The radius of the circle shown is 1 meter. Find the sum of the shaded area.

Question

Here is the link to the figure.

https://docs.google.com/document/d/1yBTlcGu4SrGuRpeY_Pvv3PeBelcI_IxSTdVECD5GehE/edit?usp=sharing

Answer 1

The area of the entire circle is π.

Each side of the square is √2 / 2. Its area is 1/2.

The area of the shaded regions is π - 1/2

Answer 2

Given

shaded A = circle A - square A (A means area)

r = 1 m

Work

circle A = pi * r^2 (^2 means squared)

circle A = pi * 1^2 (substitute r=1)

circle A = pi m^2 (remember that units, m in this case, are squared when dealing with area)

square A = s^2 (s means a side of the square)

Since the distance from the center of the square to a corner (which touches the circle circumference) is 1 m, we know that the distance from corner to corner is 2 m. Because this is a square, we know that every side is the same length and every corner is 90 degrees. We can use a 45-45-90 triangle relationship to find the length of a side of the square. In 45-45-90 triangles, the hypotenuse is always equal to the length of one other side times sqrt(2). Our hypotenuse is the distance from corner to corner, which we've established is 2 m. Use the 45-45-90 triangle relationship to set up the following proportion and solve for s:

side/hypotenuse = side/hypotenuse

1/sqrt(2) = s/2

Solving for s, you get s = sqrt(2).

Thus, square A = s^2 = sqrt(2) ^ 2 = 2 m^2.

shaded A = circle A - square A = pi - 2 m^2