top

This is ok:
The area and volume of a sphere are 1/2 that of the box it came in.
But the rest of this page is a mess...

Landmarks

volume of a sphere -

"1/2 that of the box it came in" (-5%, ±<1%)

1	exact	volume of a box of size 1
0.5236	±<<1%	volume of a sphere of diameter 1
0.52	±<1%
0.5	-5%, ±<1%

area of a sphere -

"1/2 that of the box it came in" (-5%, ±<1%)

6	exact	area of a box of size 1
3.142	±<<1%	area of a sphere of diameter 1
3	-5%, ±<1%

"pi" -

3 (-5%, ±<1%) 3.1 (-1%, ±<1%)

3.1416	±<<1%
3.14	±<<1%	(~-0.05%)
3.1	-1%, ±<1%
3	-4.5%, ±<0.01%
3	-5%, ±<1%
1	-68%, ±<1%
1	-70%, ±<2%

Notes

Currently in pre-first-draft state...
Do `vs cube of same mumble' landmarks.
Need distinct `simple closed volume' page?
Growth relations.
4.2, 4.19 concept.

Sphere vs cube

constant		side&diam.	area	volume	relation
size m	cube	1 m	6 m2	1 m3	1.91 × sphere of same size
	sphere	1 m	3.142 m2	0.5236 m3	0.524 × cube of same size
area m2	cube	0.4082 m	1 m2	0.06804 m3	0.724 × sphere of same area
	sphere	0.5642 m	1 m2	0.09403m3	1.38 × cube of same area
volume m3	cube	1 m	6 m2	1 m3	1.24 × sphere of same volume
	sphere	1.241 m	4.836 m2	1 m3	0.806 × cube of same volume

Sphere vs cube of same size (0.5, -5%)
Sphere vs cube of same area
  4/3 | 3/4 (4%)  1 1/4 | 0.8 (10%)  1.3 (6%)  "+40%" | 5/7 (2%)  1/2 (50%)
Sphere vs cube of same volume
  4/3 | 3/4 (8%)  1 1/4 | 0.8 (1%)   1.3 (5%)
c v/a  4/3 <10%
c vol  "25%" (1%)
c area "40%" (2%)
deformation image to keep the two straight

Sphere diameter shrinking vs area required to enclose 1 m³.

diam	area	vol	N spheres	total area
1 m	3.142 m2	0.5236	1.910	6
10-1 m	3.142 × 10-2 m2	5.236e-04	1.910 × 103	6 × 101
10-2 m	3.142 × 10-4 m2	5.236e-07	1.910 × 106	6 × 102
10-3 m	3.142 × 10-6 m2	5.236e-10	1.910 × 109	6 × 103
10-4 m	3.142 × 10-8 m2	5.236e-13	1.910 × 1012	6 × 104
10-5 m	3.142 × 10-10 m2	5.236e-16	1.910 × 1015	6 × 105
10-6 m	3.142 × 10-12 m2	5.236e-19	1.910 × 1018	6 × 106

Note volume increasing as r³ (10^-3 per 10^-1), but area only as r² (10^-2 per 10^-1).
[...Simplify...]

Size of shrinking spheres vs area for 1 m³ worth.

diameter	area	vol	N spheres	total area
× 1/2 (m)	× 1/4 (m2)	× 1/8 (m3)	× 8	× 2 (m2)
(1/1) 1.241 m	4.838 m2	1.000 m3	1	4.834 m2
(1/2) 0.6205 m	1.210 m2	1.251 × 10-1 m3	8	9.670 m2
(1/4) 0.3103 m	0.3024 m2	1.564 × 10-2 m3	64	19.34 m2
(1/8) 0.1551 m	0.7560 × 10-1 m2	1.955 × 10-3 m3	512	38.68 m2
(1/16) 0.7756 × 10-1 m	1.890 × 10-2 m2	2.443 × 10-4 m3	4093	77.36 m2
(1/32) 0.3878 × 10-1 m	0.4725 × 10-2 m2	3.054 × 10-5 m3	32744	154.7 m2
(1/64) 1.939 × 10-2 m	1.1812 × 10-3 m2	0.3817 × 10-5 m3	261954	309.4 m2
(1/128) 0.9695 × 10-2 m	2.953 × 10-4 m2	0.4772 × 10-6 m3	2095632	618.9 m2
(1/256) 0.4848 × 10-2 m	0.7383 × 10-4 m2	0.5965 × 10-7 m3	16765052	1238 m2
(1/512) 2.424 × 10-3 m	1.8457 × 10-5 m2	0.7456 × 10-8 m3	134120419	2475 m2
(1/1024) 1.212 × 10-3 m	0.4614 × 10-5 m2	0.9320 × 10-9-10 m3	1072963350	4951 m2
(1/2048) 0.6060 × 10-3 m	1.154 × 10-6 m2	1.165 × 10-10 m3	8583706797	9901 m2

Doables:
 In need of overhaul.
 Link to Surface volume-density.

History:
  1997.Jun.23 - Added `shrinking spheres vs area' tables.
  1997.Jun.21 - Created.