A rectangle is encribed within a circle?

Question

so that all four corners of the rectangle touch the circle. The rectangle's area is equal to the area of the circle outside of the rectangle. What is the height to width ratio of the rectangle?

Answer 1

[4+√(16-π^2)]/π

AreaC = πR^2
AreaR = A*B
R^2 = (A/2)^2 + (B/2)^2
AND 2AB = πR^2

SO 2AB = π[ (A/2)^2 + (B/2)^2 ]
8AB = π[ A^2 + B^2 ]
OR πA^2 - 8AB + πB^2 = 0

THEREFORE A = {8B + √[(-8B)^2 - 4(π)(πB^2)]}/2π

OR A = B[4+√(16-π^2)]/π

OR A/B = [4+√(16-π^2)]/π

Answer 2

Assume without loss of generality that the rectangle is aligned with the x and y axes. In this case, the displacement of the upper-right corner from the origin will be (w/2, h/2). The distance from the origin must, of course, be the radius of the circle, so (w/2)²+(h/2)²=r² or w²+h²=4r². Noting that the area of the rectangle is wh, and that this is half the area of the circle, which is πr², wh=πr²/2. This makes r² 2wh/π. Substituting, we get w²+h²=8wh/π. We rearrange to get w²-8wh/π=-h². To complete the square, note that ((-8h/π)/2)²=16h²/π². Adding this to both sides gives w²-8wh/π+16h²/π²=16h²/π²-h². Simplifying, (w-4h/π)²=h²(16-π²)/π². Taking the square root of both sides gives w-4h/π=±h√(16-π²)/π. Dividing by h and adding 4/π gives w/h=(4±√(16-π²))/π. This is the width-to height ratio of the rectangle, which evaluates to w/h≈2.0613632268 or w/h≈0.485115862. You will notice that these two numbers are reciprocals of each other - that's because it doesn't matter which side you declare to be width and which you declare to be height, the total area covered will be the same, and thus any acceptable value for w/h is also acceptable for h/w.

The final answer is w:h=(4+√(16-π²))/π≈2.0613632268 or w:h=(4±√(16-π²))/π)≈0.485115862

Answer 3

the key to this is to realise that the line between two opposite corners of the rectangle drawn in such a manner is equal to the diameter of the circle.

then you need to find out what the area of the rectangle is dependant on the angle

then you will find the further details you require

Answer 4

let the radius of the circle =1
let a and b be the sides of the rectangle(a>b)
The diagonal of the rectangle is the diameter of the circle so diagonal =2

area of rectangle =1/2 area of circle
ab = 1/2*(pi)


a^2 +b^2 = c^2
a^2 +b^2 = 4

a^2 = 4-b^2
a = squareroot(4- b^2)

ab =1/2*(pi)
b*(squareroot(4-b^2) =1/2*(pi)
square both sides
b^2*(4-b^2) =1/2*(pi)^2
4*b^2 -b^4 =1/4*(pi)^2
b^4 -4b^2 +1/4*(pi)^2
b^2 =(1/2)*(4 + squareroot(16-(pi)^2) ) =3.237981785

a^2 =4 -b^2
a^2 = 4-(1/2)*(4 + squareroot(16-(pi)^2) ) =0.762018215

a^2/b^2 =4.249218353
a/b =squareroot(4.249218353) =
a/b = 2.061363227


BTW, "Scott R" has the correct answer expressed in terms of pi and square roots

Answer 5

I agree with the answers from JP & Scott. However, in JP's case, his steps actually show that b > a, which contradicts his initial assumption of a > b. At the bottom 3 lines, the ratios should be expressed in the various forms containing b/a.

Sometimes, it is better not to make an assumption and just let the calculation take care of itself.

Answer 6

1:1

Answer 7

There can be no such thing if th rectangle is absolutly inscribed in the cirle.

Answer 8

1to1or square inside circle