6. 2D Geometry

A **composite figure ** is any figure that can be subdivided into two or more shapes. Many composite shapes can be made by combining shapes like triangles, squares, rectangles, and parallelograms in different ways.

Recall that **perimeter** is the path or distance around any plane figure. The perimeter of a circle is called the circumference. We have formulas that can be helpful with certain shapes, including the perimeter of a rectangle, P= 2\cdot \left(l + w\right), and the circumference of a circle, C=2\pi r.

We can use what we already know about finding the perimeter of a polygon to calculate the perimeter of a composite shape.

When finding perimeter, we can use the side lengths we are given to find the lengths of any unknown sides.

Let's take a look at how we can break up this shape into two rectangles to help us find the length of the two unlabeled sides.

Original composite figure

Figure divided into two rectangles

Now we can find the perimeter by finding the sum of all the side lengths.

\text{Perimeter} = 10+5+6+7+4+12

=44 \text{ cm}

Sometimes we don't have enough information to find all of the unknown side lengths by decomposing a shape. Let's look at another way we could visualize the perimeter of a composite figure.

Here we have moved the sides of the figure to create a rectangle that has the same perimeter of the original figure. We know it has the same perimeter because we did not add, remove, or change the length of any sides.

So the perimeter of the composite shape will be the same as the perimeter of the rectangle which we can calculate as:

\displaystyle \text{Perimeter} | \displaystyle = | \displaystyle 2 \cdot \left(8+13\right) |

\displaystyle = | \displaystyle 2 \cdot 21 | |

\displaystyle = | \displaystyle 42 \text{ in} |

When using this method it is important to keep track of any sides that do not get moved.

Notice that we moved the indented edge to complete the rectangle but we still need to count the two edges that weren't moved because they are part of the perimeter of the original figure.

So we can calculate the perimeter of the composite shape as the perimeter of the rectangle, plus the two 2 \text{ ft} sides:

\displaystyle \text{Perimeter} | \displaystyle = | \displaystyle 2 \cdot \left(5+11\right)+2+2 |

\displaystyle = | \displaystyle 2 \cdot 16+4 | |

\displaystyle = | \displaystyle 32+4 | |

\displaystyle = | \displaystyle 36 \text{ ft} |

With our knowledge of the perimeter of simple shapes like rectangles, triangles, and circles, we can often find creative ways to work out the perimeter of more complicated composite shapes.

Consider the following figure.

a

Find the length x.

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b

Find the length y.

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c

Calculate the perimeter of the figure.

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Find the perimeter of the figure.

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Find the perimeter of the following figure. Use the \pi button on your calculator, rounding your final answer to one decimal place.

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Here is an outline of a block of land owned by a farmer, who wants to put up fencing along the land to help keep all their cattle safe.

a

Find the length of the side labeled x\operatorname{m}.

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b

Find the perimeter of the block of land in meters.

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Idea summary

The perimeter of the composite shape can be found by determining the length of each side and then summing them up.

We can divide composite figures into more familiar figures to help with finding unknown measures.

Formulas that can help us: the perimeter of a rectangle, P= 2\cdot \left(l + w\right), and the circumference of a circle, C=2\pi r.

The following applet shows how a composite shape can be broken down into pieces of basic shapes in order to find the area.

Drag the slider slowly to reveal each basic shape and its area.

Identify the basic shapes that make up the composite shape shown in the applet.

How does the total area of the composite shape compare to the sum of the areas of the individual shapes?

To find the area of composite shapes, break them down into simpler shapes like rectangles, triangles, and circles. Calculate the area of each shape, then add them together to determine the total area of the composite shape. This is often called the addition method.

There will usually be more than one way to break up a composite shape. Some ways may be easier than others, depending on the lengths that we are given, and whether it is possible to find the missing lengths.

Sometimes we don't have enough information to calculate the area of all of the individual shapes that make a composite figure. In these cases we may need to find the area of a larger shape and subtract the area that is not part of the composite shape. This is called the subtraction method.

Here are some area formulas we find ourselves using often with composite figures:

Rectangle: A=b\cdot h

Triangle: A = \dfrac{1}{2}\cdot b \cdot h

Parallelogram: A=b\cdot h,

Trapezoid: A=\dfrac{1}{2}h\left(b_1+b_2\right)

Square: A=s^{2}

Circle: A=\pi \cdot r^2

The area of a semicircle is half the area of a circle with the same diameter or radius.

Consider the composite shape.

a

Which basic shapes make up this composite shape?

A

A rectangle minus two triangles

B

One rectangle and two trapezoids

C

Two parallelograms

D

Two trapezoids

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b

Find the area of the composite shape.

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Find the total area of the figure shown.

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Find the shaded area in the figure shown.

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Idea summary

The area of the composite shape can be found by finding the area of each of the smaller shapes, and then adding them to get the total area.

Sometimes it is easier to take a subtractive approach and find the area of a larger, familiar figure and subtract the area of shapes that are not part of the composite figure.

Some formulas we often use are:

Area of Circle: A=\pi \cdot r^2

Area of a Trapezoid: A = \dfrac{1}{2} h (b_1+b_2)

Area of a Parallelogram: A=b\cdot h

Area of a Triangle: A = \dfrac{1}{2}\cdot b \cdot h

Area of a Rectangle: A=b\cdot h

Area of a Square: A=s^2