A chord with a length of #1 # runs from #pi/8 # to #pi/2 # radians on a circle. What is the area of the circle?

The area of the circle can be found using the formula for the area of a circle, which is (A = \pi r^2). To find the area, we first need to find the radius of the circle.

The chord of length 1 spans an arc from (\frac{\pi}{8}) to (\frac{\pi}{2}) radians on the circle. This arc corresponds to an angle of (\frac{\pi}{2} - \frac{\pi}{8} = \frac{3\pi}{8}) radians.

Using the formula for the length of a chord in a circle, (L = 2r\sin\left(\frac{\theta}{2}\right)), where (L) is the length of the chord, (r) is the radius of the circle, and (\theta) is the angle subtended by the chord at the center of the circle, we can solve for (r).

Given (L = 1) and (\theta = \frac{3\pi}{8}), we have:

[1 = 2r\sin\left(\frac{3\pi}{16}\right)]

Solving for (r), we get:

[r = \frac{1}{2\sin\left(\frac{3\pi}{16}\right)}]

Once we have the radius, we can plug it into the formula for the area of the circle to find the area.