A square with perimeter 16 is inscribed in a circle. What is the area of the circle?

The answer is correct because we need to find the area of the circle in which a square is inscribed.

Here's how we arrive at that:

    
1. The perimeter of the square is given as 16. Since a square has four equal sides, each side is:
    
        
• Side = Perimeter / 4 = 16 / 4 = 4
    
    
2. Next, we find the diagonal of the square, which is also the diameter of the circle. The diagonal \(d\) of a square with side length \(s\) can be calculated using the formula:
    
        
• d = \(s \sqrt{2}\)
    
    
        
• Substituting \(s = 4\) gives:
        
• d = \(4\sqrt{2}\)
    
    
3. The radius \(r\) of the circle is half of the diameter:
    
        
• r = \(d/2 = 4\sqrt{2}/2 = 2\sqrt{2}\)
    
    
4. Now, we can find the area \(A\) of the circle using the formula:
    
        
• A = \(πr^2\)
    
    
        
• Substituting \(r = 2\sqrt{2}\) gives:
        
• A = \(π(2\sqrt{2})^2 = π(4 \times 2) = 8π\)
    

Thus, the area of the circle is 8π.

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