What is the formula for finding the area of a quadrilateral?

1 Answer
Nov 21, 2015

If a convex quadrilateral is defined by all its sides and all its interior angles, it can be broken into two triangles and the area can be expressed as a sum of the areas of these triangles.

Explanation:

Assume in a convex quadrilateral #ABCD# we know all its sides and all its interior angles:
#AB = a#
#BC = b#
#CD = c#
#DA = d#
#/_BAD = alpha#
#/_ABC = beta#
#/_BCD = gamma#
#/_CDA = delta#

Draw a diagonal #AC#. It divides our quadrilateral into two triangles, and for each one of them we know two sides and an angle between them.

For triangle #Delta ABC# we know
#AB = a#
#BC = b#
#/_ABC = beta#

Taking #a# as a base of #Delta ABC#, the altitude would be #b*sin(beta)#.
The area of this triangle is
#S_(ABC) = 1/2*a*b*sin(beta)#

For triangle #Delta ADC# we know
#AD = d#
#CD = c#
#/_ADC = delta#

Taking #c# as a base of #Delta ADC#, the altitude would be #d*sin(delta)#.
The area of this triangle is
#S_(ADC) = 1/2*c*d*sin(delta)#

The total area of a quadrilateral is, therefore,
#S = 1/2[a*b*sin(beta)+c*d*sin(delta)#]