# Areas of Combined Figures Involving Circles

## Calculate areas of irregular shapes.

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Perimeters of Composite Shapes

Learning Goal

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

Composite Figures and Perimeter

Sometimes it is useful to find the perimeter of a composite figure. A composite figure is larger shape that consists of two or more individual shapes. It is also sometimes called a combined figure. An example of this can be found in the picture below. This composite figure is made up of a square and two full circles (or 4 half circles if you see it that way).

[Figure1]

The perimeter of a two dimensional shape is the measurement of the distance around the outside of the object.

Here is a quick review video to act as a refresher on the concept of perimeter.

When dealing with a composite figure, the perimeter is still the measurement of the distance around the outside. This will usually leave lines on the shape that are not included in the perimeter. The perimeter of the shape above is highlighted in the picture below.

Note: The perimeter does not include the four straight lines of the square because they are INSIDE the shape. [Figure2]

Finding the Perimeter of a Composite Shape

To find the perimeter of a composite shape, simply identify which pieces of the shape are on the outside and add them up. This may involve the use of a variety of formulas from the formula sheet.

Example. Determine the perimeter of the following composite figures.

a)

[Figure4]

The composite shape appears to be made up of a rectangle and a triangle. The perimeter is all of the lines around the outside of the shape, so the dotted line is NOT part of the perimeter. Note the hash marks. Each line with a single hash mark has a length of 4cm and each line with the double hash mark has a length of 6cm. To find the perimeter simply add up the outside lengths.

\begin{align*}Perimeter = side + side + side + side + side\end{align*}

\begin{align*}Perimeter = 4 + 6 + 4 + 4 + 6\end{align*}

\begin{align*}Perimeter = 24cm\end{align*}

b)

[Figure5]

This composite shape is made up of two full circles and a square. However, the square is not part of the perimeter, only the circular edges are. To find the perimeter of this shape add the circumference (perimeter) of two circles. The circumference formula can be found on the formula sheet, and requires the measurement of the radius of the circle. Since the diameter of the circle is 4cm and the radiaus is half of the diameter, the radius is 2cm.

\begin{align*}Perimeter = 2\pi r + 2\pi r\end{align*}

\begin{align*}Perimeter = 2(3.14)(2)+2(3.14)(2)\end{align*}

\begin{align*}Perimeter = 12.56 + 12.56\end{align*}

\begin{align*}Perimeter = 25.12cm\end{align*}

c)

[Figure6]

This composite figure appears to be made up of two rectangles. Determining the perimeter will require one extra step first as two of the side lengths are not labeled. To determine the length of the top right side you can use the fact that 6cm - 4cm = 2cm. To find the other missing side you can use the fact that 7cm - 4cm = 3cm. These added values are found on the updated picture below.

[Figure7]

\begin{align*}Perimeter = s + s + s + s+ s +s\end{align*}

\begin{align*}Perimeter = 6 + 4 + 4 + 3 + 2 + 7\end{align*}

\begin{align*}Perimeter = 26cm\end{align*}

For a video overview of the concept, please watch the following video.

Practice

Directions: Find the perimeter of each composite figure.

1.

Solution: 31cm [Figure8]

2.

Solution: 46.56cm [Figure9]

3.

Solution: 14.05m [Figure10]

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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
9. [9]^ License: CC BY-NC 3.0
10. [10]^ License: CC BY-NC 3.0

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