Triangle Side, Angle & Area Calculator

Overview

Every planar triangle becomes solvable once you know three independent measurements, yet collecting the remaining values by hand is tedious. This calculator handles the four canonical cases in one view: three sides (SSS), two sides plus the included angle (SAS), two angles and one side (ASA/AAS), and the ambiguous two sides plus opposite angle (SSA). When the inputs define a valid triangle the tool instantly produces the missing sides and angles, the area, perimeter, heights, and both the inradius and circumradius so you can feed the numbers into a layout, cut list, or classroom exercise.

Inputs & Usage

There are six fields: sides a, b, c and angles A, B, C. Enter only the values you actually know and leave the rest blank. You need at least three entries, and one must be a side because purely angular data still lacks scale. If you know two angles, the third one is implied. The SSA case can yield two different triangles; when that happens both solutions are shown side by side. Use the "Load example triangle" button whenever you want to reset the form to a clean baseline.

How It Works

The solver combines three standard formulas:

• SSS and SAS: the law of cosines fills the missing side or angle.

  $$a=\sqrt{b^2+c^2-2bc\cos (A)}$$$$A=\arccos \frac{b^2+c^2-a^2}{2bc}$$

• ASA and AAS: the law of sines propagates the known ratio.

  $$\frac{a}{\sin (A)}=\frac{b}{\sin (B)}=\frac{c}{\sin (C)}$$

• Derived metrics: the area comes from Heron's formula, heights from A = 1/2 * b * c * sin(A), and the radii are

  $$r=\frac{2\cdot \text{Area}}{a+b+c}$$$$R=\frac{abc}{4\cdot \text{Area}}$$

Interpreting the Output

The side and angle cards confirm the shape, whereas the area and perimeter answer most sizing questions such as "how many tiles do I need." Heights are handy whenever you must draw a perpendicular or determine clearances under a roof truss. The radius values show how large a circle fits inside the triangle and how large a circumscribed circle you would need for templates. If the SSA case is ambiguous the banner reminds you to compare both solutions against your physical constraints.

Example

Assume you measured sides b = 6.2 m and c = 8.1 m plus their included angle A = 47 degrees. The calculator returns a about 5.96 m, B about 49.5 degrees, and C about 83.5 degrees. The perimeter is about 20.26 m and the area 18.36 m^2. Heights follow as h_a about 6.16 m, h_b about 5.92 m, and h_c about 4.53 m, while the inradius is roughly 1.81 m and the circumradius 4.08 m. With that you can plan anchor bolts around the circumradius yet still verify that your working space covers the needed heights.

Limitations

The tool assumes all inputs share the same unit, so make sure you do not mix millimetres and metres. SSA inputs may produce two results; discard the one that violates your real-world constraints. Extremely small angles or nearly degenerate triangles may be sensitive to rounding, so round only at the end when presenting the numbers. Finally, the solver covers flat Euclidean triangles only -- spherical triangles, survey projections, or fabrication tolerances still require domain-specific adjustments.