Question

The area of a circle is numerically equal to its circumference. Find the diameter of the circle.

• A. \( 2 \text{ unit} \)
• B. \( 4 \text{ unit} \)
• C. \( 1 \text{ unit} \)
• D. \( 5 \text{ unit} \)

Hint:

\textbf{Circle Properties.} Remember the formulas for the area and circumference of a circle in terms of its radius (\(r\)): Area \( = \pi r^2 \) and Circumference \( = 2 \pi r \). Setting these two equal to each other allows you to solve for the radius when their numerical values are the same.

Answer 1

The Correct Option is B

Let the radius of the circle be \(r\) units. The area of the circle is given by the formula \(A = \pi r^2\). The circumference of the circle is given by the formula \(C = 2 \pi r\). According to the problem, the area of the circle is numerically equal to its circumference. Therefore, we have the equation: $$ A = C $$ $$ \pi r^2 = 2 \pi r $$ To solve for \(r\), we can divide both sides of the equation by \( \pi r \) (assuming \(r \neq 0\)): $$ \frac{\pi r^2}{\pi r} = \frac{2 \pi r}{\pi r} $$ $$ r = 2 $$ So, the radius of the circle is 2 units. The diameter of the circle \(d\) is twice the radius: $$ d = 2r $$ $$ d = 2 \times 2 $$ $$ d = 4 $$ Therefore, the diameter of the circle is 4 units.