Area of More Complicated Shapes

Area of Squares and Rectangles

Area is a measure of how much space there is inside a shape

$$\textrm{area} = \textrm{height} \cdot \textrm{width}$$

Area of More Complicated Figures

Suppose we want to calculate the area of this:

We can break the shape up into rectangles and find the area of each rectangle. The total area will be the sum of these individual rectangles.

Let's start by finding the area of the green rectangle. We have the height of the green rectangle (5 inches), but we do not have the width of the rectangle. However, we do know that the width of the green rectangle plus the width of the blue rectangle plus the width of the orange rectangle is 13 inches. We also know the area of the blue and orange rectangles. We can algebraically manipulate this information to find the width of the green rectangle:

$\textrm{width of green rectangle + width of blue rectangle + width of orange rectangle} \newline \textrm{= 13 inches }$$\newline$$\textrm{width of green rectangle + 4 inches + 6 inches = 13 inches } \newline \textrm{width of green rectangle + 10 inches = 13 inches} \newline \textrm{width of green rectangle = 3 inches}$

Since we now know the height and width of the green rectangle we can find its area.

$$\textrm{area} = \textrm{height} \cdot \textrm{width} = 5 \textrm{ in.} \cdot 3 \textrm{ in.} = 15\textrm{ in.}^2$$

For the blue rectangle, we already know the width. We can find the height by adding up the lengths we are already given:

So the height of the blue rectangle is 9 inches. Using the area formula the area is:

$$\textrm{area} = \textrm{height} \cdot \textrm{width} = 9 \textrm{ in.} \cdot 34\textrm{ in.} = 36\textrm{ in.}^2$$

Finally, we are given the height and width of the orange rectangle. Its area is:

$$\textrm{area} = \textrm{height} \cdot \textrm{width} = 1 \textrm{ in.} \cdot 6 \textrm{ in.} = 6\textrm{ in.}^2$$

To find the total area of the figure we add up the area of each of the individual rectangles.

$$\textrm{area of green rectangle + area of blue rectangle + area of orange rectangle = total area} \newline 15 \textrm{ in.}^2 + 36 \textrm{ in.}^2 + 6 \textrm{ in.}^2 = 57 \textrm{ in.}^2$$

The total area is 57 $\textrm{in.}^2$ Note that we could have divided the original figure differently.

Try finding the area of the rectangles in this figure and confirming that you do get the same area as we found before.

Checkpoint

1. Find the area of the following figure:

2. Find the area of the following figure

Remember, length must be a positive quantity.