SSS Triangle Calculator
Enter three side lengths to solve the triangle's three interior angles, area, and perimeter using the Law of Cosines and Heron's formula.
Side lengths
All three values must use the same length unit and satisfy the triangle inequality: each pair of sides must add to more than the remaining side.
Results
Area 14.70 square units; angles 44.42°, 57.12°, and 78.46°; perimeter 18.00 units.
Calculation details
| Result | Method | Value |
|---|---|---|
| Angle α | Law of Cosines, opposite side a | 44.42° |
| Angle β | Law of Cosines, opposite side b | 57.12° |
| Angle γ | Law of Cosines, opposite side c | 78.46° |
| Area | Heron's formula | 14.70 square units |
| Perimeter | a + b + c | 18.00 units |
Angles are displayed to two decimal places. The underlying model keeps full floating-point precision for calculation and Excel export.
How to use the SSS triangle calculator
What this calculator does
This calculator solves a triangle when all three side lengths are known – the side-side-side, or SSS, case. It determines the interior angles opposite sides a, b, and c, then calculates the enclosed area and total perimeter. It is useful for checking geometry homework, laying out a triangular panel or plot, validating measured distances, and converting a three-side specification into the angle information needed for drafting. The result is a mathematical solution based on ideal straight sides. It does not account for measurement uncertainty, material thickness, curved edges, or construction tolerances.
When to use it
Use the calculator when you have three measured or specified side lengths but no angles. Typical cases include checking whether three survey distances can close into a triangle, finding cut angles for a triangular frame, confirming the area of a triangular piece from edge measurements, or verifying that two triangles with the same three corresponding sides have the same shape.
How to calculate
- The calculator opens with the ready-to-use example Side a = 5, Side b = 6, and Side c = 7. Its results and a validated Excel workbook are available immediately.
- Replace each sample value with your own positive decimal. Use one consistent unit – such as meters, feet, inches, or centimeters – for all three sides. Do not type the unit symbol into a field.
- Read Angle α, Angle β, and Angle γ as the angles opposite Side a, Side b, and Side c. Read Area in square units and Perimeter in the original length unit.
- Select Download Excel to create a current-state workbook containing the inputs, outputs, formulas, and calculation checkpoints. Select Reset to clear the demonstration and computed results. After Reset, the export is disabled until a complete valid triangle is entered again.
The angle calculation follows the OpenStax explanation of the Law of Cosines, which relates all three side lengths to each opposite angle.
Input guide
Side a is a required positive decimal measured in your chosen length unit; 5 is a realistic example. It is opposite Angle α. Increasing Side a while the other sides stay fixed generally increases Angle α, but the value must remain below Side b + Side c. A common mistake is assuming Side a is adjacent to α rather than opposite it.
Side b is a required positive decimal in the same unit as the other sides; 6 is the startup example. It is opposite Angle β. Increasing it tends to increase β and changes both area and perimeter. Do not mix units, such as entering one side in feet and another in inches.
Side c is a required positive decimal in the same unit; 7 is the startup example. It is opposite Angle γ. Increasing it tends to increase γ and the perimeter, but a value equal to or greater than Side a + Side b creates a degenerate or impossible triangle. The fields accept conventional dot-decimal notation, including grouped numbers such as 1,250.5; decimal-comma input is rejected to avoid ambiguity.
Output guide
Angle α, Angle β, and Angle γ are exact mathematical consequences of the three valid sides, displayed in degrees to two decimals. They are driven by the side opposite each angle and the other two sides through the Law of Cosines. A larger side is always opposite a larger angle. For a valid nondegenerate triangle, every angle is greater than 0°, less than 180°, and the three displayed values add to approximately 180° after rounding.
Area measures the enclosed two-dimensional region and is displayed in square units. It is computed from all three sides using Heron's formula. An area close to zero means the side lengths are close to a straight-line, degenerate configuration. Perimeter is the exact sum a + b + c in the original length unit; it rises one-for-one when any single side increases while the other two stay fixed.
The Calculation details table repeats every final output with its governing method. Its Angle rows use the Law of Cosines, its Area row uses Heron's formula, and its Perimeter row uses direct addition. The table and the Excel workbook read from the same full-precision model as the result cards.
Worked example
For Side a = 5, Side b = 6, and Side c = 7, the semiperimeter is s = (5 + 6 + 7) / 2 = 9. Heron's formula gives Area = √(9 × 4 × 3 × 2) = 14.6969..., displayed as 14.70 square units. Applying the Law of Cosines gives Angle α = 44.42°, Angle β = 57.12°, and Angle γ = 78.46°. The perimeter is 5 + 6 + 7 = 18.00 units. These are the exact first-open results and the values placed in the startup Excel workbook.
How the formulas work
For standard labeling, Side a is opposite Angle α, Side b is opposite Angle β, and Side c is opposite Angle γ. Solving for an angle rearranges the Law of Cosines:
α = arccos((b² + c² – a²) / (2bc))
The same pattern is applied cyclically for β and γ. Floating-point roundoff can produce a cosine value microscopically above 1 or below – 1, so the calculator safely clamps only that numerical artifact before applying arccos. The underlying identity and accepted side values are unchanged.
Area comes from Heron's formula: with s = (a + b + c) / 2, area equals √(s(s – a)(s – b)(s – c)). This is especially convenient for SSS data because no altitude or angle is required.
Validity and interpretation
Positive lengths alone are not enough. The three values must satisfy the triangle inequality: a + b > c, a + c > b, and b + c > a. Equality describes a collapsed line segment with zero area, not a proper triangle, so the calculator rejects it.
SSS data determines a triangle uniquely up to reflection. This is the geometric reason behind the SSS congruence theorem: two valid triangles with equal corresponding side lengths have equal corresponding angles and equal area. When measurements are approximate, small errors can produce larger angle changes near the degenerate boundary, so keep adequate measurement precision and avoid rounding the inputs prematurely.