Question

4. Look at the composite figure below. A vertex of the square is the center of the circle. The side
length of the square and the radius of the circle is 8 units. To the nearest whole unit, what is the
area of the figure?
8 units
F 265 square units
G66 square units
H 215 square units
J 128 square units
EKSING TOWARD STAAR © 2014
Page

Answer 1

Answer:

Approximately $215$ square units.

Step-by-step explanation:

The area of this figure is equal to:

$\begin{aligned}& \text{Area of figure} \\ =\; & \text{Area of square} \\ &+ \text{Area of circle} \\ &- \text{Area of overlap}\end{aligned}$.

The area of the square is $8^{2} = 64$ square units.

The area of the circle of radius $r = 8$ is $\pi\, r^{2} = 8^{2}\, \pi = 64\, \pi$ square units.

Refer to the diagram attached. In this figure, the overlap between the square and the circle is a sector of radius $r= 8$. The angle of this sector is $90^{\circ}$- same as the measure of the interior angle of the square.

The area of this sector would then be:

$\begin{aligned} & \pi\, r^{2} \times \frac{90^{\circ}}{360^{\circ}} = 16\, \pi \end{aligned}$.

Therefore, the area of the figure would be:

$\begin{aligned}& \text{Area of figure} \\ =\; & \text{Area of square} \\ &+ \text{Area of circle} \\ &- \text{Area of overlap} \\ =\; & 64 + 64\, \pi - 16\, \pi \\ \approx\; & 215 \end{aligned}$.