Question

If the circumference and the area of a circle are numerically equal, the radius of the circle will be:

• A. 2 units
• B. $\pi$ units
• C. 4 units
• D. 7 units

Hint:

When the circumference and area of a circle are equal, you can set up an equation using their formulas and solve for the radius.

Answer 1

The Correct Option is A

Step 1: Use the formula for the circumference and area of a circle.
The circumference $C$ of a circle is given by: \[ C = 2\pi r \] The area $A$ of a circle is given by: \[ A = \pi r^2 \]
Step 2: Set up the equation for when the circumference equals the area.
We are given that the circumference and area are numerically equal: \[ 2\pi r = \pi r^2 \]
Step 3: Simplify the equation.
Divide both sides of the equation by $\pi$: \[ 2r = r^2 \]
Step 4: Solve for $r$.
Rearrange the equation: \[ r^2 - 2r = 0 \] Factor the equation: \[ r(r - 2) = 0 \] So, $r = 0$ or $r = 2$. Since the radius of a circle cannot be zero, we have: \[ r = 2 \]
Step 5: Conclusion.
Therefore, the radius of the circle is $2$ units.