Interactive calculator
Triangle calculator (SSS)
Given three sides, derive angles, area, perimeter.
Area (Heron)-
Perimeter-
Angle A (opposite a)-
Angle B-
Angle C-
Type-
About this tool
The Triangle Calculator computes area (using Heron's formula), perimeter, all three angles (using the law of cosines), and triangle type (equilateral, isosceles, scalene, right, acute, obtuse) from three side lengths.
What this triangle calculator does
This is an SSS (side-side-side) triangle solver: you supply the three side lengths a, b, and c, and it returns the area, the perimeter, all three interior angles, and the triangle's classification (equilateral, isosceles, scalene, and right, acute, or obtuse). Because three sides uniquely fix a triangle's shape and size, no other input is needed. The whole calculation runs in your browser, so nothing you type is ever uploaded.
SSS is the most common real-world case. When you measure a physical object such as a roof gable, a garden plot, or a steel bracket, you can almost always reach a tape measure across the three edges, but the height and the angles are awkward to measure directly. This tool fills in the rest from the three lengths you already have.
How it works: the formulas
Two classical results do all the heavy lifting. Heron's formula converts the three sides into an area without ever needing the height, and the law of cosines recovers each angle from the three sides.
Semi-perimeter s = (a + b + c) / 2 Area (Heron) Area = sqrt( s (s - a)(s - b)(s - c) ) Perimeter P = a + b + c Angle A A = arccos( (b^2 + c^2 - a^2) / (2 b c) ) Angle B B = arccos( (a^2 + c^2 - b^2) / (2 a c) ) Angle C C = 180 deg - A - B Validity a + b > c and a + c > b and b + c > a
The last line is the triangle inequality. If it fails for any pairing, the three lengths cannot form a closed triangle and the tool reports an invalid input rather than a misleading number. The angle sum check (A + B + C = 180 degrees) is a useful way to confirm a hand calculation.
Worked example: the 3-4-5 triangle
Take the classic sides a = 3, b = 4, c = 5, the smallest right triangle with whole-number sides (a Pythagorean triple).
- Semi-perimeter: s = (3 + 4 + 5) / 2 = 6.
- Area: sqrt(6 (6-3)(6-4)(6-5)) = sqrt(6 x 3 x 2 x 1) = sqrt(36) = 6 square units.
- Perimeter: 3 + 4 + 5 = 12 units.
- Angle A (opposite a = 3): arccos((16 + 25 - 9) / (2 x 4 x 5)) = arccos(32/40) = arccos(0.8) = 36.87 degrees.
- Angle B (opposite b = 4): arccos((9 + 25 - 16) / (2 x 3 x 5)) = arccos(18/30) = arccos(0.6) = 53.13 degrees.
- Angle C: 180 - 36.87 - 53.13 = 90.00 degrees.
Result: area 6, perimeter 12, angles 36.87 / 53.13 / 90 degrees. The 90-degree angle confirms it is a right triangle, and because no two sides match it is also scalene.
Common triangles for reference
A few triangles appear often enough that their exact values are worth recognising at a glance.
| Triangle | Sides (a, b, c) | Angles | Type |
|---|---|---|---|
| Equilateral | 1, 1, 1 | 60, 60, 60 | Acute, equilateral |
| 3-4-5 (Pythagorean) | 3, 4, 5 | 36.87, 53.13, 90 | Right, scalene |
| 5-12-13 (Pythagorean) | 5, 12, 13 | 22.62, 67.38, 90 | Right, scalene |
| 45-45-90 | 1, 1, 1.414 | 45, 45, 90 | Right, isosceles |
| 30-60-90 | 1, 1.732, 2 | 30, 60, 90 | Right, scalene |
| Obtuse example | 2, 3, 4 | 28.96, 46.57, 104.48 | Obtuse, scalene |
Common pitfalls
- Breaking the triangle inequality. Sides like 1, 2, 5 cannot form a triangle because 1 + 2 is less than 5. The tool will say so rather than invent an area.
- Mixing units. Enter all three sides in the same unit. Combining 30 cm with 12 inches produces a shape that does not exist.
- Confusing the SSS case with SAS or ASA. This solver needs three sides. If you only have two sides and the included angle, you must first find the third side with the law of cosines.
- Reading the angle next to the wrong side. Angle A is always opposite side a, not adjacent to it. Label your sketch before trusting the largest-angle classification.
- Rounding too early. If you round the semi-perimeter before squaring, the area can drift. Keep full precision until the final step, which the calculator does internally.
- Assuming a near-90 angle is exactly right. A triangle with sides 6, 8, 10.001 looks right but is very slightly obtuse. The classification reflects the exact arithmetic, not your expectation.
Related tools
Frequently asked questions
What is Heron's formula and when do I use it?
Heron's formula gives a triangle's area from its three side lengths alone: Area = sqrt(s(s-a)(s-b)(s-c)), where s = (a+b+c)/2 is the semi-perimeter. Use it whenever you know all three sides (the SSS case) but not the height, which is exactly what this calculator does.
How does the calculator find the angles?
It uses the law of cosines. For the angle A opposite side a, cos(A) = (b^2 + c^2 - a^2) / (2bc), then A = arccos(...) converted to degrees. The same formula finds B, and C is 180 minus A minus B because the interior angles of any triangle sum to 180 degrees.
Why does it say "not a valid triangle"?
Because your sides break the triangle inequality, which requires each side to be shorter than the sum of the other two. If a + b is less than or equal to c (or any similar pairing), the three lengths cannot close into a triangle, so no area or angles exist.
How does it decide if a triangle is right, acute, or obtuse?
It looks at the largest angle. If the largest angle is exactly 90 degrees the triangle is right, if it is below 90 it is acute, and if it is above 90 it is obtuse. The tool also flags equilateral (all sides equal) and isosceles (two sides equal) classifications separately.
Can I use any units for the sides?
Yes, as long as all three sides use the same unit. The perimeter comes out in that unit and the area in that unit squared. The angles are unit-free and always reported in degrees. Mixing centimetres and inches in the same triangle will give a meaningless result.