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Relationships Among Area Formulas of Various Quadrilaterals

Instructor: Michael Quist

Michael has taught college-level mathematics and sociology; high school math, history, science, and speech/drama; and has a doctorate in education.

Can't remember all those area formulas? In this lesson we'll look at all of the area formulas for quadrilaterals and see how they're related. You'll be remembering them all in no time!

The Area of a Quadrilateral

Remembering the area formulas for different quadrilaterals can be difficult, but understanding how they all connect together can make it a lot easier. In this lesson we will fit the formulas together, so you can understand how one leads to the next one. This will help you remember all of them!

A quadrilateral is a shape made of four straight lines with no openings. The area is the amount of space inside that shape. There are several kinds of quadrilaterals, based on how their four sides fit together.

From Simple to Complex

Square

Calculating the area of a square
image of square

The square is the simplest quadrilateral. Its sides are all the same length, and its corners meet at 90-degree angles (right angles). Calculating the area is easy: just multiply the length of one side times itself.

Rectangle

Calculating the area of a rectangle
image of rectangle

The rectangle is similar to the square, only the length is different than the width. For area, you just multiply the length times the width.

Rhombus

Calculating the area of a rhombus
image of rhombus

The rhombus has four equilateral sides, like the square. The corners are not 90 degrees, however, so you have to allow for the angle. It's a little trickier to calculate. You have to know how far it's leaning. If you have the height (the perpendicular distance between opposite sides), then you can multiply the height by the length of one of the sides, like the rectangle formula.

If you know one of the interior angles between two sides, you can use your calculator to get the sine of that angle, then multiply it by the square of the sides (like with the square, only with the sine modifier). Rhombuses are actually a special type of parallelogram, which brings us to our next quadrilateral.

Parallelogram

Calculating the area of a parallelogram
image of parallelogram

The parallelogram is like the rhombus, but they can have dissimilar width and length, like the rectangle. Like the rhombus, there are two main ways to find the area. You can find the height (perpendicular distance between opposite sides) and multiply it by the length of one of those sides. Or, you can use the sine of an angle, the same way you did with the rhombus. However, this time your formula will involve multiplying the sine of the interior angle by the different side lengths.

Trapezoid

The trapezoid has two parallel sides, but the lengths of all the sides can be different. To get the area, we'll treat it like a parallelogram, but we'll need the average of the two bases (the parallel sides).

Calculating the area of a trapezoid
image of trapezoid

Calculating the area of a trapezoid is similar to the length times height formula for the parallelogram, but you have to allow for the lengths of both bases (top and bottom), so you average them, then multiply by height.

Kite

Another interesting case is the kite. It has no parallel sides, but the two sides at one corner are the same length, and the two sides at the opposite corner are the same length. To do this problem we'll need the lengths of the diagonals (lines drawn from opposite corners) or the angles.

Calculating the area of a kite
image of kite

The formula with diagonals involves multiplying the two diagonals and dividing them by two. You can also use this diagonal formula with rhombuses and squares! The formula with the sine angle is nearly identical to the parallelogram's area formula - just be sure that you're using sides of different lengths and the angle between them when working with kites!

More Complicated Quadrilaterals

The examples above are the most common quadrilaterals that appear in life and on math problems. However, there are others out there. For example, you could run into a general quadrilateral that doesn't fit any of the criteria of the special cases or a complex quadrilateral where the lines cross.

Calculating the area of a general quadrilateral
image of general quadrilateral

In those cases, calculating the area can be a little more complicated. You can divide the quadrilateral into triangles, using trigonometry to calculate the individual areas of the triangles. For example, you can calculate the area of the odd-shaped quadrilateral shown by dividing it into two separate triangles and then using triangle area formulas with the sine modifier to calculate the two parts of the quadrilateral. Adding them together will produce the original area.

Calculating the area of a complex quadrilateral
image of complex quadrilateral

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