Area of a Triangle SAS Calculator

Compute a triangle's area from two known sides and the included angle between them (Side-Angle-Side), using A = 1/2 × a × b × sin(C). Also returns the third side, perimeter, and triangle type.

Quick Facts

Formula
Area = 1/2 × a × b × sin(C)
a and b are two sides; C is the angle enclosed between them (SAS = Side-Angle-Side).
Third side
Law of Cosines: c² = a² + b² − 2ab·cos(C)
A generalization of the Pythagorean theorem for any angle.

Your Results

Calculated
Triangle Area
-
1/2 × a × b × sin(C)
Third Side (c)
-
Law of Cosines
Perimeter
-
a + b + c
Triangle Type
-
Classified by largest angle

Ready

Enter two sides and the included angle, then press Calculate.

Formula and method for the Area of a Triangle SAS Calculator

Area of a Triangle (SAS):

A = ½ × a × b × sin(C)

where a and b are two sides and C is the included angle between them

The SAS (Side-Angle-Side) method finds a triangle's area when you know two sides and the angle enclosed between them — no need to measure the height directly. The formula A = ½ × a × b × sin(C) follows from the standard base-times-height formula: if side b is treated as the base, then a × sin(C) is exactly the perpendicular height from the opposite vertex.

How the calculation works

Enter the two known sides (a and b) and the included angle (C), choosing whether C is measured in degrees or radians. The calculator computes the area directly from the SAS formula, then applies the Law of Cosines to find the missing third side, c = √(a² + b² − 2ab·cos C), and adds all three sides for the perimeter. It uses the Law of Cosines a second time to solve for the two remaining angles, then classifies the triangle as acute, right, or obtuse based on its largest angle.

Common mistakes

  • Wrong angle: C must be the angle physically between sides a and b, not one of the triangle's other two angles — using the wrong angle produces a different, incorrect triangle.
  • Degrees vs. radians: confirm the Angle unit selector matches how you entered C. An angle of 40 means something very different in radians (over 2,290°) than in degrees.
  • Degenerate angles: C must be strictly between 0° and 180° (0 and π radians). At those extremes the three points fall on a straight line and the "triangle" has zero area.

Real-world applications

  • Land surveying, where two boundary lengths and the angle between them are easier to measure directly than the far side of a plot
  • Engineering and construction layouts that specify two structural members and the angle joining them
  • Navigation and triangulation, where bearings give an included angle between two known distances
  • Trigonometry and geometry coursework whenever a problem supplies SAS data instead of three side lengths

Frequently Asked Questions

What does SAS mean in this calculator?
SAS stands for Side-Angle-Side: you provide the lengths of two sides of the triangle (a and b) and the measure of the angle between them (C). That is enough information to determine the triangle's area, its third side, and its remaining two angles uniquely.
What is the formula for the area of a triangle using SAS?
Area = ½ × a × b × sin(C), where a and b are the two known sides and C is the included angle between them. This works because a × sin(C) is the height measured relative to base b.
How is the third side calculated?
The third side c is found with the Law of Cosines: c² = a² + b² − 2ab·cos(C), then c is the square root of that value. This is a generalization of the Pythagorean theorem that works for any angle, not just 90°.
How does the calculator decide if the triangle is acute, right, or obtuse?
It solves for all three interior angles (using the Law of Cosines for the two not given directly) and looks at the largest one. A largest angle under 90° makes the triangle acute, exactly 90° makes it a right triangle, and over 90° makes it obtuse.

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