Triangle CalculatorFree online math calculator — instant results, no sign-up
Solve a triangle from three sides.
About the Triangle Calculator
This calculator solves a triangle using the lengths of its three sides (SSS configuration). It computes the perimeter, area, and all three interior angles for students, engineers, and surveyors working with geometric measurements.
- Side a
- Length of side a opposite angle A; positive real number in any consistent unit of length.
- Side b
- Length of side b opposite angle B; positive real number in the same unit as side a.
- Side c
- Length of side c opposite angle C; positive real number in the same unit as sides a and b.
Formula and method
s = (a + b + c) / 2 Area = sqrt(s × (s - a) × (s - b) × (s - c)) Perimeter = a + b + c = 2 × s Angle A = arccos((b² + c² - a²) / (2 × b × c)) × (180 / π) Angle B = arccos((a² + c² - b²) / (2 × a × c)) × (180 / π) Angle C = 180° - Angle A - Angle B
- a, b, cLengths of the three triangle sides
- sSemiperimeter of the triangle
- AreaEnclosed surface area calculated via Heron's formula
- Angle A, B, CInterior angles measured in degrees opposite sides a, b, and c respectively
The tool first validates the triangle inequality theorem: the sum of any two sides must strictly exceed the third side. If valid, it computes the semiperimeter and applies Heron's formula for the area (rounded to 6 decimal places), followed by the Law of Cosines to evaluate Angle A and Angle B in degrees, deriving Angle C by subtracting A and B from 180° (all angles rounded to 4 decimal places).
Worked example
These are the calculator's starting values and the results it actually produces for them.
Inputs
- Side a3
- Side b4
- Side c5
Results
- Area6
- Perimeter12
- Angle A36.8699°
- Angle B53.1301°
- Angle C90°
How to use the Triangle Calculator
- 1Verify the triangle inequality condition: check that a + b > c, a + c > b, and b + c > a.
- 2Calculate the semiperimeter s = (a + b + c) / 2 and total perimeter 2s.
- 3Compute the area using Heron's formula: sqrt(s(s - a)(s - b)(s - c)).
- 4Calculate Angle A and Angle B with the Law of Cosines in degrees, then subtract their sum from 180° to find Angle C.
Reading your result
Perimeter is presented in the linear units of the inputs, while Area is expressed in corresponding square units. All three angles sum exactly to 180°, allowing quick identification of whether the triangle is acute, right (one angle equals 90°), or obtuse (one angle exceeds 90°).
Limitations
The calculation requires all three side lengths as inputs and does not directly solve triangles defined by angle-side combinations. It only applies to planar (flat Euclidean) triangles, not spherical or non-Euclidean surfaces.
Common questions
Why does the calculator return 'Not a valid triangle'?+
By the triangle inequality theorem, the sum of any two side lengths must be strictly greater than the third. If the two shorter sides sum to less than or equal to the longest side, the ends cannot meet to form a closed triangle.
What units should I enter for the sides?+
You can use any unit of length (inches, meters, feet), provided all three sides use the same unit. The calculated area will be in corresponding square units.
How does Heron's formula work?+
Heron's formula determines the area of any valid triangle purely from its three side lengths without needing an altitude or angle measurement, using the semiperimeter s = (a + b + c) / 2.
Does the calculator save my information?+
Calculations run in your browser. While signed in, inputs and results are automatically recorded in account history; saving a result creates a separate saved record.
Further reading
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Results are estimates for informational purposes and are not financial, medical or legal advice.