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| Joined: | Mon Nov 21st, 2005 |
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Journey UCA
Octahedron
What is under the Pyramids of Giza??
Comparative surface area to volume ratios of shapes?
Cubes and spheres
What happens?
Question:
What happens to the surface area to volume ratio as the cube gets larger?
Answer:
SA / V decreases as the cube gets bigger
So...
Surface Area of a Regular Octahedron
Volume of a Regular Octahedron
Example Calculate the surface area and volume of an octahedron with an edge of 5 cm.
A = 2 x 1.7320508 x 5 x 5 = 86.60254
A = 3.4641016 x 5 x 5
V = 1.4142136/3 x 5 x 5 x 5 = 58.92556666666667
V = .4714045333333333 x 5 x 5 x 5
Octahedron change in ratio of surface area to volume as size increases:
Unit measure of 1 = Edge
SA for Octahedron A: 3.4641016V for Octahedron A: .4714045333333333
SA / V Ratio = 7.34843900 to 1
Unit measure of 3 = Edge
SA for Octahedron B: 31.1769144
V for Octahedron B: 12.7279223991
SA / V Ratio = 2.449489 to 1
Unit measure of 5 = Edge
SA for Octahedron D: 86.60 cm2
V for Octahedron D: 58.92 cm3
SA / V Ratio = 1.46969 to 1
Question:What happens to the surface area to volume ratio as the Octahedron gets larger?
Answer:SA / V decreases as the Octahedron gets bigger
The cube goes from 6 to 1 (at 1 unit measure for Edge) to 2 to 1 (at 3 unit measure for Edge)
The Octahedron goes from 7.3 to 1 (at 1 unit measure for Edge) to 2.4 to 1 (at 3 unit measure for Edge)
The Sphere has no edge; but the measure used often is radius in calculating volume to surface area ratio?
UCA:
To exist, something must have shape and occupy space. Both concepts have strong ties to common sense principles of geometric shape.
Some shapes are more efficient at forming volume than others. Spheres for example have a surface area to volume ratio of 4:3, meaning that there is less volume area compared to surface area.
Perfect Cubes for example have a surface area to volume of 2:1, meaning that there is one half the volume compared to surface area.
Octahedron
The most efficient shape in terms of number of points combining to create maximum volume is an octahedron (six points) combining to create eight equally proportioned triangles, expanding to a middle point and reducing to a single point. The surface area to volume of a perfect Octahedron is always 1:2. That is, an octahedron creates twice as much volume as it takes surface space to create it. Octahedrons are therefore the simplest and most efficient shapes in terms of minimum number of points for maximum volume creation.
Why do I think (intuition?) that a sphere is the most efficient shape in terms of creating the most volume for the least amount of surface area?
Why do I think that this has to do with the shortest distance between two points in a straight line?
The cube at 6 (area) is not equal to Octahedron at 7.3 (area) so that needs to be standardized for clearer picture, then get Sphere formula standardized to 6 (surface area) so as to compare Standard measures of Surface Area (cube, octahedron, and sphere) all at 6, before finding volumes inside that total measure of surface area for each shape!
1 standard cubic measure of volume requires 6 standard square measures of area with a cube shape.
Online cube calculator
1 volume (cubic unit) requires 6 square measures of surface area for a cube.
1 cubic inch or 1 cubic meter requires 6 square inches or 6 square meters of surface area. Note the same is true for 1 cubic mile and 6 square miles = changing the standard measure recalculates the ratio based upon the new size (from inches, to meters, to miles, etc.?)
How much surface area measured as the same standard square measures of area are required to encompass a sphere that measures 1 standard cubic measure.
Sphere calculator
.28204 Radius for a Sphere is .096 volume for a standard measure of 1 surface area.
1 standard measure of surface area for a sphere encloses .096 standard volume measure.
1 standard measure of surface area for a cube encloses...
This math stuff is not my hidden talent.
Back to the Sphere calculator looking for 1 standard measure of volume not a standard measure of surface area because I am looking for a standard measure of volume of 1 square inch - shown in the cube calculator.
Standard Cubic Area of 1
Cube requires 6 standard square units of surface area - 6 square units for 1 cube
Sphere requires 4.8365 standard square units of surface area to enclose the same Standard Cubic Area of 1 - a radius of .620259 Standard units (length)
Cube = 1 Volume to 6 Surface Area
Sphere = 1 Volume to 4.8365 Surface Area
What about the Octahedron?
Solving for 1 standard unit of surface area?
Octahedron Calculator
Sphere = 1 Volume to 4.8365 Surface Area
Octahedron = 1 Volume to 5.719 Surface Area
The sphere wins?
Sphere = 1 Volume to 4.8365 Surface Area
Octahedron = 1 Volume to 5.719 Surface Area <-----not the right shape?
Cube = 1 Volume to 6 Surface Area