Irregular Quadrilaterals: Definition & Area

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  • 0:04 Irregular Quadrilaterals
  • 0:56 Area of an Irregular…
  • 2:16 Irregular…
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Lesson Transcript
Instructor: Laura Pennington

Laura received her Master's degree in Pure Mathematics from Michigan State University. She has 15 years of experience teaching collegiate mathematics at various institutions.

This lesson explores irregular quadrilaterals. Take a look at some common irregular quadrilaterals and their area formulas. We will also examine a more involved example of finding the area of an irregular quadrilateral without an area formula.

Irregular Quadrilaterals

Take a look at the window shown in the image.


Window
irrquad1


Let's talk about the characteristics of the shape of this window. For starters, notice that the window has four sides and four angles. In mathematics, we call a shape such as this a quadrilateral. We can classify quadrilaterals into two different groups; regular and irregular.

A regular quadrilateral is a quadrilateral with all sides having equal length. An irregular quadrilateral is the opposite of this, so it is a quadrilateral that is not regular. In other words, an irregular quadrilateral is a quadrilateral with sides that are not all equal in length.


irrquad2


Take a look at the window again. Notice that the sides are not all equal in length. Opposite sides have equal length, but they are not all equal. Therefore, this is an irregular quadrilateral.

Area of an Irregular Quadrilateral

Suppose we want to place a glass pane in the window, but we need to determine its size. In other words, we want to know the area of the window. In this case, we are dealing with a rectangle with a length of 5 feet and a width of 2 feet. We have a well-known formula for the area of a rectangle, and that is length times width. Therefore, we find the area of our window by multiplying 5 feet by 2 feet to get 10 square feet.

  • Area = length × width = 5 × 2 = 10

This is the size of the glass pane that we need for the window.

That was a really simple process. It's great when we are working with an irregular quadrilateral that has a nice area formula like this rectangular window did.


irrquad3


However, because all irregular quadrilaterals are different, we don't have a nice universal formula that we can use for all of them. When it comes to finding the area of an irregular quadrilateral that doesn't have a known area formula, a good strategy is to use the following steps:

  1. Split the quadrilateral into two triangles by drawing in a diagonal.
  2. Use various formulas and properties to find the area of each of the triangles.
  3. Add up the areas of the triangles.

That sounds easy enough, but it can be a bit involved. Let's consider an example of this.

Irregular Quadrilaterals Example

Suppose we want to find the area of the irregular quadrilateral shown.


irrquad4


Hmmm…well the first step in our strategy is easy enough. We simply split the quadrilateral into two triangles by drawing in a diagonal.


irrquad5


Now is where things get tricky. We can't use the formula for the area of a triangle, (1/2)(base)(height), since we don't know the triangles' heights. As the second step of our strategy states, we will need to use various rules and properties to find their areas. In this instance, those are as follows:

SAS method: The area of a triangle with two adjoining side lengths a and b, and with angle θ being the angle between these sides is 1/2absin(θ).

Law of Cosines: In a triangle with sides a, b, and c, and angles A, B and C opposite their respective sides, we have the following relationships:

  • c2 = a2 + b2 - 2abcos(C)
  • b2 = a2 + c2 - 2accos(B)
  • a2 = b2 + c2 - 2bccos(A)

Heron's Formula: If triangle has side lengths a, b, and c, then its area is:

  • Area = √(s(s - a)(s - b)(s - c)) where s = (a+b+c)/2.

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