Area of a circle with diameter 'm' radius 'n' and circumference 'p' is

Options

A. `2 pi n`
B. `pi m^2`
C. `pi p^2`
D. `pi n^2`

Answer

To find the area of a circle given its diameter \( m \), radius \( n \), and circumference \( p \), we can follow these steps:

### Step 1: Understand the relationships
- The radius \( r \) of the circle is half of the diameter \( m \). Therefore, we can express the radius as:
\[ r = \frac{m}{2} \]
- The radius is also given as \( n \). Hence, we have:
\[ r = n \]

### Step 2: Formula for the area of a circle
The formula for the area \( A \) of a circle is:
\[ A = \pi r^2 \]

### Step 3: Substitute the radius
Since we know that the radius \( r \) can be represented as \( n \), we can substitute \( n \) into the area formula:
\[ A = \pi n^2 \]

### Step 4: Verify using diameter
We can also express the radius in terms of the diameter \( m \):
\[ r = \frac{m}{2} \]
Now substituting this into the area formula:
\[ A = \pi \left(\frac{m}{2}\right)^2 = \pi \frac{m^2}{4} \]

### Step 5: Verify using circumference
The circumference \( p \) of the circle is given by:
\[ p = 2\pi r \]
From this, we can express \( r \) in terms of \( p \):
\[ r = \frac{p}{2\pi} \]
Now substituting this into the area formula:
\[ A = \pi \left(\frac{p}{2\pi}\right)^2 = \pi \frac{p^2}{4\pi^2} = \frac{p^2}{4\pi} \]

### Conclusion
After evaluating the area using different parameters (diameter \( m \), radius \( n \), and circumference \( p \)), we conclude that the area of the circle can be expressed as:
\[ A = \pi n^2 \]
This is the simplest and most direct form.

### Final Answer
The area of the circle is \( \pi n^2 \).