⚡

SAS Triangle Calculator

Solve any triangle when you know two sides and the included angle. Calculate the third side, remaining angles, area, and more

The angle must be between your two known sides (the included angle).

About This Tool

The SAS (Side-Angle-Side) case is when you know two sides of a triangle and the angle between them (the included angle). This is a very common scenario — you can often measure two edges of an object and the angle where they meet. This calculator uses the Law of Cosines to find the third side, then calculates all angles using either the Law of Cosines or Law of Sines. You get the complete triangle solution: all sides, all angles, area, perimeter, heights, and a visual diagram. The SAS case always produces exactly one unique triangle, unlike SSA which can be ambiguous.

How to Use

1. Choose which angle you know (A, B, or C) 2. Enter the angle value in degrees or radians 3. Enter the two sides adjacent to that angle 4. For angle A: enter sides b and c (the sides that form angle A) 5. For angle B: enter sides a and c (the sides that form angle B) 6. For angle C: enter sides a and b (the sides that form angle C) 7. All results appear instantly with step-by-step explanation

Formula

Law of Cosines (to find the third side): If you know sides b, c and angle A: a² = b² + c² - 2bc·cos(A) a = √(b² + c² - 2bc·cos(A)) Finding remaining angles: cos(B) = (a² + c² - b²) / (2ac) cos(C) = (a² + b² - c²) / (2ab) Or use Law of Sines: sin(B) / b = sin(A) / a Area: Area = ½ · b · c · sin(A)

Frequently Asked Questions

What is SAS in triangle solving?
SAS stands for Side-Angle-Side. It means you know two sides and the angle between them (the included angle). For example, knowing sides b=5, c=7, and angle A=60° between them. This information uniquely determines the triangle.
How do you find the third side in SAS?
Use the Law of Cosines: a² = b² + c² - 2bc·cos(A). For example, if b=5, c=7, and A=60°, then a² = 25 + 49 - 2(5)(7)·cos(60°) = 74 - 35 = 39, so a = √39 ≈ 6.24.
Why is the included angle important?
The included angle is the angle between the two known sides. If the angle is not between your two known sides, you have SSA (Side-Side-Angle), which is a different case that can have 0, 1, or 2 solutions.
How do you find the area with SAS?
Use the formula: Area = ½·b·c·sin(A), where A is the included angle and b,c are the two known sides. This is often easier than using Heron's formula when you have the SAS configuration.
Is SAS always solvable?
Yes, as long as the angle is between 0° and 180° (exclusive) and both sides are positive. SAS always produces exactly one unique triangle — there's no ambiguity like in the SSA case.
When would I use SAS vs SSS?
Use SAS when you know two sides and the angle between them. Use SSS when you know all three sides. In surveying and navigation, SAS is common because you can measure distances and the angle between them with instruments.
Can I use SAS if the angle is not between my two sides?
No, that would be SSA (ambiguous case), not SAS. In SSA, the angle is opposite one of the known sides, not between them. Make sure your angle is the one formed by your two known sides meeting at a vertex.

Related Tools

SSS Triangle Calculator

Solve any triangle when you know all three sides. Calculate angles, area, perimeter, heights, and see step-by-step solution with diagram

ASA Triangle Calculator

Solve any triangle when you know two angles and the side between them. Calculate all sides, the third angle, area, and more

AAS Triangle Calculator

Solve any triangle when you know two angles and a non-included side. Calculate all sides, the third angle, area, and more

SSA Triangle Calculator

Solve the ambiguous case: two sides and a non-included angle. Find all possible triangles (0, 1, or 2 solutions) with step-by-step analysis

Circle Calculator

Enter radius, diameter, area, or circumference — instantly get all other circle measurements with step-by-step formulas