SAS Triangle Calculator
Solve any SAS (side-angle-side) triangle with two sides and the included angle. Get missing side, angles, perimeter, and area using law of cosines and sines.
About This Calculator
The SAS (Side-Angle-Side) Triangle Calculator solves triangles where you know the lengths of two sides and the measure of the included angle between them. This is a fundamental concept in geometry and trigonometry, used extensively in surveying, navigation, engineering, architecture, and physics. Our free online solver uses the law of cosines and law of sines to compute all remaining triangle properties instantly.
Given two sides a and b and the included angle C (the angle between them), the calculator applies the law of cosines formula c^2 = a^2 + b^2 - 2ab·cos(C) to find the third side c. It then uses the law of sines to determine the remaining angles A and B, and computes the area using A = 1/2ab·sin(C) and the perimeter as the sum of all three sides. The SAS configuration is unique -- unlike SSA, it always produces exactly one triangle.
This calculator is ideal for students studying geometry and trigonometry, surveyors calculating land distances, architects designing triangular structures, and anyone needing to solve triangle problems. Simply enter the two side lengths and the included angle in degrees, click Calculate, and instantly get the missing side length, all three angles, perimeter, and area. Results are displayed to two decimal places for accuracy.
Frequently Asked Questions
What is a SAS triangle?
A SAS (side-angle-side) triangle is a triangle where you know the lengths of two sides and the measure of the included angle between them. This is one of the fundamental ways to uniquely define a triangle, as the angle between the two sides constrains the shape completely.
How do you solve a SAS triangle?
To solve a SAS triangle, first use the law of cosines c^2 = a^2 + b^2 - 2ab·cos(C) to find the third side. Then use the law of sines or cosines to find one of the remaining angles. Finally, subtract the two known angles from 180 deg to find the third angle. The area can be found using A = 1/2ab·sin(C) and the perimeter is the sum of all three sides.
What is the law of cosines formula for SAS?
For a SAS triangle with known sides a and b and included angle C, the law of cosines gives the third side c = sqrt(a^2 + b^2 - 2ab·cos(C)). This formula is derived from the Pythagorean theorem extended to non-right triangles.
Can SAS triangles have multiple solutions?
No, SAS triangles are uniquely determined. Unlike SSA (side-side-angle) which can have zero, one, or two solutions, the SAS configuration always produces exactly one triangle. This is why SAS is one of the triangle congruence postulates in geometry.
What is the area of a SAS triangle?
The area of a SAS triangle can be calculated directly from the two known sides and the included angle using the formula A = 1/2ab·sin(C), where a and b are the known sides and C is the included angle in radians. This formula works because 1/2ab·sin(C) gives the area of the parallelogram formed by the two vectors.
What is the difference between SAS and SSA triangles?
In SAS (side-angle-side), the angle is between the two known sides, which uniquely determines the triangle. In SSA (side-side-angle), the angle is not between the known sides, which can result in zero, one, or two possible triangles (the ambiguous case). SAS always yields a single unique triangle.
Is this SAS triangle calculator free?
Yes, all calculators on Calculy are completely free to use with no registration required. You can solve unlimited SAS triangles and share your results with others.