SSS Triangle Calculator

Enter the three side lengths of a triangle (SSS) to solve for all three angles, the area, and the perimeter using the Law of Cosines and Heron's formula.

Quick Facts

Law of Cosines
cos(A) = (b² + c² − a²) / (2bc)
Solves for the angle opposite each side.
Heron's formula
Area = √(s(s−a)(s−b)(s−c))
Where s = (a + b + c) / 2 is the semi-perimeter.
Triangle inequality
a + b > c (and every permutation)
Each side must be shorter than the sum of the other two.
Angle sum
A + B + C = 180°
True for every triangle, once all three sides are valid.

Your Results

Calculated
Area
-
Heron's formula
Perimeter
-
a + b + c
Angles (A, B, C)
-
Opposite sides a, b, c respectively
Triangle Type
-
By sides and by largest angle

Ready

Enter three side lengths and press Calculate.

Formula and Method for the SSS Triangle Calculator

SSS ("side-side-side") means you know all three side lengths of a triangle and nothing else. That is enough information to fully determine the triangle — by the SSS congruence rule, three side lengths fix a unique triangle shape (up to reflection), so every angle, the area, and the perimeter can all be derived directly from a, b, and c.

How the calculation works

First the calculator checks the triangle inequality: each side must be strictly shorter than the sum of the other two (a + b > c, b + c > a, a + c > b). If that fails, no triangle exists. Otherwise it finds each angle with the Law of Cosines, solved for the angle opposite a given side — for example cos(A) = (b² + c² − a²) / (2bc) for the angle opposite side a. It repeats this for a second angle, then gets the third by subtracting the first two from 180° (since a triangle's interior angles always sum to 180°). For area it uses Heron's formula: compute the semi-perimeter s = (a + b + c) / 2, then Area = √(s(s−a)(s−b)(s−c)). The perimeter is simply a + b + c.

Common mistakes

  • Ignoring the triangle inequality: lengths like 2, 3, and 10 cannot form a triangle because 2 + 3 < 10 — the two short sides can never reach each other.
  • Mixing units: enter all three sides in the same unit (all in feet, or all in centimeters) before calculating; the tool does not convert between mismatched units.
  • Assuming angles are opposite the "wrong" side: angle A is always opposite side a, angle B opposite side b, and angle C opposite side c — the largest side is always opposite the largest angle.

Real-world applications

  • Surveying and construction use SSS solutions to lay out triangular plots or trusses when only distances (not angles) can be measured directly.
  • Carpentry and framing use the 3-4-5 rule (and its multiples) — a right triangle by the converse of the Pythagorean theorem — to square corners on site.
  • Navigation and triangulation reconstruct a triangle's angles from three known distances between landmarks or stations.
  • Structural engineering checks truss geometry and stability using side-length measurements alone.

Frequently Asked Questions

What is the SSS (side-side-side) method for solving a triangle?
SSS means you know all three side lengths of a triangle. Because three side lengths fix a triangle's shape and size uniquely (SSS congruence), you can solve for all three interior angles, the area, and the perimeter without knowing any angle in advance.
How do you find the angles of a triangle from its three sides?
Use the Law of Cosines solved for angle: cos(A) = (b² + c² − a²) / (2bc), where A is the angle opposite side a. Repeat for a second angle using a different pair of sides, then find the third angle as 180° minus the other two, since a triangle's angles always sum to 180°.
How is the area of a triangle calculated from its three sides? (Heron's formula)
Use Heron's formula. First find the semi-perimeter s = (a + b + c) / 2, then Area = √(s(s−a)(s−b)(s−c)). For sides 3, 4, and 5 this gives s = 6 and Area = √(6×3×2×1) = 6.
What side lengths can't form a valid triangle?
Any three lengths that fail the triangle inequality — where one side is longer than or equal to the sum of the other two (a + b ≤ c, or any permutation) — cannot form a triangle, because the two shorter sides could not reach each other to close the shape.

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