Triangle Area SAS Calculator — Two Sides and Included Angle
Calculate the area of a triangle when you know two sides and the angle between them (SAS). Enter sides a and b and the included angle C — the formula A = ½ab·sin(C) does the rest.
units
units
°
A = ½ × a × b × sin(C)
- 1
Convert angle to radians
60 × π ÷ 180 = 1.047198 - 2
sin(C)
sin(1.047198 rad) = 0.866025 - 3
Area = ½ × a × b × sin(C)
½ × 5 × 7 × 0.866025 = 15.1554
How does this calculator work?
Triangle area from two sides and their included angle: A = ½ab·sin(C). For a = 5, b = 7, C = 60°: A = ½ × 5 × 7 × sin(60°) ≈ 15.16 square units. The third side follows from the law of cosines: c = √(a² + b² − 2ab·cos C).
Formula
How this is calculated
The SAS (side-angle-side) configuration is one of the most common in applied geometry and trigonometry. When you know two sides of a triangle and the angle they enclose, you can find the area without needing a base and height measurement directly. The formula A = ½ × a × b × sin(C) comes from expressing the height of the triangle in terms of the angle: if C is the included angle, the perpendicular height from the opposite vertex to side... is h = b × sin(C), so area = ½ × a × h = ½ab·sin(C). When C = 90° the triangle is right-angled, sin(90°) = 1, and the formula reduces to the familiar A = ½ × a × b.
The calculator also derives the third side c using the law of cosines: c = √(a² + b² − 2ab·cos C). Once all three sides are known, the remaining angles A and B are computed by reapplying the law of cosines. The three angles always sum to exactly 180°, and a perimeter and height to the third side are also reported.
Inputs must be strictly positive side lengths and an angle strictly between 0° and 180°. An angle of 0° or 180° gives a degenerate flat triangle (zero area). The result is in squared units of whichever length unit you use for the sides — never mix units.
Frequently asked questions
Use the simpler formula A = ½ × base × height instead. The SAS formula is designed for when you know two sides and the angle between them — it internally computes the perpendicular height as h = b × sin(C).
The sine gives the perpendicular component of side b relative to side a. cos(C) is used to find the third side c via the law of cosines. sin(C) appears in the area formula because area = ½ × base × perpendicular height.
Yes. For an obtuse angle (90° < C < 180°), sin(C) is still positive (since sin is positive in the second quadrant), so the area formula works correctly. The resulting triangle will simply have one obtuse interior angle.
Also known as
TG we-Calculate Editorial Team. (2026). Triangle Area SAS Calculator — Two Sides and Included Angle [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/area-triangle-sas-calculator
TG we-Calculate Editorial Team. "Triangle Area SAS Calculator — Two Sides and Included Angle." TG we-Calculate. 2026. https://we-calculate.com/calculator/area-triangle-sas-calculator.
TG we-Calculate Editorial Team, "Triangle Area SAS Calculator — Two Sides and Included Angle," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/area-triangle-sas-calculator
@misc{wecalculate_area_triangle_sas_calculator, title = {Triangle Area SAS Calculator — Two Sides and Included Angle}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/area-triangle-sas-calculator}}, year = {2026}, note = {TG we-Calculate} }
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