Question

A sector of a circle with radius 8 cm has a central angle of 45°. What is the area of the corresponding segment?

• A. (8∏-16√2) cm²
• B. (8∏-32) cm²
• C. (16∏-32) cm²
• D. (16∏-64) cm²

Answer 1

Answer:

A. (8∏-16√2) cm²

The area of a segment is:

$\text{Area of segment}=\text{Area of sector}-\text{Area of triangle}.$

Step 1: Area of the sector

$\text{Area}=\frac{45^\circ}{360^\circ}\times \pi \times 8^2$
$=\frac{1}{8}\times 64\pi$
$=8\pi\ \text{cm}^2.$

Step 2: Area of the triangle

The triangle formed by the two radii has sides (8) cm and included angle (45^\circ).

$\text{Area}$
$=\frac12 r^2\sin\theta$
$=\frac12(8)^2\sin45^\circ$
$=32\cdot\frac{\sqrt2}{2}$
$=16\sqrt2\ \text{cm}^2.$

Area of the segment

$\boxed{8\pi-16\sqrt2\ \text{cm}^2}$