Triangle Area (SAS)
Area from two sides and the included angle.
Formula
Variables
Example
a=6,b=8,C=30° ⇒ 12
How it works
This formula finds the area of a triangle when you know two side lengths and the angle between them. It comes from the usual area formula "base × height ÷ 2": if one side is used as the base, the height is the other side multiplied by sin(C), because sine gives the perpendicular component of that side. So the area becomes half of a times b times sin C.
When to use it
Use this when a triangle is described by two sides and the included angle, especially in trigonometry, geometry, surveying, and physics problems involving force vectors or resultant directions. It is especially useful when the height is not given directly and would be awkward to find by other methods.
Worked example, step by step
Given a = 10, b = 7, C = 45° Area = (1/2)ab sin C Area = (1/2)(10)(7)sin(45°) Area = 35 × 0.7071 Area ≈ 24.75 Final answer: 24.75 square units
Common mistakes to avoid
- Using the formula with a non-included angle instead of the angle between the two given sides
- Forgetting to convert the angle mode or calculator setting before evaluating sine
- Leaving out the 1/2 factor and doubling the correct area
Did You Know?
The "one-half a b sine C" area formula needs no height at all — just two sides and the angle between them.
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More in Trigonometry
View allSine Ratio (SOH)
BasicRatio of the side opposite an angle to the hypotenuse.
Cosine Ratio (CAH)
BasicRatio of the side adjacent to an angle to the hypotenuse.
Tangent Ratio (TOA)
BasicRatio of the opposite side to the adjacent side.
Cosecant
IntermediateReciprocal of sine.
Secant
IntermediateReciprocal of cosine.
Cotangent
IntermediateReciprocal of tangent.