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# Comparing a rectangle to a circle

## Question

Assume we have a circle and a rectangle.  The circumference of the circle is the same as the perimeter of the rectangle and the areas of each shape is also the same.  Write the ratio of the circle's circumference to the diagonal of the rectangle as a function in terms of the width of the rectangle.

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Spoiler

Circle: Area = pi r2  Circumference = 2 pi r
Rectangle: Area = ab = pi r2  Perimeter = 2 (a + b) = 2 pi r  Diagonal = d = sqrt (a2 + b2)1/2  Width = b.

2 (a + b) = 2 pi r   =>   (a + b) = pi r   =>   (a + b)2 = pi2 r2.

[1]  (a2 + 2ab +  b2) = pi2 r2.

ab = pi r2   =>

[2]  2ab = 2 pi r2

[2] - [1]   (a2 + b2) = pi2 r2 - 2 pi r2 = pi r2 (pi - 2) = ab (pi - 2)  =>

[3]  a2 - (pi - 2) b a +  b2 = 0.

a = { (pi - 2)+/- [ (pi - 2)2 b2 - 4b4 ]1/2 } / 2 = f (b)

Ratio of circumference to diagonal = g.

Write g in terms of rectangle width b.

g = 2 pi r / d = 2 (a + b) / ( a2 + b2 )1/2

g(b) = 2 (  f (b) + b ) / (  f2 (b) + b2 )1/2

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On 4/14/2018 at 3:11 AM, bonanova said:
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Circle: Area = pi r2  Circumference = 2 pi r
Rectangle: Area = ab = pi r2  Perimeter = 2 (a + b) = 2 pi r  Diagonal = d = sqrt (a2 + b2)1/2  Width = b.

2 (a + b) = 2 pi r   =>   (a + b) = pi r   =>   (a + b)2 = pi2 r2.

[1]  (a2 + 2ab +  b2) = pi2 r2.

ab = pi r2   =>

[2]  2ab = 2 pi r2

[2] - [1]   (a2 + b2) = pi2 r2 - 2 pi r2 = pi r2 (pi - 2) = ab (pi - 2)  =>

[3]  a2 - (pi - 2) b a +  b2 = 0.

a = { (pi - 2)+/- [ (pi - 2)2 b2 - 4b4 ]1/2 } / 2 = f (b)

Ratio of circumference to diagonal = g.

Write g in terms of rectangle width b.

g = 2 pi r / d = 2 (a + b) / ( a2 + b2 )1/2

g(b) = 2 (  f (b) + b ) / (  f2 (b) + b2 )1/2

excellent work as always bonanova but should

Spoiler

{ (pi - 2)+/- [ (pi - 2)2 b2 - 4b4 ]1/2 } / 2 actually be { (pi - 2)+/- [ (pi - 2)2 b2 - 4b2 ]1/2 } / 2

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excellent work as always bonanova but should

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{ (pi - 2)+/- [ (pi - 2)2 b2 - 4b4 ]1/2 } / 2 actually be { (pi - 2)+/- [ (pi - 2)2 b2 - 4b2 ]1/2 } / 2

It certainly should. It clearly fails a units check. Good catch.

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But actually ...

Spoiler

If Rectangle (a,b) and Circle (r=1) have the same area, then ab = pi.
If they also have the same perimeter, a+b = pi.
Substituting a=pi/b into the second equation leads to

b2 - pi b + pi = 0

which has no real solutions.
So the premise of the OP is not true.

That's because circles maximize area for a given perimeter.
An equal-perimeter rectangle must have less area.

The most efficient rectangle is a square, with side pi/2 and area pi2/4.
That's (pi/4), or about 78%, of the circle's area.

Good one. You had me going for a few days.

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