Gill’s backyard has grass that is in the shape of a circle and has a diameter of 30 feet. He has decided to pour a concrete patio in the middle of the yard that is shaped as a square as in the image below.
A circle with a large square in the middle. The square has a dashed line running through the middle that is 20 feet long.
Which measurement is closest to the area of the grass that will remain in Gill’s backyard after the patio is poured?
Gill’s backyard has grass that is in the shape of a circle and has a diameter of 30 feet. He has decided to pour a concrete patio in the middle of the yard that is shaped as a square as in the image below.
A circle with a large square in the middle. The square has a dashed line running through the middle that is 20 feet long.
Which measurement is closest to the area of the grass that will remain in Gill’s backyard after the patio is poured?
100 ft2
625 ft2
676 ft2
306.5 ft2
To find the area of the grass that will remain in Gill's backyard, we need to subtract the area of the square patio from the area of the circle.
The area of a circle is given by the formula: A = πr^2, where r is the radius of the circle. Since the diameter is given as 30 feet, the radius is half of that, so r = 30/2 = 15 feet.
The area of a square is given by the formula: A = s^2, where s is the length of a side. The dashed line in the middle of the square divides it into two equal triangles, each with a base of 10 feet. Therefore, the length of a side of the square is also 10 feet.
Now we can calculate the area of the grass that will remain:
Area of circle = πr^2 = π(15)^2 ≈ 706.86 sq. ft.
Area of square patio = s^2 = (10)^2 = 100 sq. ft.
Area of grass that remains = Area of circle - Area of square patio = 706.86 - 100 ≈ 606.86 sq. ft.
The measurement closest to the area of the grass that will remain in Gill's backyard is 606.86 sq. ft., so the closest answer choice is 625 ft2.