# Formula for the Area of a Quadrilateral Given Side Lengths

TR Smith is a product designer and former teacher who uses math in her work every day.

A quadrilateral is simply any polygon with four sides. Certain quadrilaterals have special names, such as rectangle, rhombus, square (the only shape which is both a rectangle and a rhombus), trapezoid, parallelogram, kite, and cyclic quadrilateral. But in general, most quadrilaterals shapes do not fall into any particular category, rather they are irregular quadrilaterals.

You are probably familiar with the formulas for finding the areas of the special quadrilaterals above, but there are also formulas to calculate the area of any arbitrary quadrilateral, whether or not it is irregular.

With the easiest formula for the area of a quadrilateral, all you need to know are the lengths of all four sides and the angle measures of two opposite corners. With a slightly more complicated formula, all you need to know are the lengths of the four sides and one angle. These formulas also work on the special cases listed above.

(Step 1) Pick any side of the quadrilateral, and working either clockwise or counterclockwise, label the sides A, B, C, and D. The letters A, B, C, and D will also stand for the lengths of these four sides.

(Step 2) Let x be the angle between sides A and B, and on the opposite corner let y be the angle between sides C and D. Measure these angles in degrees and call these angles x and y respectively.

(Step 3) Plug the six variables A, B, C, D, x, and y into the formula

Area = 0.5[ AB*sin(x) + CD*sin(y) ]

where "sin" is the sine function. You have now computed the area of your quadrilateral.

## Why does this formula work?

Every quadrilateral can be decomposed into two triangles. The sum of these two triangles is the area of the whole quadrilateral. From geometry, you may remember the SAS (side angle side) area formula for triangles. The formula states that if two sides of a triangle are P and Q and z is the angle between them, then the area of the triangle is 0.5*PQ*sin(z).

In the case of a quadrilateral, we can split into two triangles and apply the SAS area formula to sides A and B, and to sides C and D.

Adding the areas of the two triangles gives you 0.5*AB*sin(x) + 0.5*CD*sin(y), the total area of the quadrilateral.

## Example 1

Going around clockwise, the side lengths of a quadrilateral are 12 inches, 16 inches, 17 inches, and 13 inches. The angle measure between the sides of length 12 and 16 is 72 degrees. The angle measure between the opposite sides is 63 degrees.

Here we have A = 12, B = 16, C = 17, D = 13, x = 72, and y = 63. Plugging these values into the quadrilateral area formula gives us

Area =
0.5[ AB*sin(x) + CD*sin(y) ] =
0.5[ 12*16*sin(72) + 17*13*sin(63) ] =
0.5[ 192*0.253823 + 221*0.167356 ] =
0.5[ 85.719692 ] =
42.859846 square inches.

## Example 2

The quadrilateral area formula also works on a non-convex quadrilater, a four-sided shape where one of the corners goes "inward" like an arrowhead. All you need to do is pick the two opposite angles to be at convex corners and you can apply the formula as usual. For instance, suppose you have a non-convex quadrilateral as pictured below.

The area of this quadrilateral is

Area =
0.5[ AB*sin(x) + CD*sin(y) ] =
0.5[ 20*18*sin(40) + 11*24*sin(20) ] =
0.5[ 360*0.745113 + 264*0.912945 ] =
0.5[ 509.258160 ] =
254.629080 square units.

## Alternative Formula Using 4 Sides and 1 Angle

An alternative formula for the area of a quadrilateral requires four side lengths and only one angle. If the sides are A, B, C, D in that order, and the angle between A and B is x, then the area of the quadrilateral is

Area =
0.5*AB*sin(x) + 0.25*sqrt{ 4C²D² - [C² + D² - A² - B² + 2AB*cos(x)]² }

This formula is derived from the fact that the opposite angle y is completely determined by the angle x. If you cut the quadrilateral into two triangles, they share a common edge. The length of this edge is given by two equations

Edge = sqrt[ A² + B² - 2AB*cos(x) ]
Edge = sqrt[ C² + D² - 2CD*cos(y) ]

Setting these two equations equal to each other and solving for y gives an expression for y in terms of x. Plugging this expression into the first quadrilateral area formula discussed in this article produces the formula given here in terms of one angle, x.

Although the six-variable and five-variable formulas above work for any quadrilateral, it is usually easier to calculate the area of special quadrilaterals with their special formulas.

• Area of a square with side length A:
A^2
• Area of a rectangle with width W and length L:
WL
• Area of a rhombus with side length A and acute angle x:
(A^2)*sin(x)
• Area of a cyclic quardilateral with sides E, F, G, H:
sqrt[(S-E)(S-F)(S-G)(S-H)], where S = (E+F+G+H)/2
• Area of a trapezoid with parallel sides A and B and height H:
H*(A+B)/2
• Area of a parallelogram with sides A and B and acute angle x:
AB*sin(x)
• Area of a kite with one diagonal length P and the other diagonal length Q:
PQ/2

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