Triangle Calculator: Sides, Angles & Area

Understanding Triangle Geometry

Our triangle calculator solves any triangle from three known values (at least one side). Whether you have SSS, SAS, ASA, or AAS, we apply the correct law—Law of Sines, Law of Cosines, or Heron's formula—and show every step in the Step-by-Step Calculation. Trusted by students and educators for geometry homework and surveying.

Workflow Tips

Input 3 Values

Sides (a, b, c) or angles (A, B, C in degrees). At least one side is required. The dashed mint border marks system-calculated fields.

Geometric Profile

Area & perimeter, inradius (r) and circumradius (R), and medians (mₐ, mᵦ, mᵧ) appear when the triangle is solved.

Step-by-Step Tabs

Angles/Sides tab shows Law of Sines or Cosines with substitution. Area tab shows Heron's formula.

Dynamic Schematic

The SVG triangle updates to show Equilateral, Isosceles, Scalene, Right, Obtuse, or Acute based on your inputs.

Knowledge Base

Triangle properties depend on what you know. We apply Law of Cosines (SSS/SAS), Law of Sines (angle-side pairs), and Heron's formula (area from sides):

Law of Cosines: SSS or SAS—find missing angles or the third side:

c^2 = a^2 + b^2 - 2ab \cos(C)

Law of Sines: When you have an angle and its opposite side, the ratio is constant:

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

Heron's Formula: Area from three sides only—no height needed. Semi-perimeter s = (a+b+c)/2, then:

\text{Area} = \sqrt{s(s-a)(s-b)(s-c)}

Triangle Calculator: Angles, Sides, Area | Law of Sines & Cosines

Free triangle calculator: solve any triangle from 3 known values. SSS, SAS, ASA, AAS. Law of Sines, Law of Cosines, Heron's formula. Inradius, circumradius, medians. Step-by-Step Calculation. Trusted by students and educators. No sign-up.

What This Triangle Calculator Does

Purpose & Use Cases

Input:

Any 3 of: sides a, b, c; angles A, B, C (degrees).
Output:

All 6 values, area, perimeter, inradius, circumradius, medians.

This triangle calculator solves for all sides, angles, area, perimeter, inradius, circumradius, and medians when you provide any 3 values (with at least one side). Use it to solve a triangle with 3 sides (SSS), find triangle area without height via Heron's formula, or solve SAS, ASA, and AAS cases. Ideal for geometry homework, surveying, and understanding how triangles are uniquely determined. The Step-by-Step Calculation shows every substitution so you can verify each step.

The Three Laws — When the Calculator Uses Each

Law of Sines

Used when you have a matching pair—an angle and its opposite side—plus one more. Establishes a constant ratio across the triangle:

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

Applies to ASA, AAS, and some SSA cases.

Law of Cosines

Used for SSS (find angles) and SAS (find the third side). Generalizes the Pythagorean theorem:

c^2 = a^2 + b^2 - 2ab \cos(C)

For right triangles, cos(90°)=0, giving a² + b² = c².

Heron's Formula

When all three sides are known, we compute area without height. Semi-perimeter s = (a+b+c)/2, then:

\text{Area} = \sqrt{s(s-a)(s-b)(s-c)}

Essential for finding triangle area with 3 sides only.

When to Use Each Method

SSS (Side-Side-Side): Law of Cosines finds each angle; Heron's formula gives area. SAS (two sides and included angle): Law of Cosines finds the third side; then Law of Sines for remaining angles; area via ½ab·sin(C). ASA/AAS: Third angle from sum = 180°, then Law of Sines for sides. SSA (ambiguous case): 0, 1, or 2 triangles possible depending on values—the calculator handles this automatically.

Triangle Calculator FAQ

? What combinations of sides and angles can I enter?

Enter any 3 values with at least one side: SSS (all three sides), SAS (two sides and the included angle), ASA (two angles and the included side), AAS (two angles and a non-included side), or SSA (two sides and a non-included angle). The calculator uses the Law of Sines and Law of Cosines to solve for the remaining values.

? Why does it say "Invalid triangle"?

A valid triangle requires the sum of any two sides to be greater than the third (triangle inequality). If your inputs violate this—e.g., sides 1, 2, 10—no triangle can exist. Check that your values can form a real triangle.

? How is area calculated?

Area uses Heron's formula when all three sides are known. First find the semi-perimeter s = (a+b+c)/2, then

\text{Area} = \sqrt{s(s-a)(s-b)(s-c)}

The calculator also shows inradius, circumradius, and medians in the Geometric Profile. For SAS cases, we use ½ab·sin(C).

? What is the Law of Cosines?

The Law of Cosines generalizes the Pythagorean theorem:

c^2 = a^2 + b^2 - 2ab \cos(C)

Used for SSS (find angles) and SAS (find the third side). For right triangles, cos(90°)=0, so it reduces to a² + b² = c².

? What is the Law of Sines?

The Law of Sines establishes a constant ratio:

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

Used when you know an angle and its opposite side, plus one other. Combined with A+B+C=180°, it solves ASA and AAS cases.

Triangle Calculator: Law of Sines, Cosines & Heron