About the Triangle Calculator
Given just the three side lengths of a triangle, you can find its area, perimeter, and all three interior angles using classic geometric formulas. Our Triangle Calculator does all of that in one step.
How It Works
The calculator uses Heron's formula to find area directly from the three side lengths, then applies the law of cosines to solve for each interior angle - checking first that the three sides can actually form a valid triangle (the triangle inequality: any two sides must sum to more than the third).
Formula & Methodology
Heron's formula (attributed to the ancient Greek mathematician Heron of Alexandria) finds area from side lengths alone, without needing to know any angle or height directly - it works by first computing the semi-perimeter (half the total perimeter), then combining it with each side in a specific product under a square root. The law of cosines, used for the angles, generalizes the Pythagorean theorem to non-right triangles by adding a correction term that accounts for the actual angle between two sides rather than assuming a 90-degree angle.
Step-by-Step: Calculating It By Hand
- 1Verify the triangle inequality: any two sides summed must exceed the third side.
- 2Calculate the semi-perimeter (sum of all three sides, divided by 2).
- 3Apply Heron's formula to find area from the semi-perimeter and three sides.
- 4Apply the law of cosines three times (once per angle) to find each interior angle.
Examples
Scalene triangle
Sides of 5, 6, and 7 form a valid triangle with an area of about 14.7 square units and three distinct interior angles.
Invalid triangle
Sides of 2, 3, and 10 fail the triangle inequality (2 + 3 is less than 10), so no actual triangle can be formed with those measurements.
Advantages
- Solves for area, perimeter, and all three angles from just the three sides
- Validates the triangle inequality automatically, catching impossible inputs
- Uses reliable, well-established geometric formulas
- Works for any valid triangle - scalene, isosceles, or equilateral
Common Mistakes
- Entering side lengths that don't satisfy the triangle inequality, producing no valid triangle
- Confusing this general three-sides calculator with the specialized Right Triangle Calculator, which only needs two legs
- Not double-checking measurement units are consistent across all three sides
- Rounding intermediate values manually, which can introduce small errors compared to the calculator's precise computation
Edge Cases to Watch For
- Three side lengths that fail the triangle inequality (one side longer than the sum of the other two) cannot form any valid triangle at all.
- A degenerate case where one side exactly equals the sum of the other two produces a triangle with zero area - a flat line, not a true triangle.
- Very thin, elongated triangles can produce Heron's formula calculations sensitive to small input rounding errors.
- The three calculated angles should always sum to exactly 180 degrees - a useful check on the calculation's internal consistency.
Common Use Cases
- Finding a triangle's area and angles from its three side lengths
- Geometry homework and coursework
- Construction, design, and land surveying calculations
- Verifying whether three given measurements can form a real triangle