Question

A square is inscribed in a circle of radius 1 unit, and a larger square is circumscribed about the same circle. What is the area of the region located between the two squares?

Answer 1

`A=2 " unit"^2`

Explanation:

Area of a square is given by : `A=1/2d^2`, where `d` is the length of the diagonal of the square.
Given radius of the circle `r=1`,
`=> OA=1`,
`=> "diagonal of the smaller square"=AB=2*OA=2`,
`=> "area of the smaller square " A_S=1/2*AB^2=1/2*2^2=2`
`=> OE="radius"=1, => ED=1`,
`=> CD=2ED=2`
`=> "area of the larger square " A_L= CD^2=2^2=4`
Hence, area of the region located between the two squares `=` shaded area `= A_L-A_S=4-2=2 " unit"^2`

Answer 2

Difference in areas of the two squares = 2 sq. units

Explanation:

radius of circle `r = 1`

Diagonal of inner square `d_s = 2r = 2`

Area of inner square `A_s = (1/2) (d_s)^2 = (1/2) * 2^2 = 2`

Side of outer square `a_S = 2r = 2`

Area of outer square `A_S = a^2 = 2^2 = 4`

Difference in areas = 4 - 2 = 2 sq. units