A chord with a length of #14 # runs from #pi/3 # to #pi/2 # radians on a circle. What is the area of the circle?

Question

A chord with a length of #14    # runs from #pi/3    # to #pi/2   # radians on a circle. What is the area of the circle?

Claire Akin · Accepted Answer

#=2246#

A chord travels #pi/2-pi/3=pi/6# radians of the Circle

In other words chord travels #pi/6-:2pi=1/12 th# of Circumference

Circumference#=2pir# where #r=radius#

Therefore we can write

Length of the Chord #=14=(2pir)/12#

or

#2pir=14times12#

or

#r=168/(2pi)#

or

#r=26.74#

Area of the Circle#=pir^2#

#=pi(26.74)^2#

#=2246#

Theodore Abad · Answer

undefined To find the area of the circle, we need to know the radius. Since we have a chord length and the angle it subtends, we can use trigonometry to find the radius.

1. First, note that the chord divides the circle into two segments, with one being a minor segment and the other being a major segment.
2. The chord length is the straight-line distance between the two points where the chord intersects the circle.
3. Using trigonometry, we can find half the chord length, which is the distance from the center of the circle to the midpoint of the chord.
   - This distance is $ r\sin(\frac{\theta}{2}) $, where $ r $ is the radius and $ \theta $ is the angle subtended by the chord.
4. Since the chord runs from $ \frac{\pi}{3} $ to $ \frac{\pi}{2} $ radians, the angle subtended by the chord is $ \frac{\pi}{2} - \frac{\pi}{3} = \frac{\pi}{6} $.
5. Now, we have half the chord length, which is $ 7 $ (since the full chord length is $ 14 $).
6. Solve for $ r $ using $ 7 = r\sin\left(\frac{\pi}{6}\right) $.
   - $ r = \frac{7}{\sin\left(\frac{\pi}{6}\right)} $.
7. Once you find the radius, $ r $, you can calculate the area of the circle using the formula $ A = \pi r^2 $.

Performing these calculations will give you the area of the circle.