Triangle Calculator
Solve for missing sides, interior angles, area, and perimeter using Law of Sines and Cosines.
Parameters & Inputs
Summary & Breakdown
Solved Geometry Summary
About the Triangle Calculator
A triangle is the most fundamental polygon in Euclidean plane geometry, strictly governed by the rule that its three interior angles sum to exactly 180 degrees.
This calculator solves for unknown side lengths and angles using standard geometric theorems: the Pythagorean theorem for right triangles, Heron's formula for area, the Law of Cosines, and the Law of Sines.
Simply enter any three known parameters (at least one side) to determine the complete geometric profile.
How the Calculations Work
Evaluates given parameters via Law of Cosines (c^2 = a^2 + b^2 - 2ab*cos(C)) or Law of Sines (a/sin(A) = b/sin(B) = c/sin(C)), checks the triangle inequality theorem, and derives area via Heron's formula.
Heron's Formula: Area = sqrt(s*(s-a)*(s-b)*(s-c)) where s = (a+b+c)/2
Variables & Definitions
- a, b, c: Lengths of the three opposite sides
- alpha, beta, gamma: Interior angles in degrees
- s: Semiperimeter = (a + b + c) / 2
Given sides a=7, b=8, c=9: Semiperimeter s = 24/2 = 12. Area = sqrt(12 * (12-7) * (12-8) * (12-9)) = sqrt(12 * 5 * 4 * 3) = sqrt(720) = 26.83. Angles are 48.19°, 58.41°, and 73.40°.
Key Terms Explained
Heron's Formula
Calculates triangle area directly from all three side lengths without requiring an altitude measurement.
Law of Cosines
Generalization of Pythagorean theorem relating all three sides and one angle (c^2 = a^2 + b^2 - 2ab cos C).
Triangle Inequality
Mathematical law requiring that the sum of any two sides must strictly exceed the length of the third side.
Semiperimeter
Half of the total perimeter, represented by the variable s.
Practical Tips & Pitfalls to Avoid
At least one side length is mandatory
Knowing three angles (AAA) establishes shape similarity but cannot determine absolute size.
Triangle inequality check
If a + b <= c, the sides cannot touch to enclose a polygon and no triangle exists.
Watch for obtuse angles
If c^2 > a^2 + b^2, the angle opposite side c is obtuse (> 90 degrees).
Keep precision in intermediate steps
Trigonometric functions compound rounding drift; round only final presented numbers.
Frequently Asked Questions
No. Three angles establish similar triangles, meaning infinitely many triangles have those angles but different scales.
When given two sides and an angle not between them, there may be zero, one, or two valid triangles depending on the height.
Heron's formula computes the exact area of any triangle when all three side lengths are known, without needing the height.
In flat Euclidean space, the parallel postulate dictates that the interior angles of any triangle always total 180 degrees (pi radians).
If a^2 + b^2 = c^2 (Pythagorean theorem holds), the angle opposite the longest side is exactly 90 degrees.