# A chord with a length of #4 # runs from #pi/12 # to #pi/3 # radians on a circle. What is the area of the circle?

Area of the circle

Given : Chord length

Area of circle

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The length of a chord in a circle and the angle it subtends at the center of the circle can be used to find the radius of the circle. Once the radius is known, the area of the circle can be calculated using the formula for the area of a circle.

Given the chord length ( L = 4 ) and the angle subtended by the chord ( \theta = \frac{\pi}{3} - \frac{\pi}{12} = \frac{\pi}{4} ), we can use the formula:

[ L = 2r \sin\left(\frac{\theta}{2}\right) ]

Solve for ( r ), the radius of the circle:

[ r = \frac{L}{2 \sin\left(\frac{\theta}{2}\right)} ]

Substitute the given values:

[ r = \frac{4}{2 \sin\left(\frac{\pi/4}{2}\right)} ]

[ r = \frac{4}{2 \sin\left(\frac{\pi}{8}\right)} ]

[ r = \frac{4}{2 \sin\left(22.5^\circ\right)} ]

[ r = \frac{4}{2 \sin\left(\frac{\pi}{8}\right)} ]

[ r = \frac{4}{2 \cdot \frac{\sqrt{2 + \sqrt{2}}}{2}} ]

[ r = \frac{4}{\sqrt{2 + \sqrt{2}}} ]

[ r = \frac{4\sqrt{2 - \sqrt{2}}}{2} ]

[ r = 2\sqrt{2 - \sqrt{2}} ]

Now, use the radius to find the area of the circle:

[ A = \pi r^2 ]

[ A = \pi (2\sqrt{2 - \sqrt{2}})^2 ]

[ A = 4\pi(2 - \sqrt{2}) ]

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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