Question

What is the area of the shaded region?

Answer 1

Answer:

22.5, is the area of the small area of the at the bottom of the triangles.

Step-by-step explanation:

I know that the radius of the circle is 6, which means that the area of the circle is 36$\pi$.

Now in order to find that small area, we would need to find the area of the triangle and the area of the of the two parts of the circle.

Now, we take half of one of the bottom triangle. This gives a triangle with a hypotenuse of 5 a bottom is 3 (because its total 6 and i divided it by 2) and the other side is 4. This can be found by the Pythagorean theorem. This gives us a base of 3 and height of 4.

Area of the that triangle = $\frac{bh}{2}$ = 6, we have 6 of them that means the total area of the triangles are 24.

The next step is the find the area of the parts of the circle, the only thing we need is to find the angle. Please take a look at the picture.  There are different ways in order to find the angle but the easiest way to find is $tan (angle) = \frac{opp}{adj} = \frac{3}{4}$, solving for the angle would give us 37.

Now we know that one of the angles is 37 * 4 = 148.

The total degrees of the triangle is 360 degrees, which means that 360 - 148 = 212.

Therefore, the area of the other part of the circle is $\pi 6^{2} (\frac{212}{360} )$ = 66.6

Now the area of the small part = Area of circle - (area of 4 triangles + area of part of the circle)

Area that we are looking for = 36$\pi$ - (24 + 66.6), where 24 is the area of the four triangles and 66.6 is part of the area of the circle.

Area = 22.5

Great question, thanks for sharing.