Solve any triangle from its three sides. This free triangle calculator finds all angles, the area, perimeter and type (scalene, isosceles, equilateral; acute, right, obtuse) instantly.
Enter the three side lengths and the calculator finds all three angles, the area, the perimeter and the triangle type. Sides must satisfy the triangle inequality.
Angles are found with the law of cosines and the area with Heron’s formula. The three angles always add up to 180°.
How the Triangle Calculator Works
This tool solves a triangle completely from just its three side lengths. Enter side a, side b and side c, and it works out every angle, the area, the perimeter, and whether the triangle is scalene, isosceles or equilateral, and acute, right or obtuse.
Formula: Perimeter = a + b + c. Semi-perimeter s = Perimeter ÷ 2. Area (Heron’s formula) = √(s × (s − a) × (s − b) × (s − c)). Angle A (opposite side a) = arccos((b^2 + c^2 − a^2) ÷ (2 × b × c)). Angle B (opposite side b) = arccos((a^2 + c^2 − b^2) ÷ (2 × a × c)). Angle C = 180° − Angle A − Angle B, since the three angles of any triangle always sum to 180°.
- Side a — the length of the first side of the triangle
- Side b — the length of the second side of the triangle
- Side c — the length of the third side of the triangle
Example Scenarios
| Side a | Side b | Side c | Area |
|---|---|---|---|
| 3 | 4 | 5 | 6 |
| 5 | 5 | 6 | 12 |
| 6 | 6 | 6 | 15.59 |
| 7 | 8 | 9 | 26.83 |
| 8 | 15 | 17 | 60 |
| 5 | 12 | 13 | 30 |
Triangle Calculator FAQ
What happens if my three side lengths can’t actually form a triangle?
Any three sides must satisfy the triangle inequality — the two shorter sides added together must exceed the longest side. If they don’t, no valid triangle exists and the calculator can’t return angles or an area for those lengths.How does the calculator find the angles from just three side lengths?
It uses the law of cosines, which relates each angle to the squares of the three side lengths. Rearranging that relationship for the angle itself gives arccos((b^2 + c^2 − a^2) / (2bc)) for the angle opposite side a, and similarly for the others.What’s the difference between the “sides” type and “angles” type results?
The sides-based type (scalene, isosceles, equilateral) describes whether the three side lengths are all different, two equal, or all equal. The angles-based type (acute, right, obtuse) describes whether the largest angle is under 90°, exactly 90°, or over 90°.Can I enter decimal or non-integer side lengths?
Yes, the three side fields accept decimal values as well as whole numbers, so measurements like 4.5 or 12.75 work exactly the same as whole-number sides.Why do the three angles always add up to 180 degrees?
This is a basic property of flat (Euclidean) triangles: no matter their size or shape, the interior angles of any triangle always sum to exactly 180 degrees, which the calculator uses to double-check its own results.Does it matter which side I label a, b or c?
No. The labels are just for keeping track of which angle is “opposite” which side — swapping which side you call a, b or c changes which angle label goes where, but the triangle’s area, perimeter and set of angles come out the same either way.Related Calculators
If you’re working with other 2D shapes rather than triangles, the general area calculator and rectangle area calculator cover common flat shapes, while the circumference calculator handles circles. For solving a straight line’s steepness instead of a triangle’s angles, see the slope calculator.