Question

Find the area of the shaded portion intersecting between the two circles. Show all work for full credit.

Answer 1

We can find the area using trigonometry. Given that the radii of the circles are the same, we can deduce that the shaded area in the circles are the same, so to solve the problem, we will find the shaded area in one of the circles and double our answer.

First, we know that the radius can be drawn from the center to any point on the edge of the circle. Therefore we can create an isosceles triangle by drawing the radius from the center to the other end of the chord of length 4 (see attached photo for triangle). 

Next, we can find the angle formed at the center of the circle using a trigonometry function. We know that the chord opposite of the angle is 4 long and that both adjacent sides are 4 long. If we draw the amplitude of the triangle which bisects the center angle, we will have two right triangles. The side opposite to the new center angle is 2 long. Now we can use the sine function to find the angle:

$sin^{-1}( \frac{2}{4} )=\theta$

$\theta = sin^{-1}(0.5) = 30 ^\circ$

If half of the center angle is 30 degrees, then we know that the angle between the radii is 60 degrees. Since a full circle is 360 degrees, we know that we are dealing with one-sixth of the circle.

Now we need to find the area of one-sixth of the circle and subtract the area of the triangle to find the shaded area.

To find the area of the triangle, we can use one-half base times height, but first we need to find the height. We can use the tangent  function to find it:

$tan(30) = \frac{2}{height}$

$height = \frac{2}{tan(30)}$

So the area of the shaded section is:

$\frac{1}{6} \pi (4)^2 - \frac{1}{2}(4)( \frac{2}{tan(30)} ) = 1.4494$

So multiply that by two to get  the area for both circles and the final answer is:

A = 2.8988 

Answer 2

Answer:

5 & 1/3 pi - 8 sqrt 3

Step-by-step explanation:

1) Find the area of one circle: A = pi * r^2 ----> A = 16 pi.

2) Draw or imagine another radius drawn with its endpoint on the endpoint of the chord. This creates an equilateral triangle. This also means that each angle of the triangle has a measure of 60 degrees. So, since the triangle is part of the sector, the measure of the sector's arc is 60 degrees.  

3) With this information, you can find out the area of the sector: do 60/360 = 1/6. Then multiply by the area of the circle:  

1/6 * 16 pi = 2 & 2/3 pi. So the area of the sector is 2 & 2/3 pi.

4) Then find the area of the equilateral triangle: A = 1/4 * s^2 * sqrt 3 ----> A = 4 sqrt 3. So the area of the triangle is 4 sqrt 3.

5) Finally, find the area of the segment or one of the shaded parts. You would do this by doing the area of the sector minus the area of the triangle ----> 2 & 2/3 pi - 4 sqrt 3. You also need to multiply this by 2 since there are two segments that are part of the shaded portion. So your final exact answer would be:  5 & 1/3 pi - 8 sqrt 3. The approximate answer would be 2.90.

Hope this helps! :)