Ely Ranch Stained Glass Studios

 

Stained Glass Lampshade Design Mathematics

 

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Information prepared for:
Natalie Mohamed & Lauren Urbanek
Grade 11
Holy Trinity School
Richmond Hill, ON

Print and assemble the lampshade scale model as an aid to this discussion.

The design of a pyramid, trapezoid, or cone-shaped stained glass lampshade  includes the overall desired dimensions of height, h, top width, d, bottom width, e, and the number of sides or panels, n, to produce the desired shape.
 

 

During shade construction each panel will be assembled separately on a flat surface.  All panels are then combined in an upright position  for final assembly.  The panels dimensions & angles will form the desired shape & size when fastened together.

Multi-piece panels would be pre-assembled before all panels are combined.

 

 
The dimensions of each equally-dimensioned panel is determined as follows:

First, the top and bottom widths, s and t, of the panel are calculated.  To visualize this calculation, the lamp shape is viewed from above.  Each edge is inscribed in a circle with a diameter  equal to the specified top and bottom width, d and e.

These edge lengths can be approximated by dividing each circle's circumference by the number of panels.  These dimensions, although not exact, will be close enough for stained glass work.  Using, , 3.1416, the constant ratio of a circle's circumference to its diameter / radius, and knowing the circumference of a circle equals times its diameter, , the following calculations are made: , , where n = number of sides or panels.

Last, the length of each panel, g, is determined by viewing the lamp shade shape from its side and observing that it can be divided into 2 right triangles and a rectangle.  Each triangle contains the desired dimension, g, as its hypotenuse.  Further, it can be observed that yielding .

The Pythagorean Theorem is then applied to calculate g, in this case, .  Substituting the value derived  for f above, the final formula, , is developed.

For example, for an 8-sided lamp shade with desired dimensions of 12 inch height, 6 inch top width, and 12 inch bottom width, each panel is calculated to have a top width, s, of 3.1416 x 6 / 8 = 2.36 or 2 3/8 inches, a bottom width, t, of 3.1416 x 12 / 8 = 4.71 or 4 11/16 inches, and a length, g, of sq.rt.(144 + {(12-6)/2 x (12-6)/2}) = sq.rt.(144 + 3 x 3) = sq.rt.(144 + 9) = sq.rt.(153) = 12.37 or 12 3/8 inches.

All dimensions are not exact but close enough for stained glass work.

Some stained glass lampshade self-instruction manuals and reference books express lamp patterns in terms of slope or angle of the sides to a horizontal plane.  These patterns allow for re-sizing the finished piece while maintaining the desired shape and slope.

In this case, the trigonometric SIN function is used to determine the length, g, of each panel.  Given the slope angle, A, the formula is known, and with re-arrangement, .

For example a 12-inch tall, h, lampshade with a slope of 76, the height of each panel, g, will be 12/Sin 76 = 12/.97 = 12.37 or 12 3/8 inches.


     18 Aug 2013

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