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Area of Composite Shapes

Shapes in the real world come in all sizes. Learn how to break down and calculate the area of composite shapes using the sum of areas of each part.

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Areas of Composite Shapes

Learning Goal

By the end of this lesson I will be able to find the area of a composite figure.


Composite Figures and Area

composite figure is a shape that is made up of other shapes. It is also known as a combined figure. Area is the amount of space inside of a two dimensional figure. To find the area of a composite figure you will need to find the area of the entire figure by either:

  1. Finding the area of each of the individual shapes and adding them together
  2. Finding the area of a larger shape and subtracting off the area that is not included.

It is a very useful strategy to first identify which shapes can be found within the composite figure before trying to find the area.

Formulas for the areas of different two dimensional shapes can be found on your formula sheet.

License: CC BY-NC 3.0

[Figure1]

Example. Determine the area of each of the following composite figures.

a) 

License: CC BY-NC 3.0

[Figure2]

This composite figure appears to be the combination of two rectangles. In order to determine the area, it will help to physically draw a line to separate the two shapes.

License: CC BY-NC 3.0

[Figure3]

\begin{align*}A_{TopRectangle} = lw = (7)(2) = 14cm^2\end{align*}

\begin{align*}A_{BottomRectangle}= lw = (4)(4) = 16cm^2\end{align*}

\begin{align*}Total Area = 14 + 16 = 30cm^2\end{align*}

b) 

License: CC BY-NC 3.0

[Figure4]

This composite figure is made up of a rectangle and half of a circle. The radius of the circle would be half the length of the dotted line. The dotted line measures 2.2m so the radius is 1.1m.

\begin{align*}A_{Rectangle}= lw = (4.2)(2.2) = 9.24m^2\end{align*}

\begin{align*}A_{HalfCircle}=\frac{\pi r^2}{2}=\frac{(3.14)(1.1)^2}{2}=1.9m^2\end{align*}

\begin{align*}A_{Total} = 9.24 +1.9 = 11.14 m^2\end{align*}

c) Determine the area of the shaded part of the diagram.

License: CC BY-NC 3.0

[Figure5]

This composite figure is made up of two circles. To determine the area of the shaded region, the easiest thing to do is to find the area of the larger circle and the area of the smaller circle and then subtract. Dividing the shaded part up into smaller shapes is difficult (if not impossible).

\begin{align*}A_{Large}=\pi r^2 = (3.14)(6)^2 = 113.04cm^2\end{align*}

\begin{align*}A_{small}=\pi r^2 = (3.14)(4)^2 = 50.24cm^2\end{align*}

\begin{align*}A_{Shaded} = 113.04 - 50.24 = 62.8cm^2\end{align*}

For a review of the concept covered, please watch the following video.

Practice

Directions: Determine the area of the following composite figures.

1.

License: CC BY-NC 3.0

[Figure6]

Solution: \begin{align*}Area = 600cm^2\end{align*}

2.

License: CC BY-NC 3.0

[Figure7]

Solution: \begin{align*}Area = 24m^2\end{align*}

3.Find the area of the shaded region.

License: CC BY-NC 3.0

[Figure8]

Solution: \begin{align*}Area = 201.5cm^2\end{align*}

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Image Attributions

  1. [1]^ License: CC BY-NC 3.0
  2. [2]^ License: CC BY-NC 3.0
  3. [3]^ License: CC BY-NC 3.0
  4. [4]^ License: CC BY-NC 3.0
  5. [5]^ License: CC BY-NC 3.0
  6. [6]^ License: CC BY-NC 3.0
  7. [7]^ License: CC BY-NC 3.0
  8. [8]^ License: CC BY-NC 3.0

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