  Math Topics

# Area of Composite Figures in Two Dimension

A composite figure is a figure composed of several geometric figures. It is important that you learn to recognize the different figures that make up composite figures in order to deal with them effectively. You will often have to find the area of a composite figure, and this will be easier if you can divide the figure into geometric figures that you already know how to find the area of.  In the figure above you have several known figures, and it shows a process for how to divide the large figures into smaller figures. There’s never just one way to partition a figure, so if you find another way, it’s just as correct!

Now that you understand how to split up a composite figure, let’s look at five examples where we’ll find the area of a composite figure.

Example 1

Find the area of the house: Assume the square has sides of $\text{}5\text{}\phantom{\rule{0.17em}{0ex}}\text{cm}$ and the triangle has a height of $\text{}4.33\text{}\phantom{\rule{0.17em}{0ex}}\text{cm}$ To find the total area, you have to find the area of the square and the triangle separately first, and then add them together: $\begin{array}{llll}\hfill A& =5\phantom{\rule{0.17em}{0ex}}\text{cm}\cdot 5\phantom{\rule{0.17em}{0ex}}\text{cm}+\frac{5\phantom{\rule{0.17em}{0ex}}\text{cm}\cdot 4.33\phantom{\rule{0.17em}{0ex}}\text{cm}}{2}\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill & \approx 25\phantom{\rule{0.17em}{0ex}}{\text{cm}}^{2}+10.83\phantom{\rule{0.17em}{0ex}}{\text{cm}}^{2}\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill & \approx 35.83\phantom{\rule{0.17em}{0ex}}{\text{cm}}^{2}\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\end{array}$

Example 2

Find the area of the composite figure: Just as in the previous example, you have to find the area of the square and the trapezoid separately. Then you add the two areas together to find the area of the entire figure:

$\begin{array}{llll}\hfill A& =4\phantom{\rule{0.17em}{0ex}}\text{cm}\cdot 4\phantom{\rule{0.17em}{0ex}}\text{cm}\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill & \phantom{=}+\frac{\left(5\phantom{\rule{0.17em}{0ex}}\text{cm}+3\phantom{\rule{0.17em}{0ex}}\text{cm}\right)\cdot 2\phantom{\rule{0.17em}{0ex}}\text{cm}}{2}\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill & =16\phantom{\rule{0.17em}{0ex}}{\text{cm}}^{2}+8\phantom{\rule{0.17em}{0ex}}{\text{cm}}^{2}\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill & =24\phantom{\rule{0.17em}{0ex}}{\text{cm}}^{2}\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\end{array}$

$\begin{array}{llll}\hfill A& =4\phantom{\rule{0.17em}{0ex}}\text{cm}\cdot 4\phantom{\rule{0.17em}{0ex}}\text{cm}+\frac{\left(5\phantom{\rule{0.17em}{0ex}}\text{cm}+3\phantom{\rule{0.17em}{0ex}}\text{cm}\right)\cdot 2\phantom{\rule{0.17em}{0ex}}\text{cm}}{2}\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill & =16\phantom{\rule{0.17em}{0ex}}{\text{cm}}^{2}+8\phantom{\rule{0.17em}{0ex}}{\text{cm}}^{2}\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill & =24\phantom{\rule{0.17em}{0ex}}{\text{cm}}^{2}\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\end{array}$

Example 3

Find the area of the composite figure: You divide the figure into a square and a trapezoid, and find those areas separately. Then you sum up the two areas to find the area of the entire figure:

$\begin{array}{llll}\hfill A& =5\phantom{\rule{0.17em}{0ex}}\text{cm}\cdot 5\phantom{\rule{0.17em}{0ex}}\text{cm}\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill & \phantom{=}+\frac{\left(6\phantom{\rule{0.17em}{0ex}}\text{cm}+3\phantom{\rule{0.17em}{0ex}}\text{cm}\right)\cdot 4\phantom{\rule{0.17em}{0ex}}\text{cm}}{2}\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill & =25\phantom{\rule{0.17em}{0ex}}{\text{cm}}^{2}+18\phantom{\rule{0.17em}{0ex}}{\text{cm}}^{2}\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill & =43\phantom{\rule{0.17em}{0ex}}{\text{cm}}^{2}\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\end{array}$

$\begin{array}{llll}\hfill A& =5\phantom{\rule{0.17em}{0ex}}\text{cm}\cdot 5\phantom{\rule{0.17em}{0ex}}\text{cm}+\frac{\left(6\phantom{\rule{0.17em}{0ex}}\text{cm}+3\phantom{\rule{0.17em}{0ex}}\text{cm}\right)\cdot 4\phantom{\rule{0.17em}{0ex}}\text{cm}}{2}\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill & =25\phantom{\rule{0.17em}{0ex}}{\text{cm}}^{2}+18\phantom{\rule{0.17em}{0ex}}{\text{cm}}^{2}\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill & =43\phantom{\rule{0.17em}{0ex}}{\text{cm}}^{2}\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\end{array}$

Example 4

Find the area of this figure: You need to begin by finding the area of each of the known figures: The rectangle (red), the triangle (gray) and the semicircle (blue). Then you can add the areas together:

$\begin{array}{llll}\hfill {A}_{\text{whole}}& ={A}_{\text{semicircle}}+{A}_{\text{rectangle}}\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill & \phantom{=}+{A}_{\text{triangle}}\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill & \approx 25.12\phantom{\rule{0.17em}{0ex}}{\text{cm}}^{2}+20\phantom{\rule{0.17em}{0ex}}{\text{cm}}^{2}\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill & \phantom{=}+10\phantom{\rule{0.17em}{0ex}}{\text{cm}}^{2}\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill & \approx 55.12\phantom{\rule{0.17em}{0ex}}{\text{cm}}^{2}\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\end{array}$

$\begin{array}{lll}\hfill & \phantom{\rule{2em}{0ex}}& \hfill \\ \hfill {A}_{\text{whole}}& ={A}_{\text{semicircle}}+{A}_{\text{rectangle}}+{A}_{\text{triangle}}\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill & \approx 25.12\phantom{\rule{0.17em}{0ex}}{\text{cm}}^{2}+20\phantom{\rule{0.17em}{0ex}}{\text{cm}}^{2}+10\phantom{\rule{0.17em}{0ex}}{\text{cm}}^{2}\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill & \approx 55.12\phantom{\rule{0.17em}{0ex}}{\text{cm}}^{2}\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\end{array}$

Example 5

Find the area of the composite figure You begin by finding the area of the different figures separately, and then you add the areas together:

Finding all the areas will look like this: