The Area of a Triangle Using Sine
Simple Explanation
The area of any triangle can be found from two sides and the sine of the angle between them: Area = ½ab·sinC — without needing to know the triangle's height directly.
Why Do We Need It?
This is often faster and more direct than the usual ½×base×height formula, especially when the height isn't given but two sides and the angle between them are.
See It
A triangle with vertex C at the origin, two sides a and b extending from it, and the included angle C marked between them
Formula
The Area of a Triangle Using Sine
Area = ½ab·sin C
The area of any triangle can be found from two sides and the sine of the angle between them, without needing to know the height directly.
- a, b
- — two side lengths of the triangle
- C
- — the angle included between sides a and b
When to use it: Whenever two sides and the included angle of a triangle are known, and the height is not directly given.
Worked Example
Find a triangle's area from two sides and the included angle
Find the area of a triangle with sides a = 8, b = 5, and included angle C = 30°.
Why Does This Work?
The usual area formula ½×base×height still applies — but the height (from vertex B down to base b, say) can itself be written as h = a·sinC using right-triangle trigonometry. Substituting this into ½×base×height gives ½·b·(a·sinC) = ½ab·sinC directly.
Real-Life Example
Measuring an irregular land plot
A surveyor measures two boundary edges of a triangular plot of land and the angle between them, but the perpendicular height is impractical to measure directly.
The sine area formula gives the plot's area directly from the two measured edges and the angle between them.
Practice
Find the area of a triangle with sides a = 10, b = 12, and included angle C = 90°.
MediumCommon mistake
Using two sides that do NOT actually include the given angle — the formula only works with the two sides that meet at the vertex where the known angle is measured.
Quick Review
- Area = ½ab·sinC, where C is the angle included between sides a and b.
- Especially useful when the height is not directly given.
- Derived by writing the triangle's height as h = a·sinC and substituting into ½×base×height.