Area of a Triangle from Two Sides and the Included Angle

Calculates the area from two sides and the angle between them (in degrees) as ½ × a × b × sin C. The height is not needed.

Explanation

Two sides and the angle between them are enough to get the area of a triangle. No height measurement needed.

S = \dfrac{1}{2} ab \sin C

$a$ , $b$ — the two known sides
$C$ — the angle between them, in degrees
$S$ — the area

This is the familiar base times height over two, with the height rewritten. Treat $a$ as the base; the perpendicular dropped from the far end of $b$ has length exactly $b \sin C$ . Substitute that in and the formula above falls out.

Example

With $a = 6$ , $b = 8$ and $C = 30^\circ$ , and $\sin 30^\circ = 0.5$ :

S = \dfrac{1}{2} \times 6 \times 8 \times 0.5 = 12

The area is 12.

Notes

Enter the angle in degrees, strictly between $0^\circ$ and $180^\circ$ . Anything outside that range is rejected.
$C$ must be the included angle, the one sitting between $a$ and $b$ . An angle opposite one of the sides gives the wrong area.
$\sin C$ peaks at 1 when $C = 90^\circ$ , so for two sides of fixed length the area is largest when they meet at a right angle.
Since $\sin C$ and $\sin(180^\circ - C)$ are equal, $30^\circ$ and $150^\circ$ produce the same area from the same two sides.
Keep the units consistent: sides in cm give an area in cm², sides in m give m².