An artist wishes to paint a circular region on a square poster that is

Option b.
Area of sq.=2*2=4
Area of circle=4/2=2
2=pi * r^2
rad=sqrt(2/pi)

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An artist wishes to paint a circular region on a square poster that is 2 feet on a side. If the area of the circular region is to be 1/2 the area of the poster, what must be the radius of the circular region in feet?

(A) \(\frac{1}{\pi}\)
(B) \(\sqrt{\frac{2}{\pi}}\)
(C) 1
(D) \(\frac{1}{\sqrt{\pi}}\)
(E) \(\frac{\pi}{2}\)


The side of the square is 2 feet so area of the square poster is 4. The artist can paint the circle anywhere on the poster - the only thing is that the area of the circle must be \(4/2 = 2 = \pi*r^2\) i.e. r must be \(\sqrt{\frac{2}{\pi}}\)

The diameter of a circle is equal to the side of the square only if the circle is inscribed in the square i.e. the circle touches all the four sides of the square. Here, the problem doesn't say that the circle must be inscribed in the square - it only says that it must be drawn somewhere in the square and its area must be half of the area of the square.

A circle is inside a square poster THAT IS 2 feet on a side meaning the square has a side of 2 feet each.
AREAcircle = 1/2AREAposter

Find AREA poster = s x s = 2 x 2 = 4
AREAcircle = (1/2)4 = 2
AreaCIRCLE = phi.r2
2 = phi.r2
2/phi = r2
r = \sqrt{2/phi}

The area of the poster is 4, so the area of the circle is 2. Thus:

πr^2 = 2

r^2 = 2/π

r = √(2/π)

Answer: B

To solve this problem, we can start by finding the area of the square poster. Since the side length of the square is 2 feet, the area is calculated as:

Area of the square poster = (side length)^2 = 2^2 = 4 square feet

We are given that the area of the circular region is 1/2 the area of the poster. Therefore, the area of the circular region is:

Area of the circular region = (1/2) * Area of the square poster = (1/2) * 4 = 2 square feet

The formula for the area of a circle is given by:

Area of the circle = π * (radius)^2

We can rearrange this formula to solve for the radius:

(radius)^2 = Area of the circle / π

Plugging in the value of the area of the circular region, we get:

(radius)^2 = 2 / π

radius = √(2 / π)