The equation of a circle centered at the origin is $x^2+y^2=64$. What is the radius of the circle?
A. 32
B. 8
C. 4
D. 64
Answer (1)
Recognize the standard equation of a circle centered at the origin: x 2 + y 2 = r 2 .
Compare the given equation x 2 + y 2 = 64 to the standard equation to identify r 2 = 64 .
Calculate the radius by taking the square root: $r =
r = 64 = 8
State the final answer: 8 .
Explanation
Understanding the Circle Equation The equation of a circle centered at the origin is given by x 2 + y 2 = r 2 , where r is the radius of the circle. We are given the equation x 2 + y 2 = 64 .
Comparing Equations Comparing the given equation x 2 + y 2 = 64 with the standard form x 2 + y 2 = r 2 , we can see that r 2 = 64 .
Finding the Radius To find the radius r , we need to take the square root of both sides of the equation r 2 = 64 . So, we have $r =
r = 64
Calculating the Square Root Calculating the square root of 64, we get r = 8 . Therefore, the radius of the circle is 8.
Final Answer The radius of the circle is 8, which corresponds to option B.
Examples
Understanding the equation of a circle is useful in many real-world applications. For example, when designing a circular garden, you need to know the radius to determine how much fencing you'll need. Similarly, in architecture, circular windows or domes require precise radius calculations to ensure structural integrity and aesthetic appeal. Knowing the radius helps in calculating the area and circumference, which are crucial for material estimation and design planning.
