Oblique Triangle Solver (SSS) Calculator

Enter the three side lengths to calculate all angles, area, perimeter, circumradius, and inradius of the triangle.

Created by Luis GonzalezLast updated: May 9, 2026

How to Use This Calculator

1
Input Side a
Enter the length of the first side of the triangle. Ensure it's a positive value.
2
Input Side b
Provide the length of the second side. All three side lengths must satisfy the triangle inequality theorem (e.g., a + b > c).
3
Input Side c
Enter the length of the third side. All side lengths must be in consistent units.
4
Review Your Results
The calculator will display all three interior angles (Angle A, Angle B, Angle C), the area, perimeter, circumradius, and inradius.

Example Calculation

A homeowner wants to know the angles and area of a triangular garden plot with known fence lengths.

Side a

Side b

Side c

Results

34.1970

Tips

Triangle Inequality Check

Always confirm that the sum of any two sides is greater than the third side (a+b > c, a+c > b, b+c > a). If this condition isn't met, the three lengths cannot form a valid triangle, and the calculator will indicate an error.

Law of Cosines Accuracy

The calculation of angles relies on the Law of Cosines. For maximum precision, especially with decimal side lengths, ensure your inputs are accurate to avoid cumulative errors in the angle derivations.

Consistent Units

Use a single unit of measurement (e.g., meters, feet) for all three side lengths. The calculated perimeter, circumradius, and inradius will then be in that same unit, while the area will be in square units.

The Oblique Triangle Solver (SSS) Calculator is an essential tool for unlocking all geometric properties of a triangle when only its three side lengths are known. This powerful calculator instantly determines the measures of all three interior angles, the area, perimeter, and the radii of its inscribed and circumscribed circles. For architects and structural engineers, verifying that a triangular component's angles match design specifications to within 0.05 degrees is critical for structural integrity in 2025 construction.

Verifying Triangle Validity with Side Lengths

Before any calculations can begin for an SSS triangle, it is crucial to verify its validity using the Triangle Inequality Theorem. This fundamental geometric principle states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side. That is, for sides a, b, and c:

a + b > c
a + c > b
b + c > a

If any of these conditions are not met, the three segments cannot connect to form a closed triangular shape; they would either be too short to meet or would form a degenerate triangle (a straight line). For example, if you have sides of 3, 4, and 10 units, 3+4=7, which is not greater than 10, so these lengths cannot form a triangle. This theorem is applied in structural engineering to ensure that components of a truss can actually form a stable triangular unit.

The Law of Cosines for SSS Angle Determination

The primary method for determining the angles of an SSS triangle is the Law of Cosines. This law can be rearranged to solve for each angle when all three side lengths are known.

To find Angle A:

cos(A) = (b^2 + c^2 - a^2) / (2 × b × c)
A = arccos(cos(A))

Similarly, for Angle B:

cos(B) = (a^2 + c^2 - b^2) / (2 × a × c)
B = arccos(cos(B))

And for Angle C:

cos(C) = (a^2 + b^2 - c^2) / (2 × a × b)
C = arccos(cos(C))

After calculating two angles, the third can also be found by subtracting their sum from 180° as a verification.

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Analyzing a Park Pathway Design

Consider a landscape designer planning a triangular pathway system in a new park. The three path segments are measured as Side a = 8 meters, Side b = 9 meters, and Side c = 10 meters. The designer needs to know the interior angles to ensure smooth curves and the total area for material estimation.

Here's the calculation process:

Check Triangle Inequality:
- 8 + 9 > 10 (17 > 10 - True)
- 8 + 10 > 9 (18 > 9 - True)
- 9 + 10 > 8 (19 > 8 - True) The sides form a valid triangle.
Calculate Angle A: cos(A) = (9² + 10² - 8²) / (2 × 9 × 10) = (81 + 100 - 64) / 180 = 117 / 180 = 0.65. So, A = arccos(0.65) ≈ 49.46°.
Calculate Angle B: cos(B) = (8² + 10² - 9²) / (2 × 8 × 10) = (64 + 100 - 81) / 160 = 83 / 160 = 0.51875. So, B = arccos(0.51875) ≈ 58.74°.
Calculate Angle C: C = 180° - 49.46° - 58.74° = 71.80°.
Calculate Area (Heron's Formula): Semi-perimeter s = (8 + 9 + 10) / 2 = 13.5. Area = √[13.5 × (13.5-8) × (13.5-9) × (13.5-10)] = √[13.5 × 5.5 × 4.5 × 3.5] = √1169.4375 ≈ 34.1970 square meters.

The calculated Area is approximately 34.1970 square meters.

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Limitations of SSS Triangle Solving

While the SSS method is robust for solving triangles from three side lengths, there are specific scenarios where it can be misleading or numerically unstable. Firstly, if the side lengths are extremely disproportionate (e.g., sides of 1000, 1001, and 1), the triangle becomes very "thin" or "degenerate." In such cases, the Law of Cosines might involve subtracting nearly equal large numbers, leading to significant precision loss due to floating-point arithmetic in computational systems, making the calculated angles inaccurate. Secondly, the method is inherently limited to situations where all three side lengths are available. If only two sides and an angle (SAS or SSA) or two angles and a side (ASA or AAS) are known, different trigonometric laws are required. The SSS solver cannot provide a solution if the Triangle Inequality Theorem is violated, as no valid triangle can be formed from such inputs.

The Role of SSS in Geographic Information Systems (GIS)

In Geographic Information Systems (GIS), SSS triangle solutions are fundamental for various spatial analysis tasks, particularly in triangulation. When creating digital elevation models (DEMs) or triangulated irregular networks (TINs) from discrete survey points, the SSS method is implicitly used to define the geometry of the terrain. Given the coordinates of three points, their distances (side lengths) can be calculated, and then the internal angles and area of the resulting triangle can be determined. This process is crucial for accurately representing topography, calculating surface areas, and modeling hydrological flows. GIS professionals rely on these precise geometric computations to ensure that their spatial data representations are accurate, with typical error tolerances for distances often being in the sub-meter range for high-resolution mapping.

Frequently Asked Questions

What is an SSS triangle?

An SSS (Side-Side-Side) triangle is a triangle where the lengths of all three sides are known. This specific combination of information is one of the congruence criteria, meaning it uniquely determines the shape and size of the triangle. Knowing these three side lengths allows for the calculation of all other unknown properties, such as the three interior angles, the triangle's area, its perimeter, and radii of its inscribed and circumscribed circles.

How does the Law of Cosines help solve SSS triangles?

The Law of Cosines is the primary tool for finding the angles of an SSS triangle. By rearranging the formula (e.g., `cos(A) = (b² + c² - a²) / (2bc)`), you can directly calculate the cosine of each angle, and then use the arccosine function to find the angle itself. This process is repeated for all three angles, ensuring that their sum equals 180 degrees, thereby completely solving the triangle's angular properties.

What is Heron's Formula and when is it used for SSS triangles?

Heron's Formula is a method for calculating the area of a triangle when only the lengths of its three sides are known. It first requires calculating the semi-perimeter (s), which is half the perimeter (s = (a + b + c) / 2). The area is then found using the formula `Area = √[s(s-a)(s-b)(s-c)]`. This formula is particularly useful for SSS triangles because it avoids the need to calculate any angles or heights to determine the area, making it a direct and efficient approach.