Area of Rectangle Inscribed in a Circle

Consider a circle with a radius of 10 units. A rectangle is inscribed within this circle such that the rectangle's length is twice its width. What is the area of the inscribed rectangle?

To find the area of the rectangle, we should first identify the relationship between its dimensions (length and width) and the circle's radius. The length ($l$) and width ($w$) have the relationship:

$$l = 2w$$

Additionally, the rectangle fits within the circle, meaning its diagonal must equal the circle's diameter. Therefore, we can use the Pythagorean theorem to establish:

$$ ext{Diagonal} = ext{Diameter} = 2r$$

where $r$ is the radius of the circle.