Calculating the Ratio of Areas of Two Circles

This problem asks us to find the ratio of the areas of two circles, Circle-A and Circle-B, given their respective radii. Circle-A has a radius of 4 cm, and Circle-B has a radius of 8 cm.

Understanding the Area of a Circle

The area of a circle is calculated using the formula:

$$ \text{Area} = \pi r^2 $$

where $\pi$ (pi) is a mathematical constant approximately equal to 3.14159, and $r$ is the radius of the circle.

Step-by-Step Calculation

1. Calculate the Area of Circle-A

The radius of Circle-A is $r_A = 4$ cm.

The area of Circle-A, denoted as $A_A$, is:

$$ A_A = \pi (r_A)^2 = \pi (4 \, \text{cm})^2 = \pi (16 \, \text{cm}^2) = 16\pi \, \text{cm}^2 $$

2. Calculate the Area of Circle-B

The radius of Circle-B is $r_B = 8$ cm.

The area of Circle-B, denoted as $A_B$, is:

$$ A_B = \pi (r_B)^2 = \pi (8 \, \text{cm})^2 = \pi (64 \, \text{cm}^2) = 64\pi \, \text{cm}^2 $$

3. Determine the Ratio of the Areas

We need to find the ratio of the area of Circle-A to the area of Circle-B. This ratio is $\frac{A_A}{A_B}$.

$$ \text{Ratio} = \frac{A_A}{A_B} = \frac{16\pi \, \text{cm}^2}{64\pi \, \text{cm}^2} $$

We can cancel out $\pi$ from the numerator and the denominator, and the units (cm²):

$$ \text{Ratio} = \frac{16}{64} $$

4. Simplify the Ratio

To simplify the fraction $\frac{16}{64}$, we find the greatest common divisor of 16 and 64, which is 16. Divide both the numerator and the denominator by 16:

$$ \frac{16 \div 16}{64 \div 16} = \frac{1}{4} $$

So, the ratio of the area of Circle-A to the area of Circle-B is 1:4.

Let's look at the options provided:

• 1 ∶ 2
• 1 ∶ 16
• 1 ∶ 8
• 1 ∶ 4

Our calculated ratio is 1:4, which matches one of the options.

General Relationship Between Radius and Area Ratio

Notice that if the radius of a circle is scaled by a factor $k$, the area is scaled by a factor of $k^2$. In this case, the radius of Circle-B (8 cm) is $2 \times$ the radius of Circle-A (4 cm). So, $k=2$. The ratio of the areas should be $(k)^2 = (2)^2 = 4$. Since we are comparing Area-A to Area-B, and Circle-B is larger, the ratio is $\frac{1}{4}$. If we were comparing Area-B to Area-A, the ratio would be $\frac{4}{1} = 4$. This confirms our calculation.

Revision Table: Circle Area Ratio

Additional Information: Area and Radius Relationships

Understanding how geometric properties scale is fundamental in geometry. For circles:

• Circumference: The circumference $C = 2\pi r$ scales linearly with the radius $r$. If the radius is doubled, the circumference is also doubled.
• Area: The area $A = \pi r^2$ scales with the square of the radius $r^2$. If the radius is doubled, the area is quadrupled (scaled by $2^2=4$). If the radius is tripled, the area is increased by a factor of $3^2=9$.

This principle applies to the areas of similar two-dimensional shapes in general. If two similar shapes have corresponding linear dimensions (like radius, side length, etc.) in the ratio $a:b$, then their areas will be in the ratio $a^2:b^2$. In this problem, the radii are in the ratio 4 cm : 8 cm, which simplifies to 1:2. Therefore, the areas are in the ratio $1^2:2^2$, which is 1:4.