Triangle Calculator

A triangle is a polygon with three vertices and three sides. We label the vertices A, B, C and the sides opposite them a, b, c. Triangles are often described by their side lengths (equilateral, isosceles, scalene) or by their angles (right, obtuse, acute). The drawings below are simplified, schematic guides—not to scale.

Types by sides

Small slashes (“tick marks”) on edges are a common convention: the same number of ticks indicates equal length. Angle arcs near a vertex communicate relative angle size.

Types by angles

Core theorems you’ll use often

• Angle-sum theorem: A + B + C = 180°.
• Triangle inequality: each side is shorter than the sum of the other two (a < b + c, etc.).
• Law of Sines: a/ sin A = b/ sin B = c/ sin C.
• Law of Cosines: c² = a² + b² − 2ab cos C (and cyclic).

In a right triangle, the side opposite the 90° angle is the hypotenuse, and the Pythagorean identity a² + b² = c² applies. For general triangles, the laws of sines and cosines let you solve for missing sides or angles once three compatible measurements are known.

This Triangle Calculator helps you solve any triangle by entering any three known values that include at least one side. It instantly finds the remaining sides and angles, plus useful measures such as area, perimeter, heights, inradius and circumradius. The tool supports degrees or radians, shows step-by-step math, and draws a triangle to scale so you can visually confirm results. Keywords: triangle calculator, law of sines, law of cosines, Heron’s formula, inradius, circumradius, missing angle in triangle.

Behind the scenes, the solver combines the Law of Sines, Law of Cosines, and Heron’s formula with basic identities such as the angle-sum theorem (A + B + C = 180°). It also validates inputs (for example, triangle inequality) and explains the path used to get each answer.

How to use this triangle calculator

1. Choose the angle unit (Degrees or Radians).
2. Enter any three values (e.g., two sides and an angle, or three sides). At least one must be a side.
3. The remaining measures fill in automatically. Use the diagram to confirm which angle or side you’re setting.

Finding the angle of a triangle

There are several common cases. The calculator detects which case your inputs match and applies the appropriate identity:

• SAS (two sides and included angle): use the Law of Cosines to find the opposite side, then the Law of Sines to get the unknown angles.
• ASA or AAS (two angles and a side): first compute the third angle via 180° − (A + B), then use the Law of Sines to scale sides.
• SSS (three sides): use the Law of Cosines to recover each angle.

Angle bisector of a triangle

The angle bisector splits an angle into two equal parts. If it bisects angle A, and the opposite side is a with the adjacent sides b and c, the bisector divides side a proportionally: BD/DC = c/b. The intersection of the three angle bisectors is the incenter, the center of the incircle.

Formulas used (overview)

• Law of Sines: a / sin(A) = b / sin(B) = c / sin(C).
• Law of Cosines: c² = a² + b² − 2ab cos(C) and cyclic permutations.
• Area (Heron): Area = √(s(s − a)(s − b)(s − c)), where s = (a + b + c)/2. Also Area = ½ab sin(C) when two sides and the included angle are known.
• Heights: h_a = 2·Area / a (and similarly for h_b, h_c).
• Inradius: r = Area / s. Circumradius: R = a / (2 sin A) (or any side / 2 sin its opposite angle).

Finding missing angles and sides (worked ideas)

Suppose you know SAS: b = 7, c = 9, and A = 30°. First compute the opposite side using Cosines: a = √(b² + c² − 2bc cos A). Then apply Sines to get B or C: sin B = b·sin A / a. Finish with C = 180° − (A + B).

If you know all three sides (SSS), use Cosines to recover one angle, then Sines (or Cosines again) for the others. If two angles are known (ASA/AAS), get the third by subtraction and scale the sides with Sines.

Assumptions & limits

• Euclidean (flat) geometry is assumed; on curved surfaces the angle sum may differ from 180°.
• Inputs must satisfy the triangle inequality (a < b + c, etc.). The solver warns when values are inconsistent.
• When SSA (two sides and a non-included angle) is supplied, an ambiguous case can occur. The calculator checks whether 0, 1, or 2 triangles fit and reports the valid configuration.

Why this matters

Triangles pop up in construction, navigation, surveying, graphics, robotics, physics problems, and standardized tests. A reliable triangle solver saves time, prevents arithmetic slips, and gives you an immediate visual check—especially helpful when values are close to edge cases (nearly collinear, very obtuse, or very acute).

Tip: after solving, switch between Degrees and Radians to copy the angles in the unit your workflow needs.

When to use each mode

• SSS: you measured all three sides; get every angle and the area.
• SAS: two sides and the included angle; solve the remaining side and angles.
• ASA/AAS: two angles and a side; use the angle sum and Law of Sines.

Calculations are rounded for readability. For precision work, increase decimal places in your inputs and record outputs with appropriate significant figures.