ABC Triangle Calculator (SSS)
This calculator solves any triangle when you know the lengths of all three sides, the SSS (side-side-side) case in trigonometry. Enter side a, side b and side c in any consistent unit of length (centimetres, metres, inches - it does not matter, as long as all three match), and it works out every other property of the triangle. You get all three interior angles A, B and C in degrees, calculated using the law of cosines, plus the perimeter, semi-perimeter and exact area using Heron's formula, which finds area from the three sides without the height. Results are also classified two ways: by sides (equilateral, isosceles or scalene) and by angles (acute, right or obtuse). A diagram labels each side and angle so you can match the results to your own sketch. Before solving, the calculator checks the triangle inequality rule, that each side must be shorter than the sum of the other two, and flags an error if your lengths cannot form a real triangle. Results update live as you type. This is useful for maths homework, checking land or building plot dimensions, woodworking and construction layout, or any time you have measured three lengths and need the angles or area. The outputs depend entirely on the accuracy of what you enter, so measure carefully. Figures round to three decimal places and work with any consistent unit of length.
1. Enter the Three Sides
2. Results
Worked Example
Perimeter: 5 + 6 + 7 = 18 units
Semi-perimeter (s): 18 / 2 = 9
Area (Heron's formula): sqrt(9 x (9 - 5) x (9 - 6) x (9 - 7)) = sqrt(9 x 4 x 3 x 2) = sqrt(216) = 14.697 sq units
Angle A (law of cosines): arccos((6² + 7² - 5²) / (2 x 6 x 7)) = arccos(60 / 84) = 44.415°
Angle B: arccos((5² + 7² - 6²) / (2 x 5 x 7)) = arccos(38 / 70) = 57.122°
Angle C: arccos((5² + 6² - 7²) / (2 x 5 x 6)) = arccos(12 / 60) = 78.463°
Check: 44.415 + 57.122 + 78.463 = 180.000°
Type: Scalene (all sides different), Acute (all angles less than 90°)
How to Solve an ABC Triangle (SSS)
When you know all three side lengths of a triangle, you can find everything else about it using two key formulas.
The Law of Cosines
The law of cosines extends Pythagoras' theorem to any triangle. To find angle A (the angle opposite side a):
A = arccos((b² + c² - a²) / (2bc))
Repeat the formula, cycling through the sides, to find angles B and C. This always works as long as the three sides satisfy the triangle inequality (each side must be less than the sum of the other two).
Heron's Formula for Area
Heron's formula gives the area of a triangle from its three sides alone, without needing the height:
- Calculate the semi-perimeter: s = (a + b + c) / 2
- Calculate the area: Area = sqrt(s x (s - a) x (s - b) x (s - c))
Triangle Types
| Classification | By sides | By angles |
|---|---|---|
| Equilateral | All three sides equal | All angles 60° |
| Isosceles | Two sides equal | Two angles equal |
| Scalene | All sides different | All angles different |
| Acute | - | All angles less than 90° |
| Right | - | One angle exactly 90° |
| Obtuse | - | One angle greater than 90° |
Triangle Inequality
Three lengths can only form a valid triangle if each length is strictly less than the sum of the other two. For example, sides of 1, 2, and 10 cannot form a triangle because 1 + 2 = 3, which is less than 10. The calculator will alert you if the sides you enter fail this test.
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Method: Angles calculated using the law of cosines: A = arccos((b² + c² - a²) / (2bc)). Area calculated using Heron's formula: Area = sqrt(s(s-a)(s-b)(s-c)) where s = (a+b+c)/2. Triangle inequality check: each side must be less than the sum of the other two.
Results are rounded to three decimal places for display. The calculator works with any consistent unit of length (cm, m, inches, etc.).