6.04 Perimeter and area of composite shapes

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.

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

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

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 use what we know about the opposite sides of rectangles having the same length to find the unknown side lengths.

To find the length of the right side of rectangle 1 we take the 10 \text{ cm} side and subtract the 4 \text{ cm} side to get 6 \text{ cm}.

To find the length of the bottom side of rectangle 2 we add the 5 \text{ cm} side to the 7 \text{ cm} side to get 12\text{ cm} .

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:

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:

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.

Examples

Example 1

Consider the following figure.

a

Find the length x.

Worked Solution

Create a strategy

The vertical side on the left has the same length as the two vertical sides on the right added together.

Apply the idea

The side length x will be equal to the difference between the other two vertical side lengths.

\displaystyle x	\displaystyle =	\displaystyle 9 - 2 \text{ m}	Subtract the lengths of two vertical sides
	\displaystyle =	\displaystyle 7 \text{ m}	Evaluate

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b

Find the length y.

Worked Solution

Create a strategy

The horizontal side on the bottom has the same length as the two horizontal sides on the top added together.

Apply the idea

The side length y will be equal to the difference between the other two horizontal side lengths.

\displaystyle y	\displaystyle =	\displaystyle 12 - 4 \text{ m}	Subtract the lengths of two horizontal sides
	\displaystyle =	\displaystyle 8 \text{ m}	Evaluate

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c

Calculate the perimeter of the figure.

Worked Solution

Create a strategy

Add all the side lengths in the figure.

Apply the idea

Substitute x=7 \text{ m} and y=8 \text{ m}.

\displaystyle \text{Perimeter}	\displaystyle =	\displaystyle 2+12+9+4+7+8 \text{ m}	Add all the side lengths
	\displaystyle =	\displaystyle 42 \text{ m}	Evaluate

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Example 2

Find the perimeter of the figure.

Worked Solution

Create a strategy

Use the markings on all sides of the figure to identify each side length. Add all the side lengths in the figure.

Apply the idea

Four of the sides have length 6\text{ cm} and the other four have length 7\text{ cm}.

\displaystyle \text{Perimeter}	\displaystyle =	\displaystyle 4 \cdot 6+4 \cdot 7 \text{ cm}	Add all the side lengths
	\displaystyle =	\displaystyle 24 + 28\text{ cm}	Evaluate the multiplication
	\displaystyle =	\displaystyle 52\text{ cm}	Evaluate the sum

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Example 3

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

Worked Solution

Create a strategy

Add the two straight sides of the triangle and the circumference of the semicircle.

Apply the idea

First, let's find the partial perimeter of the triangle. The tick marks show that both sides are 17\text{ cm} . Keep in mind we are not including the third, 23\text{ cm} side in this calculation because it is not part of the perimeter of the composite figure.

\displaystyle \text{Perimeter} _\triangle

\displaystyle =

\displaystyle 17 + 17

Add the two given sides of triangle

\text{Perimeter}_\triangle= 34\text{ cm}

Now let's find the circumference of the semicircle. The circumference of a circle is given by C=2\pi r, but since we have a semicircle, half of a circle, we also need to divide by C=2\pi r.

To find the radius for the formula we will find half of the diameter. The diamter is 23 \text{ cm}, so the radius is 11.5 \text{ cm}.

\displaystyle \text{Circumference}	\displaystyle =	\displaystyle \dfrac{2\pi r}{2}	Use circumference formula divided by 2
	\displaystyle =	\displaystyle \dfrac{\cancel{2}\pi \cdot 11.5}{\cancel{2}}	Substitute r=11.5
	\displaystyle =	\displaystyle \pi \cdot 11.5	Divide out the 2
	\displaystyle \approx	\displaystyle 36.1 \text{ cm}	Evaluate to one decimal place

Now that we have found the partial perimeter of the triangle and the circumference of the semicircle, we can add the two perimeters together to get the perimeter of the entire figure.

\displaystyle \text{Perimeter}	\displaystyle \approx	\displaystyle 34 \text{ cm} + 36.1 \text{ cm}	Add the two perimeters together
	\displaystyle \approx	\displaystyle 70.1\text{ cm}	Evaluate the sum correct to one decimal place

Reflect and check

Notice that our answer is approximate, indicated by \approx. This is because we used an approximate value of \pi. If we wanted an exact answer we could have written our answer in terms of \pi as \left(34 +11.5 \pi \right) \text{ cm}.

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Example 4

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}.

Worked Solution

Create a strategy

Visualize the block of land as combination of triangle and rectangle as shown.

Use the Pythagorean theorem to find the length x.

Apply the idea

The two sides would be 8 \operatorname{m} and 32-17 = 15 \operatorname{m}.

\displaystyle a^{2}+b^{2}	\displaystyle =	\displaystyle a^{2}	Write the formula of Pythagorean theorem
\displaystyle 8^{2}+15^{2}	\displaystyle =	\displaystyle x^{2}	Substitute x, \, 8 and 15 into the formula
\displaystyle \sqrt{15^2+8^2}	\displaystyle =	\displaystyle x	Take the square root of both sides
\displaystyle \sqrt{225+64}	\displaystyle =	\displaystyle x	Evaluate the square roots
\displaystyle \sqrt{289}	\displaystyle =	\displaystyle x	Evaluate the addition
\displaystyle 17 \operatorname{ m}	\displaystyle =	\displaystyle x	Evaluate the square root
\displaystyle x	\displaystyle =	\displaystyle 17 \operatorname{ m}	Symmetric property

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b

Find the perimeter of the block of land in meters.

Worked Solution

Create a strategy

Add up all the lengths of the sides.

Apply the idea

\displaystyle \text{Perimeter}	\displaystyle =	\displaystyle 17+17+8+32 \operatorname{m}	Add together the four straight side lengths
\displaystyle \text{Perimeter}	\displaystyle =	\displaystyle 74 \operatorname{m}	Evaluate

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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.

We can divide this figure into two rectangles to find the area. We see that the top rectangle has an area of 5 \text{ m}\cdot3\text{ m}=15\text{ m}^2, and the bottom rectangle has an area of 8\text{ m}\cdot4\text{ m}=32\text{ m}^2. So the total figure has an area of 15\text{ m}^2+32\text{ m}^2=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.

For example, this shape consists of a square with side lengths of 6 \text{ cm} and two semicircles. We could also look at it as the combination of a square and a circle (since each semicircle is half of a circle) giving us the area calculation:

\text{Area}=\text{Area of square}+\text{Area of circle}

\text{Area}=6 \cdot 6+ \pi \cdot 6^{2} \approx 149.1 \text{ cm}^2

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.

Examples

Example 5

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

Worked Solution

Create a strategy

We can construct a line on the composite shape to break it up into its components.

Apply the idea

As we can see there are two trapezoids in this construction.

So, the correct option is D.

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b

Find the area of the composite shape.

Worked Solution

Create a strategy

We can find the area of the composite shape by adding the areas of the two trapezoids using formula \text{A}=\dfrac{1}{2}\left(b_1+b_2\right)h.

Apply the idea

Since the two trapezoids are identical we can multiply the trapezoid formula by 2.

\displaystyle \text{Area}	\displaystyle =	\displaystyle \frac{1}{2}h\left(b_1+b_2\right) \cdot 2	Formula for the area of a trapezoid times 2
	\displaystyle =	\displaystyle \frac{1}{2}(5)\left(7+15\right)\cdot 2	Substitute the values
	\displaystyle =	\displaystyle 110\, \text{cm}^2	Evaluate

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Example 6

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\cdot h	Use the area of parallelogram formula
	\displaystyle =	\displaystyle 19\cdot6	Substitute b=19 and h=6
	\displaystyle =	\displaystyle 114	Evaluate
\displaystyle \text{Both parallelograms}	\displaystyle =	\displaystyle 114\cdot2	Multiply the area by 2
	\displaystyle =	\displaystyle 228	Evaluate

Area of rectangle:

\displaystyle A	\displaystyle =	\displaystyle l\cdot w	Use the area of rectangle formula
	\displaystyle =	\displaystyle 19\cdot15	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

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Example 7

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\cdot b\cdot h	Use the area of triangle formula
	\displaystyle =	\displaystyle \dfrac12\cdot14\cdot3	Substitute b=14 and h=3
	\displaystyle =	\displaystyle 21	Evaluate

Area of parallelogram:

\displaystyle A	\displaystyle =	\displaystyle b\cdot h	Use the area of parallelogram formula
	\displaystyle =	\displaystyle 14\cdot6	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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