Area of Composite Figures

Lesson 2: Area - Area of Composite Shapes

   Constructing Knowledge

Recall, from Lesson 1 on perimeter and circumference, that a composite figure is a shape that is made out of multiple simple shapes. The process for calculating the area of a composite figure is similar to that of calculating the perimeter of composite figures.

1. Draw and label a diagram (if there isn't one provided).
2. Separate the figure into simple shapes, whose areas can be calculated. Often there are multiple ways to split a composite figure into parts, so choose a way that makes sense to you.
3. Calculate the area of each shape and then add the areas together.

EXAMPLE 1

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Find the area of the semicircle and rectangular key of an NBA basketball court.

Solution

Step 1: Separate the figure into simple shapes.

The basketball key and semicirlce can be split into one whole rectangle and half a circle.

Step 2: Calculate the area of each shape, and then add the areas together.

\(\begin{align} \text{A}_{\text{rectangle}}&=lw \\ \\ &=12\,\text{ft}\times 19\,\text{ft} \\ \\ &=228\,\text{ft}^2 \\ \end{align}\)

The area of the rectangular key is 228 ft².

\(\begin{align} \text{A}_{\text{semicircle}}&=\frac{\pi r^2}{2} \\ \\ &=\frac{\pi\times 6^2}{2} \\ \\ &=56.5\,\text{ft}^2 \\ \end{align}\)

The area of the semi-cirle is 56.5 ft².

\(\begin{align} \text{A}_{\text{total}}&=\text{A}_{\text{rectangle}}+\text{A}_{\text{semicircle}} \\ \\ &=228\,\text{ft}^{2}+56.5\,\text{ft}^2 \\ \\ &=284.5\,\text{ft}^2 \\ \end{align}\)

The area of the basketball key and semicircle is 284.5 ft².

   Multimedia

A video describing area of composite shapes is provided.

EXAMPLE 2

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A flower bed has been designed as shown. The entire flower bed is 20 m wide and the two identical triangular sections extend halfway into the bed. What is the area of the bed?

Solution

Step 1: Add the information given in the problem to the diagram provided.



Step 2: Separate the figure into simple shapes.

The figure needs to be split into shapes whose areas can be calculated. The line shown in the diagram above splits the flower bed into two identical triangles and one trapezoid.



Step 3: Calculate the area of each shape, and then add the areas together.

The total area will be the area of the trapezoid and the area of the 2 triangles added together.

\(\begin{align} \text{A}_{\text{trapezoid}}&=\frac{\left(a+b\right)h}{2} \\ \\ &=\frac{\left(10\,\text{m}+5\,\text{m}\right)10\,\text{m}}{2} \\ \\ &=75\,\text{m}^2 \\ \end{align}\)

\(\begin{align} \text{A}_{\text{two triangles}}&=\frac{bh}{2}\times 2 \\ \\ &=\frac{5\,\text{m}\times 10\,\text{m}}{2}\times 2 \\ \\ &=50\,\text{m}^2 \\ \end{align}\)

\(\begin{align} \text{A}_{\text{total}}&=\text{A}_{\text{trapezoid}}+\text{A}_{\text{two triangles}} \\ \\ &=75\,\text{m}^{2}+50\,\text{m}^2 \\ \\ &=125\,\text{m}^2 \\ \end{align}\)

The total area of the flower bed is 125 m².




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