Real-world Math: Lampshades

My wife wants to make something like this:

The issue is that she wants the shadow part to be words, so getting the angles right is essential.

So, how should she design it in photoshop so that we can print it out and glue it on? Enter math.

She measured the top and found that it has a circumference of 26 inches. The bottom of the shade has a radius of 32 inches. The "slant height" is 7 inches. The "true" height is harder to measure because you have to keep it level and ours is already on the lamp, so it wasn't measured, but could be "if you REALLY need it".

So, what to do? Well, here's what I knew:

1. This shape is called a "frustrum." In particular, this is a conical frustrum.
2. You can create a cone by cutting a sector from a circle and gluing the ends together.
3. Since this frustrum shape is a large cone with a top cone cut off, it could be created by taking a sector from an annulus (a "ring" made by taking a large circle and cutting our a concentric, smaller circle).
4. Formulas I knew and may or may not need here:
   
   • For a sector created from a circle of radius R and sweeping out angle T (in radians), the arc length of the sector (S) is S = R*T
   • Circumference = 2 * pi * radius
   • Pythagorean theorem (in particular, using the "true" height, slant height, and radius of a cone)

So, here's what I tried first (aka The Hard Way):

(Background info: I've never taught geometry and am more of a functional analysis person, so that's why this stuff came to me first.)

Consider the sector that would form the large cone (from which we cut the top off to make the lampshade): I knew the arc length is 32, but that's about it. The angle I called T and the radius of this large circle I called x.

Things I learned about this cone by playing around with the formulas above:

• If we call the radius of this large cone R, then 32 = 2 *pi * R, so R = 16/pi.
• 32 = x * T
• If we call the "true" height of the cone H, then H^2 + R^2 = x^2 (and could use R as above).

Consider the sector that would form the cone we cut off the other one: I know the arc length is 26 and that's it for that one. This angle is also T, but the smaller circle would have a radius of y (so that y < x above). Similar to the above things, but with a couple added to relate the two cones I now know:

• If we call the radius of this smaller cone r, then 26 = 2 * pi * r, so r = 13/pi.



• 26 = y * T



• If we call the "true" height of this cone h, then h^2 + r^2 = y^2 (and can use r from above).



• x = y + 7 (the 7 was measured as the slant height of the shade, see above)



• H = h + m, where m is the measured "true" height of the shade.

Putting these pieces together, I ended up with these two equations:

• H^2 + (16/pi)^2 = (y+7)^2
• (H-m)^2 + (13/pi)^2 = y^2

Two equations with two unknowns may be solvable, so I tried to actually measure m and got about 55/8 inches. I could solve these for y, then find the angle T = 26/y and have the sector for the small circle. Adjusting a bit to get x would give the large circle.

This was messy and gross. In fact, I gave up from here and decided to find a more "elegant" solution.

Second attempt (The Easy Way):

This time I tried to go back to basics of what I remembered about geometry (bear in mind I've not had anything really geometry related since 8th grade which was almost 20 years ago now).

I know there is such a thing as "similar triangles," so maybe there's such a thing as "similar cones" (seeing as how they are just rotated triangles). So, I went with that. Variables are already defined as above, so here are the ratios I used:

x/y = R/r = (2*pi*R)/(2*pi*r) = 32/26 = 16/13

So, clearly y = (13/16)*x. We also know that y + 7 = x and using substitution, (13/16)x + 7 = x, so (3/16) x = 7, so x = 112/3. Now that I know x, I can get T = 32/x.

Whoa! That's easy, but were my assumptions ok? Well, I'm not really doing a general proof here, so I'll just check it with these numbers: To the Bat Geogebra!


This is a Java Applet created using GeoGebra from www.geogebra.org - it looks like you don't have Java installed, please go to www.java.com


I'm still getting to know this application, so I'll explain how I made this:
First I put A at the origin and defined a to be 112/3 as discovered above. Create a circle with center A and radius a (the program called this c as you can see in the variables to the left). I needed another point on this circle, so I just used the one on the x-axis by creating line b (y = 0) and then intersecting the circle with the line; this created point B.

So, create a circle centered at B and radius a; this is circle labeled as f. (I now realize this is somewhat redundant since I could just switch the roles of A and B without this extra step, but I went into it thinking A would be a point on the circle's edge and would create the center of the circle elsewhere.) Anyhow, I then figured out the angle (which I called T above, but is labeled e in this applet). 0.86 is in radians, though, and since my wife will do the measuring, I figured degrees would be better, so I converted it which is labeled as d.

Use GGB to find a point which is d degrees away from A which it labels as A'. Just to make sure, create the arc between A and A' and check the length: 32. Perfect!

Now to finish, I made a circle centered at A and radius 7 (since I knew that was the slant height of the shade). Where that intersects the x-axis is the point C which will be the beginning of the smaller arc. Make a circle centered at B and radius y (which you may remember is equal to x - 7 as above) and find where that intersects with the line between A' and B. Double check by finding the measure of arc CD (26 yay!) AND the angle CBD (49.11 degrees yay!).

So now we have it. Measurements can now be passed back to the wife who is the real Illustrator/Photoshop guru and she can bend the text she wants for the lampshade to fit.

Pshew!

I'm still not sure what she has in mind and maybe this project won't even get finished by her, but I'll post a picture if it ever gets there!