more. Sides are a, b, c and the angle opposite each side is A, B, C respectively.
The triangle calculator on this page solves a triangle completely from just three known measurements. Enter any three values — any combination of sides and angles, as long as at least one is a side — and the calculator instantly determines the three sides, the three angles, and a full set of derived properties: area, perimeter, semiperimeter, the three heights (altitudes), the three medians, the inradius, the circumradius, the coordinates of the vertices, and the locations of the centroid, incenter, and circumcenter. It even draws the triangle to scale and identifies its type.
Triangles are the most fundamental shape in geometry, and "solving" a triangle — finding all of its unknown parts from a few known ones — is a classic problem in trigonometry, surveying, engineering, navigation, and construction. This tool automates the law of sines, the law of cosines, and Heron's formula so you get accurate answers in a single click, without working through the algebra by hand.
Decide which angle unit you are working in — degrees or radians — using the selector at the top. Then fill in exactly three of the six fields (Side a, Side b, Side c, Angle A, Angle B, Angle C) and press Calculate. At least one of your three values must be a side, because angles alone determine a triangle's shape but not its size. The standard labeling convention applies: side a is opposite angle A, side b is opposite angle B, and side c is opposite angle C.
Depending on which three values you supply, the calculator automatically chooses the correct method to solve the triangle. In one special situation — described below as the "ambiguous case" — two different triangles can satisfy the same three measurements, and the calculator displays both possible solutions.
There are several standard combinations of three measurements that determine a triangle. Each has a name based on the order of sides (S) and angles (A) around the triangle:
When all three sides are known, the triangle is fully determined (provided the sides satisfy the triangle inequality). The calculator uses the law of cosines to find each angle. For example, angle A is found from cos A = (b² + c² − a²) / (2bc).
If you know two sides and the angle between them, the third side comes directly from the law of cosines, after which the remaining angles follow. SAS always yields exactly one triangle.
Knowing two angles immediately gives the third, since the angles of a triangle sum to 180° (or π radians). With all three angles and any one side, the law of sines scales the triangle to the correct size. ASA (the side lies between the two angles) and AAS (the side is not between them) both produce a single, unique triangle.
This is the famous ambiguous case. When you know two sides and an angle opposite one of them, there may be zero, one, or two triangles that fit. The calculator detects this automatically: if two valid triangles exist it shows both, if only one is geometrically possible it shows that one, and if no triangle can be formed it reports that the values are invalid.
Three classical relationships do most of the work in solving triangles.
Law of Sines: a / sin A = b / sin B = c / sin C
Law of Cosines: c² = a² + b² − 2ab·cos C
Heron's Formula: Area = √[ s(s−a)(s−b)(s−c) ], where s = (a+b+c)/2
The law of sines is ideal when you have an angle paired with its opposite side. The law of cosines handles cases where the law of sines cannot start — namely three sides (SSS) or two sides with the included angle (SAS). Heron's formula gives the area directly from the three side lengths, which is convenient once all sides are known.
Beyond the three sides and three angles, the calculator computes a complete profile of the triangle:
Angles are shown in degrees, in degrees-minutes-seconds, and in radians, and when an angle is a clean fraction of π (such as π/3 or π/2) that exact form is shown as well.
The calculator classifies every triangle two ways at once: by its sides and by its angles.
So a triangle might be described as "acute scalene," "right isosceles," or "obtuse scalene," combining both classifications. An equilateral triangle is always acute, so it is simply called equilateral.
Not every set of three numbers can be the sides of a triangle. The triangle inequality states that the sum of the lengths of any two sides must be greater than the length of the third side. If a + b is less than or equal to c (or any similar combination fails), the three lengths cannot close into a triangle, and the calculator will report that the values are invalid. Likewise, the three angles must add up to exactly 180°, so if you enter two angles that already sum to 180° or more, no triangle is possible.
The SSA configuration deserves a closer look because it is the one situation where three measurements may not pin down a unique triangle. Suppose you know sides a and b and angle A (which is opposite side a). Using the law of sines, sin B = b·sin A / a. If this value is greater than 1, no triangle exists. If it equals 1, there is exactly one right triangle. If it is less than 1, there are two candidate angles for B — an acute one and its obtuse supplement — and each may lead to a valid triangle. When both are valid, the calculator presents them as "Possible 1" and "Possible 2," because both genuinely satisfy the given measurements. This is why a careful solver always checks for a second solution in the SSA case rather than stopping at the first answer.
Triangle solving is far from an abstract exercise. Surveyors use triangulation to measure distances and map land without crossing every point directly. Navigators and pilots rely on triangle relationships to plot courses and account for wind and current. Engineers and architects analyze trusses, roofs, ramps, and braces, all of which are built from triangles because the triangle is the only rigid polygon. Astronomers use parallax — a giant triangle — to estimate distances to stars. Even in everyday life, knowing how to find a missing side or angle helps with carpentry, landscaping, and DIY projects where you can measure some parts but not others.
A few triangles appear so often that their proportions are worth memorizing. The 30-60-90 triangle has sides in the ratio 1 : √3 : 2, so once you know the short leg you can find the others instantly. The 45-45-90 triangle (an isosceles right triangle) has legs in the ratio 1 : 1 : √2. The classic 3-4-5 triangle is the smallest right triangle with whole-number sides, and any multiple of it (6-8-10, 9-12-15, and so on) is also a right triangle — a fact carpenters use to square corners. The equilateral triangle, with all sides equal and all angles 60°, is the most symmetric of all and the basis for many tilings and trusses. Entering any of these into the calculator confirms their angles, area, and other properties at a glance.
Suppose a surveyor measures two sides of a triangular plot as 50 and 70 meters with an included angle of 65° between them — an SAS setup. The calculator applies the law of cosines to find the third side, then the law of sines (or cosines) for the remaining angles, and finally Heron's formula for the area. In one step it returns the missing side, both unknown angles, the area of the plot, its perimeter, and even the radius of the largest circle that would fit inside it. Doing the same work by hand would mean several careful trigonometric calculations, each an opportunity for a rounding slip. This is exactly the kind of repetitive, error-prone arithmetic that a triangle calculator is built to eliminate, letting you focus on the problem rather than the algebra.
You need exactly three, and at least one of them must be a side. Three angles alone fix the triangle's shape but not its size, so the calculator requires at least one length to determine the actual dimensions.
This happens in the ambiguous SSA case — when you provide two sides and an angle opposite one of them. Geometry allows two different triangles to share those exact measurements, so the calculator shows both "Possible 1" and "Possible 2." You then choose the one that matches your problem.
It means the numbers cannot form a real triangle. Common causes are violating the triangle inequality (one side as long as or longer than the other two combined), entering angles that sum to 180° or more, or an SSA setup where the given side is too short to reach and close the triangle.
A height (altitude) is the perpendicular distance from a vertex to the opposite side, while a median runs from a vertex to the midpoint of the opposite side. They coincide only in special triangles, such as the median and altitude from the apex of an isosceles triangle.
The inradius is the radius of the largest circle that fits inside the triangle (the inscribed circle), equal to the area divided by the semiperimeter. The circumradius is the radius of the circle that passes through all three vertices (the circumscribed circle), equal to the product of the sides divided by four times the area.
Yes. Enter a 90° angle (or three sides that form a right triangle) and it will solve it like any other, identifying it as a right triangle. The familiar Pythagorean theorem is simply the law of cosines applied to a 90° angle.
This Triangle Calculator is provided for educational and informational purposes. Results are computed using standard trigonometric formulas and rounded for display, so values may differ slightly from hand calculations that round at different stages. Always verify critical measurements independently for engineering, construction, or safety-related work.