Evaluate the difference between the areas of a circle and a square.
≈ 223.75 square units
Practice makes perfect
Let's take a look at the given figure.
Notice that the area of the shaded region is the difference between the area of the circle and the area of the square. First, let's recall the formula for the area of a circle A_c. In this formula r is the radius.
A_c=π r^2Now we will evaluate the area of our circle that has a radius of 14.
The area of the circle is approximately 615.75 square units. Next let's recall that we can divide a square into four congruent 45^(∘)-45^(∘)-90^(∘) triangles, as diagonals are perpendicular and bisect each other.
Recall that in a 45^(∘)-45^(∘)-90^(∘) triangle the length of the hypotenuse is sqrt(2) times the leg length. In our exercise this means that the hypotenuse of this triangle &mdahs; which is also a side length of the square — is 14 sqrt(2).
Next we can evaluate the area of the square A_s. Let's recall that the area of a square is a squared side length.
The area of the square is 392 square units. Finally, we will evaluate the difference between the area of the circle and the area of the square to find the area of the shaded region.
A=A_c-A_s=615.75-392=223.75
The area of the shaded region is approximately 223.75. Notice that this is only an approximation, as we used an approximate value of A_c to evaluate it.
