Question

12 feet

the largest possible circle is cut out of a square whose side length is 12 feet. what will be the approximate area, in square feet, of the remaining board? \\(a = \pi r^2\\) where \\(\pi = 3.14\\)

\\(15\text{ ft}^2\\)
\\(30\text{ ft}^2\\)
\\(60\text{ ft}^2\\)
\\(90\text{ ft}^2\\)

Explanation:

Calculate the area of the square

The side length of the square is \(s = 12\text{ ft}\).
The area of the square is:

$$ A_{\text{square}} = s^2 = 12^2 = 144\text{ ft}^2 $$

Determine the radius of the circle

The largest circle cut out of the square has a diameter equal to the side length of the square.

$$ d = 12\text{ ft} $$

The radius \(r\) is half of the diameter:

$$ r = \frac{12}{2} = 6\text{ ft} $$

Calculate the area of the circle

Using the formula \(A = \pi r^2\) with \(\pi = 3.14\):

$$ A_{\text{circle}} = 3.14 \times 6^2 = 3.14 \times 36 = 113.04\text{ ft}^2 $$

Calculate the remaining area

Subtract the area of the circle from the area of the square:

$$ A_{\text{remaining}} = A_{\text{square}} - A_{\text{circle}} = 144 - 113.04 = 30.96\text{ ft}^2 $$

Round to the nearest option

Using the Rounding Decimals knowledge point

$$ 30.96\text{ ft}^2 \approx 30\text{ ft}^2 $$

Answer:

• (A) \(15\text{ ft}^2\)
• (B) \(30\text{ ft}^2\) (Correct answer)
• (C) \(60\text{ ft}^2\)
• (D) \(90\text{ ft}^2\)