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

Question

Evelyn Arboleda · Accepted Answer

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.

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)#]

Theodore Abad · Answer

undefined The formula for finding the area of a quadrilateral depends on the type of quadrilateral. Here are the formulas for some common types:

1. For a rectangle or parallelogram: Area = base × height

2. For a square: Area = side × side (or side squared)

3. For a trapezoid: Area = (1/2) × (sum of parallel sides) × height

4. For a kite: Area = (1/2) × product of diagonals

5. For a rhombus: Area = (1/2) × product of diagonals

6. For a general quadrilateral, if you know the lengths of all four sides and one angle (or the lengths of two diagonals and an angle between them), you can use the formula for the area of a general quadrilateral, known as Brahmagupta's Formula:

Area = √((s - a)(s - b)(s - c)(s - d)), where s is the semiperimeter of the quadrilateral, and a, b, c, d are the lengths of its sides.

These are some common formulas for finding the area of different types of quadrilaterals.