Why can't I solve a triangle using only three angles (AAA)?

Three angles (AAA) determine the shape and proportions of a triangle, but not its size. Infinite similar triangles share the exact same set of angles.

What is the ambiguous case (SSA) in triangle calculations?

The SSA (Side-Side-Angle) condition occurs when you know two sides and a non-included angle. Depending on the dimensions, this can result in zero valid triangles, one unique triangle, or two different valid triangles.

How is the area of a triangle calculated without knowing the height?

When all 3 side lengths are known, Heron's Formula calculates the area: Area = √(s(s - a)(s - b)(s - c)), where s is the semi-perimeter (a + b + c) / 2.

Geometry Solver

Triangle Calculator

Q: How does the Triangle Calculator work?

The triangle calculator uses trigonometric laws including the Law of Cosines, Law of Sines, Pythagorean theorem, and Heron's formula to solve for all remaining sides, interior angles, area, and heights given any 3 known values.

Enter any 3 known values (sides, angles) to solve for all sides, interior angles, area, perimeter, heights, inradius, circumradius, and view the rendered triangle.

Input Values

Provide any 3 known values (at least 1 side required):

Side Lengths

Side a:

Side b:

Side c:

Angles (Degrees)

Angle A (°):

Angle B (°):

Angle C (°):

Interactive Visual Diagram

Type: Scalene

Calculated Properties

Side a 5.0000

Angle A (α) 55.77°

Side b 6.0000

Angle B (β) 82.82°

Side c 4.0000

Angle C (γ) 41.41°

                        Area
                        9.9216
                    

                        Perimeter
                        15.0000
                    

Height h_a 3.9686

Semi-Perimeter (s) 7.5000

Height h_b 3.3072

Inradius (r) 1.3229

Height h_c 4.9608

Circumradius (R) 3.0237

How to Solve Triangle Calculations

A triangle is defined by six basic elements: 3 side lengths (a, b, c) and 3 interior angles (A, B, C). Knowing any 3 of these elements (provided at least one is a side) is sufficient to solve for all other properties using classical geometric laws:

Law of Cosines: Used for SSS and SAS cases ($c^2 = a^2 + b^2 - 2ab \cos(C)$).
Law of Sines: Used for ASA, AAS, and SSA cases ($\frac{a}{\sin(A)} = \frac{b}{\sin(B)} = \frac{c}{\sin(C)}$).
Heron's Formula: Computes area directly from all 3 sides: $\text{Area} = \sqrt{s(s - a)(s - b)(s - c)}$.

Frequently Asked Questions

The triangle calculator applies fundamental trigonometric laws including the Law of Cosines, Law of Sines, Pythagorean theorem, and Heron's formula to solve for all remaining sides, interior angles, area, perimeter, and heights given any 3 known values.

You must provide at least 3 known values, and at least one of those values must be a side length (such as SSS, SAS, ASA, AAS, or SSA combinations).

Three angles (AAA) determine the shape and relative proportions of a triangle, but not its physical scale or size. Infinite similar triangles share the exact same set of interior angles.

The SSA (Side-Side-Angle) condition occurs when you know two sides and an angle that is not between them. Depending on the proportions, this can lead to zero valid triangles, one unique right triangle, or two distinct valid triangles.

When all 3 side lengths are known, Heron's Formula calculates the area: $\text{Area} = \sqrt{s(s - a)(s - b)(s - c)}$, where $s$ is the semi-perimeter $\frac{a + b + c}{2}$.