Triangle area with sine
When two sides of a triangle and the included angle are known, the area can be found with
[A=\tfrac12 ab\sin(C)]
where (a) and (b) are the two given sides and (C) is the angle between them.
Method
- Identify the two sides that form the given angle.
- Substitute those side lengths into (\tfrac12 ab\sin(C)).
- Evaluate the sine of the angle.
- Multiply everything and simplify the result exactly when possible.
Important details
- Use the angle between the two sides, not an opposite angle.
- If the angle is special, simplify the sine exactly when you can, such as (\sin 30^\circ=\tfrac12) or (\sin 60^\circ=\tfrac{\sqrt3}{2}).
- Keep radicals and fractions in simplified form if the answer is exact.
Check your work
- The area must be positive.
- A quick reasonableness check is to compare with (\tfrac12\times\text{base}\times\text{height}); the sine formula should give the same area.
- If your answer is numerical, round only if the problem asks for it.