Area of a Triangle

In triangle , given two side lengths and the included angle, the area of triangle is:

Explanation:

Consider triangle where the foot of the perpendicular from vertex to side is . The area of triangle can be analyzed based on three cases for the angle : acute, right, or obtuse.

Acute angle	Right angle	Obtuse angle

1. When (acute angle):
   
2. When (right angle):
   
3. When (obtuse angle):
   

In all cases, regardless of the size of , we have , so:

Similarly, we can derive:

Note:

In a quadrilateral , where the diagonals have lengths and and the angle between them is , the area of the quadrilateral is:

Explanation:

As shown in the diagram, we draw lines parallel to diagonal through points and , and lines parallel to diagonal through points and . If are the intersections of these lines, quadrilateral is a parallelogram.

The area of quadrilateral is half the area of parallelogram , and the area of triangle is also half the area of . Therefore, the area of quadrilateral is equal to the area of triangle :