Find the area of the shaded regions below. give your answer as a completely simplified exact value in terms of π (no approximations).

See below

Step-by-step explanation:

Outline

First find the diameter, LN. Then find the radius, NO Then find the area of the semi circle LMNFinally the area of the triangle.Subtract semi circle and triangle.

Step One

The Pythagorean Theorem says a^2  +  b^2 = r^2

<N = 180 - 90 - 45 Property of a trianglea = b = 4                 The triangle is isosceles Two 45o angles4^2 + 4^2 = c^2      Expand16 + 16 = c^2           Add32 = c^2                  Take the square root of both sidessqrt(c^2) = sqrt(32) c = sqrt(16 * 2)c = 4 * sqrt(2)The diameter of the circle is 4 sqrt(2)The radius =d/2 = 4 * sqrt(2)/2 = 2 sqrt(2)

Step Two

Find the area of the of the Semi Circle

Formula

Area = pi r^2 / 2

Solution

Area = pi * (2*sqrt(2) )^2 / 2Area = pi * 4 * 2 / 2Area = 4 pi

Step 3

Find the area of the triangle.

Area

The two legs are at right angles and they both = 4

Area = b*h/2Area = 4*4/2Area = 8

Step Four

Area shaded part = 4pi - 8

There are all kinds of possible answers. Here are a couple

Area= 4(pi - 2)Area = 4pi - 8Area= 4.56

If none of these are among your choices, leave a note.