# 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.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.

The blue rectangle has a width of

$12$

and a length of $4$

. The red rectangle has a width of $2$

, 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 $4$

units long, the length of the red rectangle must be $6$

units.The area of the figure is

$60$

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

146946### example

Find the area of the shaded region.Show Solution

Solution

We can break this irregular figure into a triangle and rectangle. The area of the figure will be the sum of the areas of triangle and rectangle.

The rectangle has a length of

$8$

units and a width of $4$

units.We need to find the base and height of the triangle.

Since both sides of the rectangle are

$4$

, the vertical side of the triangle is $3$

, which is $7 - 4$

.The length of the rectangle is

$8$

, so the base of the triangle will be $3$

, which is $8 - 4$

.Now we can add the areas to find the area of the irregular figure.

The area of the figure is

$36.5$

square units.### try it

[ohm_question]146949]### example

A high school track is shaped like a rectangle with a semi-circle (half a circle) on each end. The rectangle has length$105$

meters and width $68$

meters. Find the area enclosed by the track. Round your answer to the nearest hundredth.Show Solution

Solution

We will break the figure into a rectangle and two semi-circles. The area of the figure will be the sum of the areas of the rectangle and the semicircles.

The rectangle has a length of

$105$

m and a width of $68$

m. The semi-circles have a diameter of $68$

m, so each has a radius of $34$

m.### try it

[ohm_question]146952[/ohm_question][ohm_question]146953[/ohm_question]

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