A circle has a chord that goes from #( pi)/3 # to #(2 pi) / 3 # radians on the circle. If the area of the circle is #16 pi #, what is the length of the chord?

Because we are given the area, we can find the radius:

If we imagine two radii, one the starting point of the chord and the other on the ending point of the chord, then they form a triangle with the chord. The angle between to the two radii is:

At this point, one can realize that the triangle formed is equilateral and the length of the chord is 4.

But, if this special case does not exist, then the following is how the length of the chord is computed.

We know lengths of two sides and the angle between those two sides, therefore, we can use the Law of Cosines, to find the length of the third side:

To find the length of the chord, we can use the formula for the length of a chord in a circle:

[ \text{Length of chord} = 2 \times \text{radius} \times \sin\left(\frac{\theta}{2}\right) ]

Where:

• ( \theta ) is the angle subtended by the chord at the center of the circle,
• The radius of the circle is given by ( r = \sqrt{\frac{\text{Area of circle}}{\pi}} ).

Given that the chord spans from ( \frac{\pi}{3} ) to ( \frac{2\pi}{3} ) radians, the angle ( \theta ) subtended by the chord is ( \frac{2\pi}{3} - \frac{\pi}{3} = \frac{\pi}{3} ).

Using the given area of the circle, we can calculate the radius: [ r = \sqrt{\frac{16\pi}{\pi}} = \sqrt{16} = 4 ]

Now, substituting the values into the formula for the length of the chord: [ \text{Length of chord} = 2 \times 4 \times \sin\left(\frac{\pi}{3}\right) ]

[ \sin\left(\frac{\pi}{3}\right) = \frac{\sqrt{3}}{2} ]

[ \text{Length of chord} = 2 \times 4 \times \frac{\sqrt{3}}{2} = 4\sqrt{3} ]

So, the length of the chord is ( 4\sqrt{3} ) units.