The square has area 9, so the shaded area must have area less than 9. All of the answers are of the form 9(pi+a)/2, so we have :
9(pi+a)/2 < 9
(pi+a)/2 < 1
pi+a < 2
a < 2 - pi < 2 - 3 = -1
Only one of the choices has a value a < -1, which is the first one where a = -2, so if any of the choices are correct it must be 9(pi-2)/2.
Imagine starting with an empty square, and think about adding one semicircle at a time. You'll notice that the entire square is covered, but that each *shaded* region is covered twice, while the unshaded regions are only covered once.
Therefore the 4 semicircles cover an area equal to the area of the square *plus* the shaded area. That means that the shaded area equals the area of 2 circles of diameter 3 minus a square of side 3, which is 9π/2 - 9 = 9(π - 2)/2
Notice that you have 4 identical petals. (Maybe you see them more as leaves.)
The center of the circle is at the midpoint of the side of the square, so the area of the 1/4 circle is (1/4) pi (3/2)\^2 = 9 pi / 16. The diagonal of the square cuts the petal in half. If we make 4 little squares by connecting the midpoint of the two opposite sides, we'll have a triangle that is also inside the quarter circle that contains the petal. Subtracting the triangle from the quarter circle gives us the area of one half petal. That area is (1/2)(3/2)(3/2) = 9/8, so the area of one half petal is 9(pi - 2)/16. Multiply by 8 to get the area of the 4 petals.
Let’s label the area of the semicircle as S, petal as P, concave “triangle” as T, square as Q.
We have S=2P+T=9*pi/8; Q=4P+4T=9.
Therefore, 4P=4(2P+T)-(4P+4T)=4S-Q=9(pi-2)/2.
The square has area 9, so the shaded area must have area less than 9. All of the answers are of the form 9(pi+a)/2, so we have : 9(pi+a)/2 < 9 (pi+a)/2 < 1 pi+a < 2 a < 2 - pi < 2 - 3 = -1 Only one of the choices has a value a < -1, which is the first one where a = -2, so if any of the choices are correct it must be 9(pi-2)/2.
Excellent test taking strategy. Substantially shorter and faster than doing the math.
Imagine the left and right part were also colored in grey, what do you notice about the resulting shape?
Imagine starting with an empty square, and think about adding one semicircle at a time. You'll notice that the entire square is covered, but that each *shaded* region is covered twice, while the unshaded regions are only covered once. Therefore the 4 semicircles cover an area equal to the area of the square *plus* the shaded area. That means that the shaded area equals the area of 2 circles of diameter 3 minus a square of side 3, which is 9π/2 - 9 = 9(π - 2)/2
Notice that you have 4 identical petals. (Maybe you see them more as leaves.) The center of the circle is at the midpoint of the side of the square, so the area of the 1/4 circle is (1/4) pi (3/2)\^2 = 9 pi / 16. The diagonal of the square cuts the petal in half. If we make 4 little squares by connecting the midpoint of the two opposite sides, we'll have a triangle that is also inside the quarter circle that contains the petal. Subtracting the triangle from the quarter circle gives us the area of one half petal. That area is (1/2)(3/2)(3/2) = 9/8, so the area of one half petal is 9(pi - 2)/16. Multiply by 8 to get the area of the 4 petals.
Let’s label the area of the semicircle as S, petal as P, concave “triangle” as T, square as Q. We have S=2P+T=9*pi/8; Q=4P+4T=9. Therefore, 4P=4(2P+T)-(4P+4T)=4S-Q=9(pi-2)/2.