Investigation: Area with holes and cutaways

We can think of a composite shape as one that is made from a number of smaller shapes. Many complicated shapes can be made by combining simple shapes like triangles, squares, rectangles, and parallelograms in different ways.

The rectangle on the left is a composite shape built from two smaller triangles.

Often of the properties of these simpler shapes can be used to understand the composite shape. For example, knowing the total area of all the smaller shapes is the same as knowing the area of the whole composite shape.

Dashed lines can be used to visualize which simple shapes make up a composite shape.

We may not always initially know the length of every edge of a shape, but we can use the given information to work out missing lengths. This can be useful if a missing length is needed to find the area of a composite shape.

In the figure to the left, the vertical side on the left has a length of 7\text{ m}. Since the horizontal sides are parallel, the two vertical sides on the right must have a combined length of 7\text{ m} as well.

Using this fact, we know that the missing vertical length will add to 4\text{ m} to give 7\text{ m}. This number is found by taking the difference between the known vertical lengths, which gives 7-4=3\text{ m}.

Now that we know all the side lengths of the figure, we can break it up into two rectangles to find the area. In the figure below we see that the top rectangle has an area of 5\times13=15\text{ m}^2, and the bottom rectangle has an area of 8\times4=32\text{ m}^2. So the total figure has an area of 15+32=47\text{ m}^2.

There will usually be more than one way to break up a composite shape. Some ways may be easier than others, depending on the information that we start with, and whether it is possible to determine initially unknown information.

Examples

Example 1

Find the area of the figure shown.

Worked Solution

Create a strategy

Calculate the area of the triangle and rectangle, then add them.

Apply the idea

Area of the triangle:

\displaystyle A	\displaystyle =	\displaystyle \dfrac12\times b\times h	Use the area of triangle formula
	\displaystyle =	\displaystyle \dfrac12\times15\times4	Substitute b=15 and h=4
	\displaystyle =	\displaystyle 30	Evaluate

Area of the rectangle:

\displaystyle A	\displaystyle =	\displaystyle l\times w	Use the area of triangle formula
	\displaystyle =	\displaystyle 15\times9	Substitute b=15 and h=9
	\displaystyle =	\displaystyle 135	Evaluate

Total area:

\displaystyle A	\displaystyle =	\displaystyle 30+135	Add the areas
	\displaystyle =	\displaystyle 165\text{ mm}^2	Evaluate

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Example 2

Find the total area of the figure shown.

Worked Solution

Create a strategy

Divide the shape into three. Use the area formulas for a parallelogram and rectangle.

Apply the idea

Area of parallelograms:

\displaystyle A	\displaystyle =	\displaystyle b\times h	Use the area of parallelogram formula
	\displaystyle =	\displaystyle 19\times6	Substitute b=19 and h=6
	\displaystyle =	\displaystyle 114	Evaluate
\displaystyle \text{Both parallelograms}	\displaystyle =	\displaystyle 114\times2	Multiply the area by 2
	\displaystyle =	\displaystyle 228	Evaluate

Area of rectangle:

\displaystyle A	\displaystyle =	\displaystyle \times l\times w	Use the area of rectangle formula
	\displaystyle =	\displaystyle 19\times15	Substitute l=19 and w=15
	\displaystyle =	\displaystyle 285	Evaluate

Total Area:

\displaystyle A	\displaystyle =	\displaystyle 228+285	Add the areas
	\displaystyle =	\displaystyle 513\text{ cm}^2	Evaluate

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The examples above have explored how adding areas of simple shapes can help us determine the area of a complicated shape. A similar method involves subtracting the area of constituent shapes, and this is particularly useful for composite shapes that have holes and cutaways.

Imagine a rectangular sheet of paper. The area of the paper can be found easily by calculating the product of its length and width. Now picture taking a pair of scissors and cutting a small triangle off one of the corners of the sheet of paper. If we discard the small triangle, what is the area of the remaining paper?

The remaining area is what we get after having taken away the area of the small triangle from the original sheet. That is, we subtract the area of the small triangle from the area of the larger rectangle, and the result is the area of the composite shape that is the remaining paper.

Notice that we could have found the same area by breaking up the composite shape into two rectangles and one triangle, as shown in the figure below. But this method involves adding together three areas, while the method that uses negative area only involves taking the difference between two areas.

Two alternative ways to build the shape using only addition of areas.

Examples

Example 3

Find the shaded area in the figure shown.

Worked Solution

Create a strategy

Subtract the area of the triangle from the area of the parallelogram.

Apply the idea

Area of triangle:

\displaystyle A	\displaystyle =	\displaystyle \dfrac12\times b\times h	Use the area of triangle formula
	\displaystyle =	\displaystyle \dfrac12\times14\times3	Substitute b=14 and h=3
	\displaystyle =	\displaystyle 21	Evaluate

Area of parallelogram:

\displaystyle A	\displaystyle =	\displaystyle b\times h	Use the area of rectangle formula
	\displaystyle =	\displaystyle 14\times6	Substitute b=14 and h=6
	\displaystyle =	\displaystyle 84	Evaluate

Area of composite shape:

\displaystyle A	\displaystyle =	\displaystyle 84-21	Subtract the area of triangle from the area of parallelogram
	\displaystyle =	\displaystyle 63\text{ cm}^2	Evaluate

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