Question

What is the area of the segment formed by a chord in a circle of radius 8 cm, if the angle subtended at the center is 60°?

• A. $$\frac{32\pi}{3}-16\sqrt3$$
• B. $$64∏.3-16\sqrt3$$
• C. $$\frac{32\pi}{3}-8\sqrt3$$
• D. $$\frac{64\pi}{3}-32\sqrt3$$

Answer 1

Answer:

$$\frac{32\pi}{3}-16\sqrt3$$

The area of a segment is:

$\text{Area of Segment}=\text{Area of Sector}-\text{Area of Triangle}$

Given:

• Radius (r=8) cm

• Central angle (\theta=60^\circ)

Step 1: Area of the sector

$\frac{60}{360}\times\pi(8)^2$
$=\frac{1}{6}\times64\pi$
=\frac{32\pi}{3}\ \text{cm}^2$<br></p><p data-pxy="true" style="color: rgb(0,0,0); margin-top: 2px; margin-bottom: 2px;"> Area of the triangle</p><p data-pxy="true" style="color: rgb(0,0,0); margin-top: 2px; margin-bottom: 2px;">The two radii and the chord form an <b>equilateral triangle</b> (since the central angle is (60^\circ)), with side (8) cm.</p><p data-pxy="true" style="color: rgb(0,0,0); margin-top: 2px; margin-bottom: 2px;">$\text{Area}=\frac{\sqrt3}{4}(8)^2$<br>$=16\sqrt3\ \text{cm}^2$<br></p><p data-pxy="true" style="color: rgb(0,0,0); margin-top: 2px; margin-bottom: 2px;"> Area of the segment</p><p data-pxy="true" style="color: rgb(0,0,0); margin-top: 2px; margin-bottom: 2px;">$\frac{32\pi}{3}-16\sqrt3\ \text{cm}^2$<br></p><p data-pxy="true" style="color: rgb(0,0,0); margin-top: 2px; margin-bottom: 2px;"><b>Answer:</b></p><p data-pxy="true" style="color: rgb(0,0,0); margin-top: 2px; margin-bottom: 2px;">$\boxed{\frac{32\pi}{3}-16\sqrt3\ \text{cm}^2}$<br></p><p data-pxy="true" style="color: rgb(0,0,0); margin-top: 2px; margin-bottom: 2px;">Approximate value:</p><p style="color: rgb(0,0,0);"></p><p data-pxy="true" style="color: rgb(0,0,0); margin-top: 2px; margin-bottom: 2px;">$\frac{32\pi}{3}-16\sqrt3$