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

One vertical rectangle and two right triangles facing each other to form a rectangle.

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

Composite shape composed of a rectangle with triangle on top, parallelogram below, and small square on the lower right.

Dashed lines can be used to visualize what 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.

This image shows a composite shape. Ask your teacher for more information.

In the figure, 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 equal a total of 7\text{ m} as well.

Using this fact, we know that the missing vertical length will add to 4\text{ m} to equal 7\text{ m}. Find the difference by subtracting, 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 we see that the top rectangle has an area of 5\times3=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 lengths that we are given, and whether it is possible to find the missing lengths.

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

Example 2

Find the total area of the figure shown.

Worked Solution

Create a strategy

Divide the shape into three shapes, two paralellograms and one 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 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 of the parallelograms and the rectangle
	\displaystyle =	\displaystyle 513\text{ cm}^2	Evaluate

Idea summary

Composite shapes are made from a number of smaller shapes. The area of the composite shape can be found by finding the area of each of the smaller shapes, and the adding them to get the total area.

Outcomes

6.G.A.1

Find area of right triangles, other triangles, special quadrilaterals, and polygons by composing into rectangles or decomposing into triangles and other shapes; apply these techniques in the context of solving real-world and mathematical problems.

7.04 Area of composite shapes

Ideas

Composite shapes

Examples

Example 1

Example 2

Outcomes

6.G.A.1

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