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No, I'm not smarter than a fifth grader!

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- Thread starter fauxtex
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In summary, the conversation discusses the process of dividing a circular dome with a 14' diameter and a gradual slope to 2' high in the center into 6 or 8 equal "pie" shapes for the purpose of creating a design template for a mural. The conversation covers different methods and formulas for finding the exact shape and size of each piece, including using a flat piece of paper, approximating with smaller pieces, and using geometry calculations. The final solution involves using a dymaxion map to simplify the dome into a polyhedron and then transferring the design onto it.

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No, I'm not smarter than a fifth grader!

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I do believe that I can lay a flat piece of paper on it but it has to be cut into a shape that has curved sides. Kind of like a top of a globe flattened out or a flat map of the world that has equal longitude lines with curved sides. (I'm talking about the world maps that look cut up into 'slices", If you cut the flat map of the world on the lines you could wrap it around a globe) I want to cut the dome into 6 or 8 equal sizes. I am painting the same thing on each part of the ceiling though so I only need one for a template. There must be some kind of formula to figure these shapes. No?

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Ok, Maybe I'm not wording it correctly. Let's say it is a 14' diameter round pool and it slopes to 2' deep in the center. I would like to divide it into 8 equal size and shaped pieces. (Starting in the center and cutting out to the edge similar to a pie, I know they won't have straight edges though) With out having the pool in front of me, How do I figure out how big and what shape the peices will be? Anyone? :yuck:

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You could find the lengths of the sides you want and the angles between them with some simple but tedious geometry (with some assumptions about what the shape is, like whether it's a spherical cap), but you'd get a shape that couldn't be cut out of a flat piece of paper. The smaller the pieces, the less you'll have to modify these values to get something flat.

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I expect what you want is a template that is the same linear dimensions as your curved slice of dome. If you actually want to match it up with the curved dome you may need to slice up the template after you have made the design.

It's a fairly simple bit of geometry to work out the shape assuming the dome really is spherical as you described it. The picture is a section through the centre of the dome.

To find the radius of the dome r, use Pythagoras theorem

r^2 = (r-2)^2 + 7^2

r = 11.25 ft

To find max value of theta:

sin theta = 7/r, theta = 38.5 degrees

At any point P, the "height" of your template is the distance round the circle from the top of the dome

= r.theta (working in radians)

= 2 pi r theta / 360 (working in degrees)

The radius of the horizontal circle round the dome through P is r sin theta

So the "width" of the template at that point is 1/8 of the circumference of the circle = (2 pi r sin theta)/8

Work out the width and length from the top at say 10 different values of theta between 0 and 38.5 degrees and plot them out to make your template.

As a check, the angle between the two curved sides at the top of the template must be 360/8 = 45 degrees. At the bottom the width should by 1/8 of a 14 ft diameter circle = 14 pi /8

It's a fairly simple bit of geometry to work out the shape assuming the dome really is spherical as you described it. The picture is a section through the centre of the dome.

To find the radius of the dome r, use Pythagoras theorem

r^2 = (r-2)^2 + 7^2

r = 11.25 ft

To find max value of theta:

sin theta = 7/r, theta = 38.5 degrees

At any point P, the "height" of your template is the distance round the circle from the top of the dome

= r.theta (working in radians)

= 2 pi r theta / 360 (working in degrees)

The radius of the horizontal circle round the dome through P is r sin theta

So the "width" of the template at that point is 1/8 of the circumference of the circle = (2 pi r sin theta)/8

Work out the width and length from the top at say 10 different values of theta between 0 and 38.5 degrees and plot them out to make your template.

As a check, the angle between the two curved sides at the top of the template must be 360/8 = 45 degrees. At the bottom the width should by 1/8 of a 14 ft diameter circle = 14 pi /8

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Thank you AlephaZero. This is what I was looking for! OOOO's

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A little late,

But this might help those struggling with the same problem.

Use a dymaxion map--simplify a sphere into a polyhedron:

http://en.wikipedia.org/wiki/Dymaxion_map

[URL]http://upload.wikimedia.org/wikipedia/commons/b/bb/Dymaxion_2003_animation_small1.gif[/URL]

and then transfer your design!

But this might help those struggling with the same problem.

Use a dymaxion map--simplify a sphere into a polyhedron:

http://en.wikipedia.org/wiki/Dymaxion_map

[URL]http://upload.wikimedia.org/wikipedia/commons/b/bb/Dymaxion_2003_animation_small1.gif[/URL]

and then transfer your design!

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