Compare Circle Areas: Finding the Ratio When r₁ = 6cm and d₂ = 12cm

1. Restate the problem: We are given two circles. Circle 1 has a radius of 6 cm, and Circle 2 has a diameter of 12 cm. We need to determine how many times greater the area of Circle 2 is compared to Circle 1. 2. Key information: - Radius of Circle 1: 6 cm - Diameter of Circle 2: 12 cm, which implies the radius is half of the diameter, i.e., 6 cm. 3. Potential approach: Calculate the area of both circles and find the ratio of the area of Circle 2 to the area of Circle 1. 4. Key formula: The area of a circle is given by $A = \pi r^2$ where $r$ is the radius. 5. Appropriate approach: Use the formula for the area of a circle to find the areas of the two circles and compare them. 6. Steps: - Compute the area of Circle 1 using its radius. - Compute the area of Circle 2 using its radius. - Calculate the ratio of the areas of Circle 2 to Circle 1. 7. Assumptions: Circles are perfect geometrical circles, and $\pi$ is a constant. 8. Break down: Since the radii of both circles are equal (both 6 cm), their areas will be identical. 9. Special conditions: None are expected beyond confirming equal areas due to equal radii. 10. Instructions: Ensure clarity and correctness in the solution. 11. Compare with choices: Verify if the areas are indeed equal as suggested by choice 4. 12. Common mistakes: Misunderstanding "diameter" and "radius" could lead to incorrect calculations. 13. Changing variables: Altering the radius or diameter affects circle areas proportionally to the square of the radius ratio.

To solve the problem, let's follow the necessary steps:

• Step 1: Identify the given values for each circle.
• Step 2: Use the formula for the area of a circle, $A = \pi r^2$, to calculate both areas.
• Step 3: Compare the areas to determine the ratio.

Now, let's go through each step:
Step 1: We know:

• Circle 1 has a radius $r_1 = 6$ cm.
• Circle 2 has a diameter of 12 cm, so its radius $r_2 = \frac{12}{2} = 6$ cm.

Step 2: Using the formula $A = \pi r^2$, calculate the area of each circle:

• For Circle 1: $A_1 = \pi (6)^2 = 36\pi$ square centimeters.
• For Circle 2: $A_2 = \pi (6)^2 = 36\pi$ square centimeters.

Step 3: Compare the areas by calculating the ratio:
The ratio of the area of Circle 2 to Circle 1 is:

$\frac{A_2}{A_1} = \frac{36\pi}{36\pi} = 1$

This means that the areas of Circle 1 and Circle 2 are identical.

Therefore, the solution to the problem is that the areas are equal.