The area of the rectangle circumscribed by a circle is 32 cm² and the length of one side of the rectangle is 8 cm. The length of the diameter of the circle is

### Concept & Properties of Inscribed Rectangles When a rectangle is circumscribed by a circle, it means the rectangle is inscribed *inside* the circle. The diagonal of this rectangle is the diameter of the circle. We first find the missing side using the area. $$Diagonal = \sqrt{length^2 + width^2}$$ ### Step-by-Step Solution * Given area of the rectangle = $32$ cm². * Given length of one side ($l$) = $8$ cm. * Find the other side ($w$): $Area = l \times w \Rightarrow 32 = 8 \times w \Rightarrow w = 4$ cm. * The diameter of the circumscribing circle is the diagonal of the rectangle. * Calculate the diagonal: $Diameter = \sqrt{8^2 + 4^2}$ $Diameter = \sqrt{64 + 16}$ $Diameter = \sqrt{80}$ * Simplify the square root: $\sqrt{80} = \sqrt{16 \times 5} = 4\sqrt{5}$ cm. ### Exam Strategy & Shortcut Once you find the sides are $8$ and $4$, you need $\sqrt{8^2 + 4^2}$. Notice you can factor out the common multiple $4^2$ before squaring: $\sqrt{4^2(2^2 + 1^2)} = 4\sqrt{4 + 1} = 4\sqrt{5}$. This mental math trick speeds up square root simplifications. ### Common Pitfall A common misinterpretation is thinking the circle is inscribed *in* the rectangle (which is impossible unless it's a square). "Rectangle circumscribed by a circle" means the circle is on the outside, acting as the circumcircle. ### Final Answer Therefore, the correct answer is **$4\sqrt{5}$ cm**.