Quick answer
With two sides b and c and the included angle A, the third side is a = √(b² + c² − 2bc·cos A) and the area is ½bc·sin A. Sides 3 and 4 with a 90° angle give a = 5, angles of 36.87° and 53.13°, area 6 and perimeter 12.
Enter two sides and the angle between them. The calculator returns the third side, the other angles, area and perimeter. Use this sas triangle calculator to check your own figures.
Related tools: Triangle Calculator: Area, perimeter, angles and type of a triangle from its three sides. 45-45-90 Triangle Calculator: Sides of a 45°-45°-90° triangle from one leg. 30-60-90 Triangle Calculator: Sides of a 30°-60°-90° triangle from the short leg.
How to use the SAS Triangle Calculator
- Enter side b (above 0).
- Enter side c (above 0).
- Enter included angle A (°) (between 0 and 179.999999).
- Press Calculate to see the result, and use Reset to start again.
| Input | What to enter | Allowed values |
|---|---|---|
| Side b | A number | > 0 |
| Side c | A number | > 0 |
| Included angle A | A number, in ° | > 0, ≤ 179.999999 |
Formula and method
Formula
a² = b² + c² − 2bc·cos A; area = ½ bc sin A
IEEE-754 double precision; shown to at most 6 decimal places.
Law of cosines for the third side, then the law of sines for the remaining angles.
Worked examples
This example is verified: each one is an automated test case that the live calculator must reproduce exactly.
Example 1
Inputs: Side b = 3; Side c = 4; Included angle A = 90.
| Result | Value |
|---|---|
| Side a | 5 |
| Angle B | 36.869898° |
| Angle C | 53.130102° |
| Area | 6 |
| Perimeter | 12 |
The first line is the headline result (side a). Below it the sAS Triangle Calculator lists angle B, angle C, area and perimeter, which describe the same case from other angles. IEEE-754 double precision; shown to at most 6 decimal places.
Steps
The law of cosines gives a; then the law of sines or cosines gives the remaining angles. Area = ½ × 3 × 4 × sin 90° = 6.
Condition
The angle must lie strictly between 0° and 180° and be the one between the two known sides. SAS fixes a unique triangle (unlike SSA).
How the result changes with side b
Holding every other input at the first worked example, this table changes only side b. It shows how sensitive the answer is to that one value; each row was produced by running this calculator.
| Side b | Side a | Angle B | Angle C |
|---|---|---|---|
| 1.5 | 4.272002 | 20.556045° | 69.443955° |
| 3 | 5 | 36.869898° | 53.130102° |
| 6 | 7.211103 | 56.309932° | 33.690068° |
| 12 | 12.649111 | 71.565051° | 18.434949° |
What happens with invalid input
These messages are produced by the calculator itself, so you know what to fix.
- Included angle A: entering "180" shows "Included angle A must be at most 179.999999." beside that field and no result.
Choosing between the SAS Triangle Calculator and similar calculators
Each row shows what a related calculator on this site asks for and what it returns, so you can pick the one that matches your question.
| Calculator | You enter | It returns |
|---|---|---|
| [Triangle Calculator](/math/triangle-calculator/) | side a, side b and side c | Area |
| [45-45-90 Triangle Calculator](/math/45-45-90-triangle-calculator/) | leg length | Leg |
| [30-60-90 Triangle Calculator](/math/30-60-90-triangle-calculator/) | short leg length | Short leg |
| [Right Triangle Calculator](/math/right-triangle-calculator/) | leg a and leg b | Hypotenuse c |
| [Triangle Area Calculator](/math/triangle-area-calculator/) | base and height | Area |
Background: geometry measurements
Geometry results carry units that depend on the quantity: a perimeter or length is in linear units such as cm, an area or surface area is in square units (cm²) and a volume is in cubic units (cm³). Use one length unit for every input, because the calculator multiplies the numbers as given.
Flat (2D) shapes have an area and a perimeter; solids (3D) have a surface area and a volume. Two familiar checks: a circle of radius r has area πr² and circumference 2πr, and a sphere has volume (4/3)πr³ and surface area 4πr².
Assumptions and limitations
- Angle strictly between 0° and 180°
If a required value is empty, not a number, or outside its allowed range, the form marks that field with a message and shows no result instead of a misleading number. The automated checks deliberately try invalid entries for included angle A to confirm this.
Frequently asked questions
What does SAS stand for?
SAS means side, angle, side: two sides and the angle between them.
How is it different from SSS?
SSS gives three sides.
What if the angle is not included?
Use the sine rule with care.
What does the SAS Triangle Calculator calculate?
Solve a triangle from two sides and the included angle. It returns Side a, Angle B, Angle C, Area and Perimeter.
What formula does the SAS Triangle Calculator use?
a² = b² + c² − 2bc·cos A; area = ½ bc sin A. Law of cosines for the third side, then the law of sines for the remaining angles.
What do I need to enter in the sAS Triangle Calculator?
You enter Side b, Side c and Included angle A (°).
How is the result rounded or limited?
IEEE-754 double precision; shown to at most 6 decimal places. Angle strictly between 0° and 180°.
What is an example of a sAS Triangle result?
With side b 3, side c 4 and included angle A 90, the side a is 5.
Related calculators
- Triangle Calculator — Area, perimeter, angles and type of a triangle from its three sides.
- 45-45-90 Triangle Calculator — Sides of a 45°-45°-90° triangle from one leg.
- 30-60-90 Triangle Calculator — Sides of a 30°-60°-90° triangle from the short leg.
- Right Triangle Calculator — Solve a right triangle from its two legs: hypotenuse, angles, area and perimeter.
- Triangle Area Calculator — Area of a triangle from base and height.
- Equilateral Triangle Calculator — Area, height and perimeter of an equilateral triangle.
Sources
- OpenStax, Prealgebra 2e (Rice University): perimeter, area, volume, circumference and the Pythagorean theorem
Content last reviewed 2026-11-04. Formulas and worked examples are checked by automated tests against independently computed values.