Triangle Area SAS Calculator
Calculate triangle area from two sides and included angle
Input Values
Results
Step-by-Step Solution
Understanding the SAS Area Formula
The SAS (Side-Angle-Side) area formula calculates the area of a triangle when you know two sides and the angle between them. It's derived from the standard area formula A = ½ × base × height.
The Formula
Variables
- a, b - The lengths of two sides of the triangle
- C - The angle between sides a and b (the included angle)
- sin(C) - The sine of angle C
Derivation
If we draw an altitude from vertex C to side c, its height h can be expressed as:
Since the area is ½ × base × height, and we can use either side as the base, we get Area = ½ × a × b × sin(C) when using the included angle.
Frequently Asked Questions
What does SAS stand for?
SAS stands for Side-Angle-Side, referring to a triangle configuration where you know two sides and the angle between them (the included angle). This is one of the triangle congruence theorems.
When should I use the SAS formula?
Use the SAS formula when you know two sides of a triangle and the angle between them, but don't know the third side or the height. It's more direct than calculating the height first.
Can I use any angle in the formula?
No, you must use the included angle - the angle that is between the two known sides. Using a different angle will give incorrect results.
Should I use degrees or radians?
Most scientific calculators and programming functions use radians for trigonometric calculations. This calculator accepts both and converts degrees to radians automatically when needed.
How do I find the third side?
Once you have two sides and the included angle, you can find the third side using the Law of Cosines: c² = a² + b² - 2ab·cos(C). This calculator shows this calculation as a bonus.
What if the angle is 90 degrees?
If the included angle is 90° (π/2 radians), sin(90°) = 1, so the formula simplifies to Area = ½ × a × b, which is the standard right triangle area formula.
Why use sine instead of cosine?
The sine function relates to the perpendicular height when an angle is given, which is essential for calculating area. The cosine is used in the Law of Cosines for finding side lengths.
What are practical applications?
This formula is used in surveying when measuring land plots, in engineering for structural analysis, in navigation for triangulation, and in computer graphics for rendering triangular surfaces.