Question

What is the maximum area of a rectangle which can be inscribed in a circle of radius 2 cm?

• A. 2√2 cm²
• B. 8 cm²
• C. 4√2 cm²
• D. 4 cm²

Answer 1

The Correct Option is B

The problem involves finding the maximum area of a rectangle inscribed in a circle of radius 2 cm. To do this, we need to understand that the diagonal of the rectangle is equal to the diameter of the circle.

1. The diameter of the circle is twice the radius, so for our circle, the diameter is 2 * 2 = 4 cm.
2. Consider a rectangle with length \( l \) and width \( w \). The diagonal \( d \) of the rectangle (which is also the diameter of the circle) can be expressed as:
   \( d = \sqrt{l^2 + w^2} \).
   Given, \( d = 4 \),
   we have:
   \( \sqrt{l^2 + w^2} = 4 \)
3. Square both sides:
   \( l^2 + w^2 = 16 \)
4. The area \( A \) of the rectangle is given by:
   \( A = l \times w \).
   To maximize \( A \), consider using \( l = w \) (i.e., it is a square), because for a fixed diagonal, a square has the maximum area.
5. With \( l = w \):
   \( l^2 + l^2 = 16 \)
   \( 2l^2 = 16 \)
   \( l^2 = 8 \)
   \( l = \sqrt{8} \)
6. The maximum area \( A \) becomes:
   \( A = l \times l = (\sqrt{8})(\sqrt{8}) = 8 \) cm².

Therefore, the maximum area of the rectangle inscribed in the circle is 8 cm².