A chord with a length of #7 # runs from #pi/8 # to #pi/3 # radians on a circle. What is the area of the circle?

To find the area of the circle given the length of the chord and the angle it subtends, you can use the formula for the area of a circle segment.

First, let's find the radius of the circle. The chord divides the circle into two segments. The radius, the chord, and the perpendicular bisector of the chord form a right triangle.

The central angle that the chord subtends is ( \frac{\pi}{3} - \frac{\pi}{8} = \frac{5\pi}{24} ) radians.

Using the sine formula for the triangle:

[ \sin\left(\frac{5\pi}{48}\right) = \frac{7}{2R} ]

where ( R ) is the radius of the circle.

Solving for ( R ), we get:

[ R = \frac{7}{2 \sin\left(\frac{5\pi}{48}\right)} ]

Now, the area of the circle segment can be calculated as:

[ A = R^2 \left( \frac{\theta - \sin(\theta)}{2} \right) ]

where ( \theta ) is the central angle in radians. Plugging in the values, we get:

[ A = \left( \frac{7}{2 \sin\left(\frac{5\pi}{48}\right)} \right)^2 \left( \frac{5\pi}{24} - \sin\left(\frac{5\pi}{24}\right) \right) ]

Calculating this expression will give you the area of the circle segment, which is the area of the circle enclosed by the chord.