A square has a side that measures 11 units. What is the area of a circle with a circumference that equals the perimeter of the square? Use 3.14 for [pi], and round your answer to the nearest hundredth.
A. 1,519.76 units²
B. 379.94 units²
C. 616.56 units²
D. 154.14 units²
Answer (2)
Calculate the perimeter of the square: P = 4 × 11 = 44 .
Determine the circumference of the circle: C = 44 .
Calculate the radius of the circle: r = 2 × 3.14 44 ≈ 7.006369 .
Calculate the area of the circle and round to the nearest hundredth: A = 3.14 × ( 7.006369 ) 2 ≈ 154.14 .
Explanation
Problem Analysis First, let's analyze the information we have. We know that the square has a side of 11 units. We need to find the area of a circle whose circumference is equal to the perimeter of this square.
Calculate the Perimeter of the Square The perimeter of a square is calculated by multiplying the length of one side by 4. So, the perimeter of the square is: P s q u a re = 4 × s i d e = 4 × 11 = 44 Thus, the perimeter of the square is 44 units.
Determine the Circumference of the Circle Since the circumference of the circle is equal to the perimeter of the square, the circumference of the circle is 44 units. C c i rc l e = P s q u a re = 44
Calculate the Radius of the Circle The formula for the circumference of a circle is C = 2 π r , where r is the radius of the circle. We can rearrange this formula to solve for the radius: r = 2 π C Substituting the given values: r = 2 × 3.14 44 = 6.28 44 ≈ 7.006369 So, the radius of the circle is approximately 7.006369 units.
Calculate the Area of the Circle Now that we have the radius, we can calculate the area of the circle using the formula A = π r 2 . Substituting the values: A = 3.14 × ( 7.006369 ) 2 A = 3.14 × 49.08921 A ≈ 154.1401
Round the Answer Finally, we round the area to the nearest hundredth: A ≈ 154.14 Therefore, the area of the circle is approximately 154.14 square units.
Final Answer The area of the circle with a circumference equal to the perimeter of the square is approximately 154.14 square units.
Examples
Imagine you're designing a circular garden and want its fence (circumference) to be the same length as the perimeter of a square sandbox. By calculating the area of this circular garden, you know how much space you have for planting flowers. This ensures you utilize the space efficiently and aesthetically, creating a balanced and visually appealing garden layout.
The area of the circle, which has a circumference equal to the perimeter of a square with a side length of 11 units, is approximately 154.14 square units. The calculation involves finding the perimeter of the square, which is 44 units, and then using this to find the radius of the circle and calculating the area. The final answer is rounded to 154.14 square units.
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