SAS Triangle Calculator computes the missing side lengths and angles when you know two sides and the included angle between them. It uses the Law of Cosines to find the third side, then the Law of Sines to find the remaining angles.
What “SAS” Means in Triangle Math
In triangle geometry, SAS stands for Side–Angle–Side. You are given two sides and the angle between them. This is enough information to determine the triangle uniquely (assuming a valid, non-degenerate triangle).
- Given: side a, side b, and included angle γ (the angle between them).
- Find: third side c, plus the other angles α and β.
Variables Used by the SAS Triangle Calculator
To keep formulas consistent, this article uses the standard convention:
- a and b are the two known sides.
- γ is the included angle between sides a and b.
- c is the side opposite angle γ.
- α and β are the remaining angles.
Angle units can be entered as degrees (default) or radians. The calculator converts internally so the results are correct.
Core Formulas (Law of Cosines + Law of Sines)
1) Find the third side with the Law of Cosines
When you know two sides and their included angle, the third side comes directly from the Law of Cosines:
c² = a² + b² − 2ab cos(γ)
Then:
c = √(a² + b² − 2ab cos(γ))
2) Find the remaining angles with the Law of Sines
Once you have c, you can compute angles using the Law of Sines:
a / sin(α) = b / sin(β) = c / sin(γ)
Rearrange to solve for each unknown angle:
- sin(α) = a · sin(γ) / c
- sin(β) = b · sin(γ) / c
The calculator uses arcsin and then applies the triangle angle sum check (α + β + γ = 180°) to ensure consistent results.
Angle Conversion: Degrees vs. Radians
Trigonometric functions in math software typically expect radians. The calculator accepts either input and performs the conversion automatically.
- Degrees → Radians: θ(rad) = θ(deg) × π / 180
- Radians → Degrees: θ(deg) = θ(rad) × 180 / π
Output angles are shown in the unit you choose, so you can plug results back into your work without extra steps.
How to Use the SAS Triangle Calculator
Follow these steps to get correct results every time:
- Enter side a and side b.
- Select the unit for sides (e.g., meters, inches, feet). The calculator keeps the same unit for the output side c.
- Enter the included angle γ and choose its unit (degrees or radians).
- Choose your preferred output angle unit (degrees or radians).
- Click Calculate to get c, plus α and β.
If the inputs do not form a valid triangle (for example, an included angle of 0° or 180°), the calculator shows an error and highlights the field.
Practical Examples (Real-World Use)
Example 1: Engineering layout with a known included angle
Suppose you have two beams meeting at a joint. You know their lengths (a = 2.4 m, b = 1.6 m) and the angle between them (γ = 52°). The SAS triangle method gives the connector length c and the remaining joint angles α and β.
This avoids guesswork and reduces measurement error when you can measure two lengths and the included angle more reliably than the third side.
Example 2: Navigation and triangulation
In navigation or mapping, you may know distances from a point to two landmarks and the angle between those paths. With a and b as the distances and γ as the included angle, you can compute the distance between the landmarks (c) and the other angles to refine your route planning.
Because the included angle is part of the input, the solution is directly suited to SAS scenarios common in surveying and robotics.
Validation Tips: When SAS Inputs Don’t Make a Triangle
Even with correct formulas, not every set of inputs is geometrically valid. Use these checks:
- Angle must be between 0 and 180 degrees (exclusive). If γ is 0° or 180°, the triangle collapses.
- Sides must be positive. Zero or negative lengths are not valid geometric sides.
- Third side must be real. If the Law of Cosines expression becomes negative due to inconsistent inputs, the triangle cannot exist.
The calculator performs these validity checks and reports what needs fixing.
Frequently Asked Questions
What does the SAS triangle formula use to find the third side?
With SAS, you use the Law of Cosines to compute the third side. The formula is c = √(a² + b² − 2ab cos(γ)), where γ is the included angle between sides a and b. This works because it converts angle information into a side length.
How do you find the remaining angles after finding the third side?
After calculating c, use the Law of Sines. Specifically, sin(α) = a·sin(γ)/c and sin(β) = b·sin(γ)/c. Then convert arcsin results into your chosen angle unit. The triangle angle sum helps confirm the correct angle measures.
Can SAS produce two different triangles (ambiguous case)?
SAS does not have the classic SSA ambiguity because the included angle between the two known sides fixes the triangle’s shape. As long as the inputs form a valid, non-degenerate triangle, the solution for c, α, and β is unique. The calculator still validates for impossible cases.
What units can I enter for sides and angles?
You can enter side lengths in any consistent linear unit (meters, inches, feet, etc.). The calculator keeps that same unit for the output side c. For angles, enter degrees or radians; the calculator converts internally and outputs angles in your selected unit.
Why does my calculator show an error?
An error usually means the inputs cannot form a real triangle. Common causes are an included angle of 0° or 180°, non-positive side lengths, or a combination that makes the Law of Cosines expression negative. The calculator highlights the problem field so you can correct it.
Summary: Get Missing Sides and Angles Fast
The SAS Triangle Calculator gives you a reliable way to solve triangles when you know two sides and the included angle. It applies the Law of Cosines to find the third side, then the Law of Sines to determine the remaining angles.
Use it for engineering tasks, surveying, navigation, and any geometry problem where SAS data is available.
