Why this matters
Triangles are the fundamental building block of geometry, engineering, and physics. From calculating the area of a triangular plot of land to determining the forces in a truss bridge, the ability to derive every property of a triangle from its three side lengths is a skill that applies across disciplines. Heron's formula gives you the area without needing to know the height, and the Law of Cosines gives you every interior angle without needing a protractor or diagram. Together, these two formulas unlock the complete geometric profile of any triangle from just three numbers.
Manually computing square roots and inverse cosines is tedious and error-prone, especially when you need precise results to four decimal places. This calculator validates the triangle inequality first, ensuring your three sides can actually form a triangle, and then produces the area, perimeter, semi-perimeter, and all three angles instantly. For students working through geometry problem sets, architects verifying structural calculations, or game developers computing collision normals, having these results in under a second eliminates a significant source of friction.
Computed properties and formulas
| Property | Formula | Description |
|---|---|---|
| Semi-perimeter | s = (a+b+c)/2 | Half the perimeter |
| Area | sqrt(s(s-a)(s-b)(s-c)) | Heron's formula |
| Perimeter | a + b + c | Sum of all sides |
| Angle A | cos^-1((b^2+c^2-a^2)/(2bc)) | Law of Cosines |
| Angle B | cos^-1((a^2+c^2-b^2)/(2ac)) | Law of Cosines |
| Angle C | cos^-1((a^2+b^2-c^2)/(2ab)) | Law of Cosines |
How to use it
Enter the three side lengths labeled a, b, and c in any order.
The calculator immediately checks the triangle inequality: the sum of any two sides must be greater than the third.
If the sides are valid, the area (via Heron's formula), perimeter, semi-perimeter, and all three angles are displayed.
If the sides are invalid, a clear warning explains which inequality is violated.
Results are rounded to four decimal places for readability.
Testing your result
The best verification method is to use a well-known triangle and confirm the output matches expected values. A 3-4-5 right triangle should produce an area of 6, a perimeter of 12, and angles of approximately 90, 53.13, and 36.87 degrees. An equilateral triangle with sides of 10 should produce an area of approximately 43.3013 and three angles of exactly 60 degrees. These two test cases cover both right-angled and non-right-angled scenarios and will quickly confirm the calculator is working correctly.
Common mistakes
Entering side lengths that violate the triangle inequality, such as 1, 2, and 5, which cannot form a closed shape.
Mixing up which side is a, b, or c when the triangle has specific angle assignments in your problem.
Forgetting that the angles are displayed in degrees, not radians, when plugging them into other formulas.
Rounding intermediate values manually before entering them, which can compound errors in the final area or angle calculation.
Assuming the calculator accepts two sides and an angle; it requires all three sides as input.
Edge cases and options
The triangle inequality check ensures that the sum of every pair of sides exceeds the remaining side. When two sides sum to exactly the third, the result is a degenerate triangle (a straight line) with zero area, and the tool warns accordingly. Very thin triangles where one side is nearly equal to the sum of the other two produce angles approaching 0 and 180 degrees, which are mathematically valid but may appear unusual. The calculator accepts decimal inputs of any precision, and all positive numbers are valid candidates as long as they satisfy the triangle inequality.
Real-world use cases
Calculating the area of an irregularly shaped parcel of land by triangulating it into smaller triangles.
Computing angles in a roof truss or structural frame from the known lengths of its members.
Determining the coverage area of a triangular garden bed for soil or mulch ordering.
Solving navigation problems where two known distances and a calculated angle determine a position.
Verifying geometric proofs in mathematics education by checking numerical examples.
Frequently asked questions
Q: What is Heron's formula?
A: Area equals the square root of s times (s-a) times (s-b) times (s-c), where s is the semi-perimeter (a+b+c)/2. It computes the area from side lengths alone without needing the height.
Q: How are the angles computed?
A: Using the Law of Cosines: cos A = (b^2 + c^2 - a^2) / (2bc), and similarly for angles B and C. The three angles always sum to exactly 180 degrees.
Q: What if my three sides do not form a triangle?
A: The tool displays a clear warning if the triangle inequality (a+b>c, a+c>b, b+c>a) is violated by any combination of sides.
Q: Can I enter decimal values?
A: Yes. Any positive numbers are supported, and the output is rounded to four decimal places for readability.
Q: Does this work for right triangles?
A: Yes. For a right triangle, one angle will be exactly 90 degrees. The 3-4-5 triangle is a good test case.
Q: Can I input angles instead of sides?
A: No. This calculator requires three side lengths as input. If you have two sides and an included angle, you would need to compute the third side first using the Law of Cosines.
Start using it now
Try the Triangle Calculator tool. See also Circle Calculator and Pythagorean Theorem Calculator and Quadratic Equation Solver.
