Free SSS Triangle Calculator

Understanding SSS Triangles and How to Solve Them

An SSS triangle refers to any triangle where the lengths of all three sides are known — the acronym stands for Side-Side-Side. When you have these three measurements but no angles, the triangle is fully determined in shape and size, but the interior angles and area still need to be computed. This is where a dedicated Triangle Side Side Side Calculator becomes invaluable, as it automates the otherwise tedious trigonometry.

The core technique for solving an SSS triangle relies on the Law of Cosines, which relates the sides of a triangle to the cosine of one of its angles. By applying this law twice, you can work out two of the internal angles, and the third angle follows naturally from the fact that the sum of the three angles in any triangle is always $180^{\circ}$.

Step-by-Step Solution Using the Law of Cosines

Given side lengths $a$, $b$, and $c$ (where $c$ is typically the longest side), the process is:

1. Determine angle $\gamma$ opposite side $c$ using the Law of Cosines:

   $$\gamma = \arccos\!\left(\frac{a^{2} + b^{2} - c^{2}}{2ab}\right)$$

2. Determine angle $\beta$ opposite side $b$ by applying the Law of Cosines again:

   $$\beta = \arccos\!\left(\frac{a^{2} + c^{2} - b^{2}}{2ac}\right)$$

3. Compute the remaining angle $\alpha$ as:

   $$\alpha = 180^{\circ} - \beta - \gamma$$

With a Triangle Angle Calculator that works on the SSS principle, you don’t have to manually evaluate the arccos function — the tool delivers all three angles instantly once you enter the three side lengths.

Calculating the Area of an SSS Triangle

Two reliable methods exist for finding the area when only sides are known:

• Heron’s Formula — a classic that works for any triangle. First compute the semi‑perimeter

  $$s = \frac{a + b + c}{2}$$

  then the area is

  $$\text{Area} = \sqrt{s(s-a)(s-b)(s-c)}$$

  This formula is elegant and requires no angle information.

• Using the sine of an angle — once an angle (such as $\gamma$) is known from the Law of Cosines, the area can also be expressed as

  $$\text{Area} = \frac{1}{2}ab\sin\gamma$$

A Heron’s Formula Calculator and a Triangle Area Calculator are often built into the same tool, so you get both the area and the complete angle set in one go.

Worked Example: Sides 2, 3, 4

To illustrate, take a triangle with sides $a = 2$, $b = 3$, and $c = 4$:

• Angle $\gamma$ (opposite $c$): $$\gamma = \arccos\!\left(\frac{2^{2}+3^{2}-4^{2}}{2\cdot2\cdot3}\right) \approx 104.48^{\circ}$$
• Angle $\beta$ (opposite $b$): $$\beta = \arccos\!\left(\frac{2^{2}+4^{2}-3^{2}}{2\cdot2\cdot4}\right) \approx 46.57^{\circ}$$
• Angle $\alpha = 180^{\circ} - 104.48^{\circ} - 46.57^{\circ} \approx 28.95^{\circ}$.

These three angles fully define the triangle’s shape. The area can then be found using Heron’s formula:
$s = \frac{2+3+4}{2} = 4.5$,

$$\text{Area} = \sqrt{4.5(4.5-2)(4.5-3)(4.5-4)} \approx 2.9047 \ \text{square units}.$$

Are SSS Triangles Congruent?

Yes — if two triangles have exactly the same three side lengths, they are congruent by the SSS congruence criterion. This means one can be mapped onto the other by rigid motions (rotation, reflection, translation). The calculator implicitly checks for consistency: if the entered side lengths cannot form a valid triangle (violating the triangle inequality), the tool will alert you.

Why Use a Dedicated SSS Triangle Solver?

Performing the Law of Cosines and Heron’s formula manually is straightforward but error‑prone, especially when dealing with non‑integer sides or when you need quick results. A SSS Triangle Calculator like this one streamlines the entire process: you input the three side lengths, and it outputs all angles, the area, and even the perimeter with no extra effort. Whether you are a student verifying homework, a professional doing geometry work, or just curious about triangle properties, this tool eliminates the busywork and lets you focus on the interpretation.