SSS Triangle Calculator (Side-Side-Side)
Enter the lengths of all three sides and this calculator solves the triangle completely. You get all three interior angles (via the Law of Cosines), the area (Heron's formula), the perimeter, the three altitudes, the three medians, all three angle bisectors, the circumscribed circle radius, and the inscribed circle radius. Results update as you type, and the Show Your Work panel walks through each formula step by step.
What is an SSS triangle?
SSS stands for Side-Side-Side, one of the five standard triangle congruence cases. When all three side lengths are known, the triangle is uniquely determined (up to congruence): there is exactly one shape and size that those three sides can form, so every angle, every altitude, every median, the area, and both special circle radii are fixed. This contrasts with cases like SAS or ASA where you know fewer measurements. The two key tools for solving an SSS triangle are the Law of Cosines, which turns the three side lengths into angles, and Heron's formula, which delivers the area without needing any angle at all.
How to solve an SSS triangle step by step
Step 1: Verify the triangle inequality. Each side must be strictly less than the sum of the other two; if that fails, no triangle is possible. Step 2: Find the largest angle first using the Law of Cosines to avoid the ambiguous-case rounding issues that arise near 180 degrees. For angle A (opposite side a): cos A = (b squared + c squared - a squared) divided by (2bc), then A = arccos of that value. Repeat for B and C, or use A + B + C = 180 degrees to find the third angle. Step 3: Compute the semi-perimeter s = (a + b + c) / 2, then plug into Heron's formula: Area = square root of s(s - a)(s - b)(s - c). All remaining properties follow directly from these values.
Altitudes, medians, angle bisectors, inradius, and circumradius
Once the area and side lengths are known, six more lengths follow immediately. The altitude (height) to side a is h = 2A / a, where A is the area. The median from the opposite vertex to the midpoint of a is m = 0.5 times the square root of (2b squared + 2c squared - a squared). The angle bisector from the vertex opposite a to side a is t = 2bc cos(A/2) divided by (b + c). The inradius (radius of the inscribed circle that touches all three sides) is r = Area / s, where s is the semi-perimeter. The circumradius (radius of the circumscribed circle that passes through all three vertices) is R = (abc) / (4 times Area). This calculator returns all six of these for all three sides and vertices.
Checking your answer with the Pythagorean theorem
If the calculator classifies your triangle as a right triangle, you can cross-check with the Pythagorean theorem: the square of the longest side must equal the sum of squares of the other two. The classic 3-4-5 example: 3 squared + 4 squared = 9 + 16 = 25 = 5 squared. Similarly, 5-12-13 and 8-15-17 are exact integer right triangles (Pythagorean triples). For any other triangle type, the Law of Cosines generalises the Pythagorean theorem: c squared = a squared + b squared - 2ab cos C. When C equals 90 degrees the cosine term vanishes and the familiar theorem re-emerges.
Triangle classification by sides and angles
| Side classification | Angle classification | Description |
|---|---|---|
| Equilateral | Acute (60 deg each) | All three sides equal, all angles 60 deg |
| Isosceles | Acute | Two sides equal, all angles below 90 deg |
| Isosceles | Right | Two sides equal, one 90 deg angle (45-45-90 triangle) |
| Isosceles | Obtuse | Two sides equal, one angle above 90 deg |
| Scalene | Acute | All sides different, all angles below 90 deg |
| Scalene | Right | All sides different, one 90 deg angle (e.g. 3-4-5 triangle) |
| Scalene | Obtuse | All sides different, one angle above 90 deg |
Any SSS triangle falls into exactly one cell of this grid.
Frequently asked questions
Can any three lengths form a triangle?
No. The triangle inequality requires that each side be strictly shorter than the sum of the other two: a + b > c, a + c > b, and b + c > a. If any one of those three conditions fails, the sides are either too long to close into a triangle or collinear (degenerate). This calculator flags that case automatically.
Why does the Law of Cosines work for SSS triangles?
The Law of Cosines (c squared = a squared + b squared - 2ab cos C) is a generalisation of the Pythagorean theorem that works for any triangle, not just right triangles. Rearranged to cos C = (a squared + b squared - c squared) / (2ab), it lets you recover any angle from the three known sides. Applying it three times - once per angle - completely solves the triangle.
What is Heron's formula and when is it useful?
Heron's formula computes the area of a triangle directly from its three side lengths without needing any angle: Area = square root of s(s - a)(s - b)(s - c), where s is the semi-perimeter (a + b + c) / 2. It is especially useful in SSS problems because you have side lengths but no heights to plug into the base-times-height approach. For very large or very small triangles, prefer numerically stable variants to avoid floating-point cancellation errors.
What is the difference between inradius and circumradius?
The inradius r is the radius of the largest circle that fits entirely inside the triangle and touches all three sides. It equals the area divided by the semi-perimeter. The circumradius R is the radius of the unique circle that passes through all three vertices (the circumscribed circle). It equals the product of all three sides divided by four times the area. For an equilateral triangle R = 2r; for all other triangles R > 2r by Euler's inequality.
How do I find the height (altitude) of a triangle from SSS?
First solve for the area using Heron's formula. Then, because the area also equals half the base times the height, you can isolate the height for any base: height to side a = 2 times Area divided by a, and similarly for sides b and c. This calculator returns all three altitudes at once.
What is a median of a triangle?
A median connects a vertex to the midpoint of the opposite side. Every triangle has three medians and they all meet at a single point called the centroid, which is the triangle's centre of mass. The length of the median from the vertex opposite side a is given by m = half the square root of (2b squared + 2c squared - a squared).