Question

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

Answer 1

4π-8

Explanation:

Answer 2

See below

Explanation:

Outline

• First find the diameter, LN. Then find the radius, NO
• Then find the area of the semi circle LMN
• Finally 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 triangle
• a = b = 4 The triangle is isosceles Two 45o angles
• 4^2 + 4^2 = c^2 Expand
• 16 + 16 = c^2 Add
• 32 = c^2 Take the square root of both sides
• sqrt(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 / 2
• Area = pi * 4 * 2 / 2
• Area = 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/2
• Area = 4*4/2
• Area = 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 - 8
• Area= 4.56

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