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

Approximately
`215` square units.

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) * (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}`.