If you know two non-congruent side lengths and the size of the angle between those two sides, use the formula: area = a * b * sin(α) , where α is the angle between sides a and b .
In order to calculate the height (distance BC in the diagram) you need to know the distance from point D (the closer of the two measurement points to the kite) to the point directly under the kite, B. Once you know this distance then multiplying this by the tangent of angle b will give you the height.
