Four circles, each with a radius of 2 inches, are removed from a square. What is the remaining area of the square?
A. $(16-4 \pi)$ in. $^2$
B. $(16-\pi)$ in $^2$
C. $(64-16 \pi)$ in. $^2$
D. $(64-4 \pi)$ in $^2$
Answer (2)
Calculate the side length of the square: s i d e = 4 × 2 = 8 inches.
Calculate the area of the square: A re a s q u a re = 8 2 = 64 square inches.
Calculate the total area of the four circles: A re a c i rc l es = 4 × π × 2 2 = 16 π square inches.
Calculate the remaining area: R e mainin g A re a = 64 − 16 π square inches. The remaining area of the square is ( 64 − 16 π ) in. 2 .
Explanation
Problem Analysis Let's analyze the problem. We have a square, and four circles are removed from it. We need to find the remaining area of the square. The radius of each circle is given as 2 inches. We'll assume the four circles are arranged in a 2x2 grid inside the square.
Find the side length of the square First, we need to find the side length of the square. Since the circles are arranged in a 2x2 grid, the side length of the square is equal to 4 times the radius of one circle. Given that the radius is 2 inches, the side length of the square is:
s i d e = 4 × r a d i u s = 4 × 2 = 8 inches.
Calculate the area of the square Next, we calculate the area of the square:
A re a s q u a re = s i d e 2 = 8 2 = 64 square inches.
Calculate the area of one circle Now, we calculate the area of one circle:
A re a c i rc l e = π × r a d i u s 2 = π × 2 2 = 4 π square inches.
Calculate the total area of the four circles Since there are four circles, the total area of the circles is:
T o t a l A re a c i rc l es = 4 × A re a c i rc l e = 4 × 4 π = 16 π square inches.
Calculate the remaining area Finally, we subtract the total area of the circles from the area of the square to find the remaining area:
R e mainin g A re a = A re a s q u a re − T o t a l A re a c i rc l es = 64 − 16 π square inches.
Final Answer Therefore, the remaining area of the square after removing the four circles is ( 64 − 16 π ) square inches.
Examples
Imagine you're designing a decorative tile pattern for your kitchen backsplash. You start with a square tile and decide to inlay four circular designs within it. Knowing the radius of the circular inlays and using the formula we derived, you can calculate exactly how much of the original square tile will remain visible, allowing you to plan your design precisely and avoid material waste. This ensures your backsplash is both beautiful and cost-effective.
The remaining area of the square, after removing four circles with a radius of 2 inches each, is given by the formula (64 - 16π) square inches. To determine this, we calculated the area of the square and the total area of the circles. Hence, the correct answer is (64 - 16π) in².
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