A chord with a length of #9 # runs from #pi/8 # to #pi/6 # radians on a circle. What is the area of the circle?

Question

A chord with a length of #9  # runs from #pi/8   # to #pi/6    # radians on a circle. What is the area of the circle?

Violet Aldrich · Accepted Answer

#A ~~ 14872.29 units²#

Let #theta = # the measure of the angle:

#theta = pi/6 - pi/8#

#theta = (8pi - 6pi)/48#

#theta = (2pi)/48#

#theta = (pi)/24#

This angle, the chord and two radii (one on either end of the chord) form a triangle. Therefore, we can use the law of cosines:

#c² = a² + b² - 2(a)(b)cos(theta)#

where #c = 9, a = b = r and theta = pi/24#:

#9² = r² + r² -2(r)(r)cos(pi/24)#

#9² = 2r² -2(r²)cos(pi/24)#

#9² = r²(2 -2cos(pi/24))#

#r² = (pi9²)/(2 - 2cos(pi/24))#

The area of the circle, A, is the above multplied by #pi#:

#A = (pi9²)/(2 - 2cos(pi/24))#

#A ~~ 14872.29 units²#

Maverick Smith · Answer

undefined The area of the circle can be calculated using the formula:

$$ \text{Area} = \frac{1}{2} r^2 (\theta - \sin \theta) $$

Given that the chord length is 9 and the chord spans from $ \frac{\pi}{8} $ to $ \frac{\pi}{6} $ radians, we can find the radius using the chord's length and the central angle:

$$ r = \frac{c}{2\sin(\frac{\theta}{2})} $$

$$ r = \frac{9}{2\sin(\frac{\pi/6 - \pi/8}{2})} $$

$$ r = \frac{9}{2\sin(\frac{\pi}{48})} $$

$$ r \approx \frac{9}{2\sin(0.0654)} $$

$$ r \approx \frac{9}{2 \times 0.0654} $$

$$ r \approx \frac{9}{0.1308} $$

$$ r \approx 68.73 $$

Now, we can plug the values of $ r $ and $ \theta $ into the area formula:

$$ \text{Area} = \frac{1}{2} \times (68.73)^2 \times (\frac{\pi}{6} - \sin(\frac{\pi}{6})) $$

$$ \text{Area} = \frac{1}{2} \times (68.73)^2 \times (\frac{\pi}{6} - \frac{1}{2}) $$

$$ \text{Area} \approx \frac{1}{2} \times (68.73)^2 \times (\frac{\pi}{6} - \frac{1}{2}) $$

$$ \text{Area} \approx \frac{1}{2} \times (68.73)^2 \times (\frac{\pi}{6} - \frac{3}{6}) $$

$$ \text{Area} \approx \frac{1}{2} \times (68.73)^2 \times \frac{\pi}{6} $$

$$ \text{Area} \approx \frac{1}{2} \times 68.73^2 \times \frac{\pi}{6} $$

$$ \text{Area} \approx \frac{1}{2} \times 4716.752 \times \frac{\pi}{6} $$

$$ \text{Area} \approx 1106.454 $$

So, the area of the circle is approximately 1106.454 square units.