The perimeter of a square is 1 m less than the circumference of a circle. The side of the square is 109.75 m. Find the diameter of the circle.
Understand the Problem
The question asks to find the diameter of a circle, given that the perimeter of a square is 1 meter less than the circumference of the circle, and the side length of the square is 109.75 meters. We need to calculate the perimeter of the square, use the given relationship to find the circumference of the circle, and then calculate the diameter from the circumference.
Answer
The diameter of the circle is approximately $140.03 \text{ m}$.
Answer for screen readers
The diameter of the circle is approximately $140.03 \text{ m}$.
Steps to Solve
- Calculate the Perimeter of the Square The perimeter $P$ of a square is calculated using the formula:
$$ P = 4 \times \text{side length} $$
Given the side length is 109.75 m:
$$ P = 4 \times 109.75 = 439 \text{ m} $$
- Set Up the Equation for the Circle's Circumference According to the problem, the perimeter of the square is 1 m less than the circumference $C$ of the circle:
$$ P = C - 1 $$
Substituting the perimeter value we found:
$$ 439 = C - 1 $$
- Solve for the Circumference of the Circle To find $C$, add 1 to both sides of the equation:
$$ C = 439 + 1 = 440 \text{ m} $$
- Calculate the Diameter from the Circumference The circumference $C$ of a circle is related to the diameter $D$ by the equation:
$$ C = \pi D $$
To find the diameter, rearrange this equation:
$$ D = \frac{C}{\pi} $$
Substituting the circumference we found:
$$ D = \frac{440}{\pi} \approx 140.03 \text{ m} $$
The diameter of the circle is approximately $140.03 \text{ m}$.
More Information
Knowing the relationship between a square and a circle can be useful in various real-world scenarios, such as when designing spaces or materials that are shaped differently but required to fit together.
Tips
- Confusing the perimeter of the square with its area.
- Not accounting for the "+1" in the relationship between the square's perimeter and the circle's circumference.
- Forgetting to change the units or miscalculation in prior steps.
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