Area of a Triangle (SAS) Calculator
Enter two sides of a triangle and the angle between them to get the area instantly using the SAS (side-angle-side) formula. The calculator also derives the third side via the law of cosines, finds all three angles, the perimeter, and all three altitudes. Switch between metric and imperial units, and see every step of the working below the result.
The SAS area formula
When you know two sides of a triangle and the angle between them, the area is given by the Side-Angle-Side (SAS) formula: Area = (1/2) × a × b × sin(γ), where a and b are the two known sides and γ (gamma) is the included angle. The sine function converts the angle into a ratio that scales the parallelogram formed by the two sides, and dividing by two gives the triangle. This formula works for any triangle, whether acute, right, or obtuse, as long as the angle is strictly between 0 and 180 degrees.
How the law of cosines completes the triangle
Knowing two sides and the included angle fully determines the triangle (the SAS congruence theorem). The third side follows from the law of cosines: c = sqrt(a^2 + b^2 - 2ab × cos(γ)). From there, the law of sines (sin(α)/a = sin(γ)/c) gives the remaining two angles, and the three altitudes are each 2 × Area divided by the corresponding base. This calculator works through all of these steps automatically, so you always get the complete picture of your triangle from just three inputs.
Why the maximum area occurs at 90 degrees
The factor that controls area is sin(γ). The sine function reaches its peak value of 1 at exactly 90 degrees, so for any fixed pair of side lengths a and b, the area is largest when the angle between them is a right angle. Below or above 90 degrees the sine falls, making the triangle thinner. A useful consequence: two triangles with the same pair of sides but supplementary angles (for example 60 and 120 degrees) have identical areas, because sin(60) = sin(120). When you are choosing an angle in a design problem, the chart above shows exactly how area scales across the full range of valid angles.
Metric and imperial length units
This calculator accepts any consistent length unit. Select metric to work in metres (area in m^2) or imperial to work in feet (area in ft^2). If your measurements are in centimetres, millimetres, inches, or kilometres, enter the values directly and interpret the area in the corresponding squared unit (cm^2, mm^2, in^2, or km^2). The formula is dimensionally consistent: as long as both sides use the same unit, the result is correct.
Area factor by included angle
| Angle γ (deg) | sin(γ) | Area factor vs. right angle (%) |
|---|---|---|
| 10 | 0.174 | 17.4 |
| 20 | 0.342 | 34.2 |
| 30 | 0.500 | 50.0 |
| 45 | 0.707 | 70.7 |
| 60 | 0.866 | 86.6 |
| 75 | 0.966 | 96.6 |
| 90 | 1.000 | 100.0 |
| 105 | 0.966 | 96.6 |
| 120 | 0.866 | 86.6 |
| 135 | 0.707 | 70.7 |
| 150 | 0.500 | 50.0 |
| 165 | 0.259 | 25.9 |
The multiplier sin(γ) applied to (1/2 × a × b). At 90 degrees it reaches its maximum of 1.000. Supplementary angles (e.g. 60 and 120 degrees) share the same sine and therefore the same area.
Frequently asked questions
What does SAS stand for in a triangle?
SAS stands for Side-Angle-Side. It describes a triangle specification where two sides are known along with the angle formed between those two sides (the included angle). Given these three values, the triangle is uniquely determined, meaning no other triangle has the same two sides with the same included angle.
What is the formula for the area of a triangle given two sides and the included angle?
Area = (1/2) × a × b × sin(γ), where a and b are the two known side lengths and γ is the included angle in degrees. Convert the angle to radians before applying the sine if your calculator requires it (radians = degrees × pi / 180), or use a calculator that accepts degrees directly.
Can I use the SAS formula if the angle is obtuse (greater than 90 degrees)?
Yes. The formula Area = (1/2) × a × b × sin(γ) is valid for any angle strictly between 0 and 180 degrees. The sine of an obtuse angle is positive (sin is positive in the first and second quadrants), so the result is still a positive area. For example, sin(120°) = sin(60°) = about 0.866, so a triangle with a 120-degree included angle has the same area as one with a 60-degree included angle (for the same pair of sides).
How do I find the third side and the other angles after calculating the area?
Once you have two sides and the included angle, the third side c is found with the law of cosines: c = sqrt(a^2 + b^2 - 2ab × cos(γ)). The remaining angles follow from the law of sines: sin(α)/a = sin(γ)/c, so α = arcsin(a × sin(γ) / c), and β = 180 - γ - α. This calculator performs all of these steps automatically.
Why is the area zero when the included angle is 0 or 180 degrees?
At 0 degrees the two sides point in the same direction and the triangle collapses to a line segment, giving zero area. At 180 degrees the sides point in opposite directions and again collapse to a line. Because sin(0) = sin(180) = 0, the formula correctly returns zero in both degenerate cases. The calculator excludes these exact boundary values because they do not form a valid triangle.
Does the formula work if both sides have different units?
No. Both sides must be expressed in the same unit for the formula to give a meaningful result. If side a is in metres and side b is in centimetres, convert one of them first. The resulting area will then be in the square of that unit (m^2, cm^2, ft^2, etc.).