Triangle Calculator - Solve Any Triangle
Enter any 3 values that describe a triangle (at least one must be a side length) and this calculator will solve for all remaining sides, angles, area, perimeter, semi-perimeter, inradius, circumradius, and all three altitudes. It handles SSS, SAS, ASA, AAS, and SSA cases automatically. Angles are in degrees and lengths can be any consistent unit.
What is the ABC triangle calculator?
This tool solves any triangle from just three known measurements, provided at least one of them is a side length. Triangles have six measurements in total: three sides (a, b, c) and three interior angles (A, B, C). Knowing any valid combination of three is sufficient to determine the remaining three, plus derived quantities such as area, perimeter, inradius, circumradius, and the three altitudes and medians. The calculator uses the law of sines, the law of cosines, and Heron's formula to produce exact results.
How to enter values: solving modes
Enter any three values that describe your triangle (at least one side). Zero or blank fields are treated as unknowns. The calculator detects the solving mode automatically: SSS (all three sides) - uses the law of cosines to find each angle. SAS (two sides and the included angle) - the law of cosines finds the third side, then the law of sines finds the remaining angles. ASA or AAS (two angles and any side) - the third angle is derived from the 180-degree rule, then the law of sines finds both remaining sides. SSA (two sides and an angle not between them) - uses the law of sines; note that this configuration can sometimes produce two valid triangles, called the ambiguous case. This calculator returns the first valid solution. AAA (all three angles, no side) is not solvable to a unique triangle: only the shape is defined, not the scale.
Formulas used
Law of sines: a / sin(A) = b / sin(B) = c / sin(C). This relates each side to the sine of its opposite angle. Law of cosines: c^2 = a^2 + b^2 - 2ab cos(C). Generalises the Pythagorean theorem to any triangle. Heron's formula for area: K = sqrt(s(s-a)(s-b)(s-c)), where s = (a+b+c)/2 is the semi-perimeter. Inradius: r = K / s. Circumradius: R = a / (2 sin A). Altitude from vertex X to opposite side x: h_x = 2K / x. Median from vertex A to midpoint of a: m_a = 0.5 * sqrt(2b^2 + 2c^2 - a^2).
Practical uses
Triangle calculations appear throughout architecture (roof pitch and rafter lengths), surveying (triangulation for land area and distances), navigation (bearing and distance problems), engineering (truss and frame analysis), and physics (vector decomposition). Knowing the formulas behind the numbers, and being able to check them instantly, is useful any time you have a three-point geometric problem.
Triangle classification by sides and angles
| Type | Sides | Angles | Special property |
|---|---|---|---|
| Equilateral | All equal (a = b = c) | All 60 deg | Also equiangular; most symmetric triangle |
| Isosceles | Two equal | Two base angles equal | Axis of symmetry along the perpendicular bisector |
| Scalene | All different | All different | No lines of symmetry |
| Right | Any | One angle = 90 deg | Pythagorean theorem: a^2 + b^2 = c^2 |
| Acute | Any | All angles < 90 deg | Circumcenter lies inside the triangle |
| Obtuse | Any | One angle > 90 deg | Circumcenter lies outside the triangle |
Quick reference for identifying triangle types from their properties.
Frequently asked questions
Can I solve a triangle if I only know two sides and no angle?
No. Two sides alone are not enough to determine a unique triangle: the angle between them can vary, producing infinitely many different triangles with those same two side lengths. You need at least three pieces of information, including at least one side, for the triangle to have a unique solution.
What is the ambiguous case (SSA)?
When you know two sides and an angle that is not between them, it is sometimes possible to form two different valid triangles. This is called the ambiguous case or the SSA case. For example, if the side opposite the known angle is shorter than the other known side, two triangles may fit. This calculator returns the principal (acute) solution. If you need both solutions, solve once, then check whether pi minus the found angle still produces a valid triangle.
What are the inradius and circumradius?
The inradius (r) is the radius of the largest circle that fits entirely inside the triangle, touching all three sides. It equals the area divided by the semi-perimeter. The circumradius (R) is the radius of the circle that passes through all three vertices of the triangle. By Euler's inequality, R is always at least twice r (R greater than or equal to 2r), with equality only for the equilateral triangle.
What are altitudes and medians?
An altitude is the perpendicular distance from a vertex to the opposite side (or the line containing it). Each triangle has three altitudes; they all meet at a single point called the orthocenter. A median connects a vertex to the midpoint of the opposite side. The three medians intersect at the centroid, which is the triangle's center of mass. The length of each median is computed with the formula m_a = 0.5 * sqrt(2b^2 + 2c^2 - a^2).
How do I verify my triangle is valid?
Two rules must hold: the triangle inequality (each side must be shorter than the sum of the other two), and the angles must sum to exactly 180 degrees. If either rule is violated, no triangle with those measurements exists. This calculator returns blank results if the inputs describe an impossible triangle.
Does the calculator work in any units?
Yes, for side lengths. Sides can be in any unit (metres, feet, inches, etc.) as long as all three are in the same unit. The outputs, including area, altitudes, medians, inradius and circumradius, will be in the same unit or its square. Angles are always in degrees.