A Table of Isoperimetric Ratios
version March 5, 2017
Matthew M. Conroy
conroy “at” math dot washington dot edu
For a simple closed plane curve, the isoperimetric ratio is
P2
A
where P is the length of the curve, and A is the area enclosed by the curve. This ratio is a dimensionless constant
that is invariant under scaling: changing the size of the shape does not change this ratio.
The table below gives this ratio for a variety of shapes.
One can prove that the circle is the plane curve with the smallest (and hence, smallest possible) isoperimetric ratio,
and so the table begins with the circle.
4π ≈ 12.56637061
circle
regular 11-gon
square with quarter circle
rounded corners (radius of
corners is one-third of overall
width)
regular 10-gon
regular 9-gon
π
44 tan 11
≈ 12.91956568
(4+2π)2
5+π
≈ 12.98810988
π
40 tan 10
≈ 12.99678784
36 tan π9 ≈ 13.10292843
three-quarter truncated circular
arc
≈ 13.14170265
regular octagon (equivalent to
square augmented by four optimal isosceles triangles)
32 tan π8 ≈ 13.25483399
square with hacked off corners
(horizontal and vertical edges
are one-third overall width)
√
(4+4 2)2
7
≈ 13.32211914
≈ 13.35374554
ellipse, 3:2
regular septagon
cardioid
28 tan π7 ≈ 13.48408932
128
3π
≈ 13.58122181
semicircle with optimal adjoined isosceles triangle
≈ 13.71361987
square with three optimal adjoined isosceles triangles
≈ 13.75752837
≈ 13.77927647
16 pointed star
equilateral triangle with three
adjoined semicircles
regular hexagon
semi-circle with adjoined right
isosceles triangle
≈ 13.78342300
24
√
3
≈ 13.85640646
√
(2 2+π)2
1
2 π+1
≈ 13.86385058
≈ 13.96032741
15 pointed star
equilateral triangle with two adjoined semicircles
4(1+π)2
√
π+ 3
≈ 14.07800126
≈ 14.18654408
14 pointed star
semicircle with optimal adjoined rectangle
(4+π)2
2+ π
2
≈ 14.2831853071
≈ 14.35072219
square with two optimal adjoined isosceles triangles
≈ 14.47485646
13 pointed star
√
regular pentagon
square with two adjoined semicircles
100
√
25+10 5
(2+π)2
1+ π
4
≈ 14.53085056
≈ 14.80676722
≈ 14.85125168
12 pointed star
≈ 14.93924249
ellipse, 2:1
square with
semicircles
three
adjoined
rectangle, 2:1, with four adjoined semicircles
square with adjoined semicircle
(1+ 32 π)2
1+ 83 π
36π 2
8+5π
(π+6)2
1
2 π+4
≈ 14.98160283
≈ 14.98676855
≈ 15.00121550
square with optimal adjoined
isosceles triangle
≈ 15.07344594
square with single optimally
rounded corner (radius is one
half the side length)
12 + π ≈ 15.14159265
square with four adjoined semicircles
4π 2
1+ π
2
≈ 15.35649370
≈ 15.35751730
11 pointed star
equilateral triangle with adjoined semicircle
(4+π)2
√
π
2+ 3
≈ 15.44193344
16
square
16
circular sector, two radians
≈ 16.06491327
10 pointed star
quarter circular sector
semicircle augmented by two
semicircles, I
semicircle
equilateral triangle with adjoined optimal rectangle
2
4(2+ π
2)
π
16
3 π
≈ 16.75516081
2(2+π)2
π
4(6 −
≈ 16.23455083
√
≈ 16.82966439
3) ≈ 17.07179676
≈ 17.10465828
9 pointed star
rectangle, 2:1
18
18
circular sector, one radian
18.31061221
rectangle, 2:1, with two adjoined
semicircles
64 − (16)43/4 ≈ 18.74516600
8 pointed star
isosceles right triangle with √
optimal adjoined rectangle (1 : 22 )
isosceles right triangle with adjoined square
equilateral triangle
rectangle, 3:1
√
8 + 8 2 ≈ 19.31370849
√
2(4+ 2)2
3
36
√
3
≈ 19.54247233
≈ 20.78460969
64
3
= 21.3̄
≈ 21.62680012
7 pointed star
isosceles triangle, 2:1 sides, with
adjoined semicircle
(8+π)2
√
π
2 + 15
≈ 22.80310635
2(2 +
√
2)2 ≈ 23.31370849
isosceles right triangle
2(2 +
square with two adjoined
isosceles right triangles
√
64
√
1+ 3
square with four adjoined equilateral triangles
2)2 ≈ 23.31370849
≈ 23.42562584
25
rectangle, 4:1
8π ≈ 25.13274122
semicircle augmented with two
semicircles, II
24
√
3
≈ 25.85640646
√
68+48 2
5
≈ 27.17645019
12 +
right triangle, 30-60-90
square with three adjoined right
isosceles triangles
(3 +
√
√
5)2 = 14 + 6 5 ≈ 27.41640786
right triangle, 2:1 legs
√
6 pointed star (Koch curve, stage
1)
768 ≈ 27.71281292
5-square cross
rectangle, 5:1
isosceles triangle, 2:1 sides
13-square diamond
5-square cross with four adjoined semicircles
square with four adjoined
isosceles right triangles
= 28.8
144
5
= 28.8
50
√
3
≈ 28.86751345
400
13
≈ 30.76923076
8(4+π)2
π+10
16 +
32
3
√
≈ 31.04789318
2 ≈ 31.08494466
784
25
25-square diamond
98
3
rectangle, 6:1
isosceles triangle, 3:1 sides
144
5
98
√
8
= 31.36
= 32.6̄
≈ 34.64823227
rectangle, 7:1
256
7
81
2
rectangle, 8:1
Koch curve, stage 3
384
√
5 3
9-square cross
rectangle, 9:1
5 pointed star
semicircle augmented by two
semicircles, III
Koch curve, stage 4
≈ 36.57142857
= 40.5
≈ 44.34050067
400
9
= 44.4̄
400
9
= 44.4̄
√
10−2 5
sin π
5
≈ 47.02282018
16π ≈ 50.26548245
6144
√
47 3
≈ 75.47319263
General formulas
Rectangles
For a rectangle with aspect ratio m (i.e., the length of a long side is m times the length of a short side), the ratio is
(2m + 2)2
4
= 4m + 8 + .
m
m
Regular polygons
For a regular polygon with n sides, the ratio is 4n tan nπ .
Circular sectors
For a circular sector with angle θ, the ratio is
2(2 + θ)2
θ
This has a minimum of 16 at θ = 2.
Regular polygonal stars
For a star based on a regular polygon with n sides, the area is
sin θ sin θ/2
θ
sin 3θ
2 − sin 2
A = nr2
where θ =
2π
n .
!
The perimeter of the star is
p
P = 2n (b − a)2 + w2
where a = r cos θ2 , w = r sin θ2 , and
b=
r sin θ
.
sin − sin θ2
3θ
2
n-square diamond/cross
The ratio is
(8m + 12)2
2m2 + 6m + 5
where m is the order of the diamond (m = 0 is the 5-square cross, m = 1 is the 13-square diamond, etc.)
This approaches 32 from below as m tends to infinity.
Koch curve
The ratio is
√
180
3(8 −
4 2n
3
n
3 49 )
for the n-th iteration (i.e., n = 0 is an equilateral triangle, n = 1 is a six-pointed star, etc.)