Triangle Calculator Online Free Tool
Triangle Calculator
Triangle Properties Calculator
Sides
Angles (°)
Given any combination of sides and angles, this calculator finds all remaining measurements of a triangle: the missing sides, missing angles, area, and perimeter. It applies the Law of Sines, Law of Cosines, and other relationships to solve both right and oblique (non-right) triangles completely.
Solving Triangles
A triangle has 6 measurements: 3 sides (a, b, c) and 3 angles (A, B, C). You need at least 3 known values (with at least one being a side) to solve the triangle completely. The angle sum rule (A + B + C = 180°) always applies.
Law of Sines: a/sin(A) = b/sin(B) = c/sin(C) Law of Cosines: c² = a² + b² - 2ab × cos(C) Area = ½ × a × b × sin(C)
Use Law of Sines when you know two angles and a side, or two sides and an opposite angle. Use Law of Cosines for other combinations.
Triangle Types
Triangles are classified by their angles (acute, right, obtuse) and by their sides (equilateral, isosceles, scalene). These properties affect which formulas are most convenient.
| Type | Property | Notes |
|---|---|---|
| Equilateral | All sides equal, all angles 60° | Simplest case |
| Isosceles | Two sides equal, two angles equal | Common in geometry |
| Scalene | All sides different | General case |
| Right | One 90° angle | Use Pythagorean theorem |
Frequently Asked Questions
Can any three side lengths form a triangle?⌄
No. The Triangle Inequality Theorem requires that the sum of any two sides must be greater than the third side. For sides 3, 4, 8: since 3 + 4 = 7 < 8, these cannot form a triangle. For sides 3, 4, 5: since 3+4=7>5, 3+5=8>4, and 4+5=9>3, this is valid (it is also a right triangle).
What is the ambiguous case (SSA) in triangle solving?⌄
When you know two sides and an angle opposite one of them (SSA), there may be 0, 1, or 2 valid triangles depending on the values. This is called the ambiguous case because the given information alone does not uniquely determine the triangle. The calculator handles this automatically and will show all valid solutions.
What is Heron's formula for triangle area?⌄
When you know all three sides (SSS) but no height, use Heron's formula: s = (a+b+c)/2 (semi-perimeter), then Area = √(s(s-a)(s-b)(s-c)). This avoids needing to calculate any angles first.
What does it mean for triangles to be congruent vs similar?⌄
Congruent triangles are identical in size and shape (all corresponding sides and angles are equal). Similar triangles have the same shape but different sizes (corresponding angles are equal, sides are proportional). Similar triangles are identified by AA, SAS, or SSS similarity criteria.