Ratio: Impact of a radius change on the area of a circle

To solve this problem, follow these steps:

• Step 1: Calculate the radius of each circle.
• Step 2: Use the formula for the area of a circle to find the areas.
• Step 3: Find the ratio of the areas to determine how many times larger circle 2's area is compared to circle 1's.

Step 1:
The diameter of circle 1 is 4 cm. Therefore, the radius of circle 1 is $\frac{4}{2} = 2$ cm.

The diameter of circle 2 is 10 cm. Therefore, the radius of circle 2 is $\frac{10}{2} = 5$ cm.

Step 2:
The area of a circle is given by $A = \pi r^2$.
Area of circle 1 is $A_1 = \pi (2)^2 = 4\pi$ square cm.

Area of circle 2 is $A_2 = \pi (5)^2 = 25\pi$ square cm.

Step 3:
To find out how many times larger circle 2's area is than circle 1's area, we compute the ratio of the areas:
$\text{Ratio} = \frac{A_2}{A_1} = \frac{25\pi}{4\pi} = \frac{25}{4}$

The ratio $\frac{25}{4}$ simplifies to $6\frac{1}{4}$, indicating that the area of circle 2 is $6\frac{1}{4}$ times larger than the area of circle 1.

Therefore, the solution to the problem is $6\frac{1}{4}$.

The area of a circle is calculated using the following formula:

where r represents the radius.

Using the formula, we calculate the areas of the circles:

Circle 1:

$π\cdot4² =$

$π16$

Circle 2:

$\pi\cdot10²=$

$π100$

To calculate how much larger one circle is than the other (in other words - what is the ratio between them)

All we need to do is divide one area by the other.

$\frac{100}{16}=$

$6\frac{1}{4}$

Therefore the answer is 6 and a quarter!

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:

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.