The radius of circle A is twice that of circle B and the radius of circle B is twice that of circle C. Their area will be in the ratio

Options

A. `16:4:1`
B. `4:2:1`
C. `1:2:4`
D. `1:4:16`

Answer

To solve the problem, we need to find the ratio of the areas of three circles (A, B, and C) based on their radii. Let's break it down step by step.

### Step 1: Define the radii of the circles
Let the radius of circle C be \( r \). According to the problem:
- The radius of circle B is twice that of circle C:
\[ r_B = 2r \]
- The radius of circle A is twice that of circle B:
\[ r_A = 2r_B = 2(2r) = 4r \]

### Step 2: Write down the radii
Now we have:
- Radius of circle A: \( r_A = 4r \)
- Radius of circle B: \( r_B = 2r \)
- Radius of circle C: \( r_C = r \)

### Step 3: Calculate the areas of the circles
The area \( A \) of a circle is given by the formula:
\[ A = \pi r^2 \]
Now we can calculate the areas of each circle:

1. Area of circle A:
\[ A_A = \pi (r_A)^2 = \pi (4r)^2 = \pi (16r^2) = 16\pi r^2 \]

2. Area of circle B:
\[ A_B = \pi (r_B)^2 = \pi (2r)^2 = \pi (4r^2) = 4\pi r^2 \]

3. Area of circle C:
\[ A_C = \pi (r_C)^2 = \pi (r)^2 = \pi (r^2) = \pi r^2 \]

### Step 4: Write the areas in ratio form
Now we have the areas:
- Area of circle A: \( 16\pi r^2 \)
- Area of circle B: \( 4\pi r^2 \)
- Area of circle C: \( \pi r^2 \)

To find the ratio of the areas, we can express them without the common factor \( \pi r^2 \):
\[ \text{Ratio} = 16 : 4 : 1 \]

### Step 5: Simplify the ratio
The ratio \( 16 : 4 : 1 \) is already in its simplest form.

### Final Answer
Thus, the ratio of the areas of circles A, B, and C is:
\[ \boxed{16 : 4 : 1} \]

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