Triangle Calculator

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Solve any triangle with our free online triangle calculator. Enter any 3 known values (at least 1 side) and get all sides, angles, area, perimeter, and a visual representation. Uses the Law of Cosines and Law of Sines for precise calculations.

Universal Triangle Solver

Our calculator handles all triangle types: SSS (side-side-side), SAS (side-angle-side), ASA (angle-side-angle), and AAS (angle-angle-side). Simply input your known values and let the calculator do the rest.

Core Trigonometry Formulas

For those interested in the math (or checking homework), here are the primary laws used:

Law of Sines

a/sin(A) = b/sin(B) = c/sin(C)

Useful when you know a side and its opposite angle.

Law of Cosines

c² = a² + b² - 2ab·cos(C)

Essential for SSS and SAS cases.

Heron's Formula (Area)

Area = √[s(s-a)(s-b)(s-c)]

Where 's' is the semi-perimeter: (a+b+c)/2.

The "Ambiguous Case" (SSA)

One of the trickiest parts of trigonometry is the SSA (Side-Side-Angle) case. If you know two sides and a non-included angle, there might be:

  • No solution: The sides never meet.
  • One solution: A nice right-angled or obtuse triangle.
  • Two solutions: The swinging side can hit the base in two places, creating two valid triangles.

Our calculator automatically detects this and will alert you if multiple solutions exist.

Triangle Inequality Theorem

Not every set of three numbers can form a triangle. The Triangle Inequality Theorem states that the sum of any two sides must be strictly greater than the third side ($a + b > c$).

Imagine trying to connect two short sticks (length 2 and 3) to a long stick (length 10). They will never touch! This simple rule is the first check our calculator performs.

Why Heron's Formula is Magic

Before Heron of Alexandria (c. 60 AD), calculating area required knowing the height (altitude). But measuring height creates a "chicken and egg" problem if you only have sides. Heron proved you can calculate area solely from the three side lengths, a discovery that revolutionized ancient land surveying.

Area = √(s(s-a)(s-b)(s-c))

Triangle Classification Guide

By Sides

  • Equilateral: All 3 sides are equal.
  • Isosceles: 2 sides are equal.
  • Scalene: No sides are equal.

By Angles

  • Acute: All angles < 90°.
  • Right: One angle = 90°.
  • Obtuse: One angle > 90°.

Real-World Applications

Trigonometry isn't just for classrooms; it builds our world:

  • Architecture & Construction: Calculating roof slopes, ramp angles, and structural stability relies entirely on triangle properties.
  • Video Games (CGI): Every 3D character you see is composed of thousands of tiny triangles (polygons). This calculator helps developers understand the geometry of rendering.
  • Navigation (GPS): Triangulation uses distances from multiple satellites to pinpoint your exact location on Earth.

Complete Results

Get all 6 values (3 sides + 3 angles), plus area (using Heron's formula), perimeter, triangle type classification, and a visual SVG representation of your triangle.

Need to work with fractions in your geometry problems? Try our Fraction Calculator. Or use the Percentage Calculator for ratio problems. For advanced sine/cosine functions, visit the Scientific Calculator.

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