The numerical value of the area of a circla is greater than the numerical value of its circumference. Is this statement true ? Why ?

Explanation

To determine whether the statement "The numerical value of the area of a circle is greater than the numerical value of its circumference" is true, we will analyze the formulas for the area and circumference of a circle.

### Step 1: Write down the formulas
The area \( A \) of a circle is given by:
\[ A = \pi r^2 \]
The circumference \( C \) of a circle is given by:
\[ C = 2 \pi r \]

### Step 2: Set up the inequality
We need to check if the area is greater than the circumference:
\[ \pi r^2 > 2 \pi r \]

### Step 3: Simplify the inequality
We can divide both sides of the inequality by \( \pi \) (assuming \( \pi > 0 \)):
\[ r^2 > 2r \]

### Step 4: Rearrange the inequality
Rearranging gives us:
\[ r^2 - 2r > 0 \]

### Step 5: Factor the inequality
Factoring the left side, we get:
\[ r(r - 2) > 0 \]

### Step 6: Determine the intervals
To find the values of \( r \) that satisfy this inequality, we can find the roots of the equation \( r(r - 2) = 0 \):
- The roots are \( r = 0 \) and \( r = 2 \).

Now we can test the intervals:
1. \( r < 0 \)
2. \( 0 < r < 2 \)
3. \( r > 2 \)

### Step 7: Test the intervals
- For \( r < 0 \): Choose \( r = -1 \) → \( (-1)(-1 - 2) = (-1)(-3) = 3 > 0 \) (True)
- For \( 0 < r < 2 \): Choose \( r = 1 \) → \( (1)(1 - 2) = (1)(-1) = -1 < 0 \) (False)
- For \( r > 2 \): Choose \( r = 3 \) → \( (3)(3 - 2) = (3)(1) = 3 > 0 \) (True)

### Step 8: Conclusion
The inequality \( r(r - 2) > 0 \) holds true for:
- \( r < 0 \)
- \( r > 2 \)

Thus, the area of the circle is greater than the circumference when the radius \( r \) is either less than 0 (not physically meaningful) or greater than 2. Therefore, the statement "The numerical value of the area of a circle is greater than the numerical value of its circumference" is **not true** for \( 0 < r < 2 \).

### Summary
The statement is incorrect because for \( 0 < r < 2 \), the circumference is greater than the area.

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