Finding the Area of Irregular Figures

Learning Outcomes

  • Combine area of regular shapes to find the area of irregular shapes.

So far, we have found area for rectangles, triangles, trapezoids, and circles. An irregular figure is a figure that is not a standard geometric shape. Its area cannot be calculated using any of the standard area formulas. But some irregular figures are made up of two or more standard geometric shapes. To find the area of one of these irregular figures, we can split it into figures whose formulas we know and then add the areas of the figures.

example

Find the area of the shaded region.

An image of an attached horizontal rectangle and a vertical rectangle is shown. The top is labeled 12, the side of the horizontal rectangle is labeled 4. The side is labeled 10, the width of the vertical rectangle is labeled 2.

Solution
The given figure is irregular, but we can break it into two rectangles. The area of the shaded region will be the sum of the areas of both rectangles.

An image of an attached horizontal rectangle and a vertical rectangle is shown. The top is labeled 12, the side of the horizontal rectangle is labeled 4. The side is labeled 10, the width of the vertical rectangle is labeled 2.
The blue rectangle has a width of [latex]12[/latex] and a length of [latex]4[/latex]. The red rectangle has a width of [latex]2[/latex], but its length is not labeled. The right side of the figure is the length of the red rectangle plus the length of the blue rectangle. Since the right side of the blue rectangle is [latex]4[/latex] units long, the length of the red rectangle must be [latex]6[/latex] units.

An image of a blue horizontal rectangle attached to a red vertical rectangle is shown. The top is labeled 12, the side of the blue rectangle is labeled 4. The whole side is labeled 10, the blue portion is labeled 4 and the red portion is labeled 6. The width of the red rectangle is labeled 2.

The first line says A sub figure equals A sub rectangle plus A sub red rectangle. Below this is A sub figure equals bh plus red bh. Below this is A sub figure equals 12 times 4 plus red 2 times 6. Below this is A sub figure equals 48 plus red 12. Below this is A sub figure equals 60.
The area of the figure is [latex]60[/latex] square units.
Is there another way to split this figure into two rectangles? Try it, and make sure you get the same area.

 

try it

The following video gives another example of how to find the area of an “L” shaped polygon using the dimensions of two rectangles.

example

Find the area of the shaded region.

A blue geometric shape is shown. It looks like a rectangle with a triangle attached to the top on the right side. The left side is labeled 4, the top 5, the bottom 8, the right side 7.

 

try it

 

example

A high school track is shaped like a rectangle with a semi-circle (half a circle) on each end. The rectangle has length [latex]105[/latex] meters and width [latex]68[/latex] meters. Find the area enclosed by the track. Round your answer to the nearest hundredth.

A track is shown, shaped like a rectangle with a semi-circle attached to each side.

 

try it

The next video example is similar to the previous example, but the object for which we find area only contains one semi-circle.