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Ely Ranch
Stained Glass Studios
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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.
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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.
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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.
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The dimensions of each equally-dimensioned panel is determined as
follows:
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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.
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