Friday Fluff: Using Trigonometry while Decorating

A look this perfect requires MATH! Image from Home Decor Store

Thanksgiving is over and that means the Christmas season is upon us! Christmas time is a wonderful time and I'm really looking forward to, for the very first time, decorating my own house. Every year, once I had moved out of my parents' house after college, I put up a big plastic Christmas tree. Every year I added fancy ornaments to the tree to replace a few of the cheap plastic balls I bought the first year. But that was the limit of my apartment's festive adornments because my apartment complex had an HOA that banned outdoor or window decorations. So no wreaths on the doors or windows, no garland, no plug-in candles, no fun at all!

This year I own my house and I can make it as fancy and festive as I want (well, as I can afford). And as part of that I have decided to use lots of lights and garland. And that is what this post is really about: how to figure out how much garland or how many strings of light are needed to candy-cane wrap or swag across a column or rail. To do this we will use trig!

Most of the credit for pointing this out to me goes to Mr. Dr. Kris :)

Straight across:

For this look, we don't need any trig at all! Just measure along the edge you want to mount the garland to and buy garland that length.

If you want lighted garland you may find yourself struggling to find some the exact length. In that case, buy unlighted garland and cut it to length. Then twist the garland with a string of lights. In this picture, you can see they did that, and they twisted the garland with a red ribbon.

That was too easy.

Candy-Cane Wrap:

To figure out how much garland (or how long a string of lights) you need for this look you're going to need some trig. We'll start by calculating this the easy way. For this method we will choose a wrap angle, and just accept the number of spirals and spacing between the spirals we get. For this method the diameter of your column or rail does not matter!

Imagine a right triangle. The base sits on your patio, the other leg goes up the column, and the hypotenuse is where we will put the garland. Imagine the triangle is made of paper, we can wrap it around our column or flatten it out.

I've drawn a picture to approximate what you should have envisioned. Now to calculate how much garland we need we are going to solve for an the hypotenuse (garland) length of this angle, angle, side triangle problem. If you remember from geometry or trig, each triangle has six traits (3 angles, and 3 side lengths) and if you know the right combination of 3 of those, you can calculate the other three. This is a right triangle. So, once we choose a wrap angle, we will have all the angles. (All three angles of any triangle must add to 180.). And we will use the column height as the side (the patio length is totally irrelevant for this). Enough theory; it's time for a formula.

Column Height / Sin (wrap angle) = garland length / sin (90 degrees)
The sine of 90 degrees is 1, so that means

Column Height / Sin (wrap angle) = garland length

Let's say you want to climb 1 ft of column at a 30 degree angle

1ft / Sin (30d) = g --> 1 / 0.5 = g --> g = 2 feet

That means that for a wrap angle of 30 degrees you need a garland twice as long as your column is tall. (You can just remember this cheat and skip doing your own trig if you want)

Candy-Cane Wrap II:

Let's say that the first method wasn't right for you. You don't care what your wrap angle is. You want your wraps of garland to be spaced precisely so many inches apart. Ok. We can do that. For this we will need the circumference of the column/rail we're wrapping around. That will replace the "patio" length in the figure above. We will replace column height with wrap spacing. Let's see that formula.

square root (column circumference² + wrap spacing²) = garland length per wrap
column height / wrap spacing = number of wraps
garland length per wrap * number of wraps = total garland length needed

Let's say you have a round column that is 6 inches in diameter and 10ft tall and you want the wraps to be spaced 6 inches apart (center to center).

cc = 3.14*6 --> cc = 18.84"
sqrt (18.84² + 6²) = g / wrap --> sqrt (354.95 + 36) = g / wrap --> 19.77in garland per wrap
10ft / 6in = number of wraps --> 10ft / 0.5ft = number of wraps --> 20 wraps
19.77in garland per wrap * 20 wraps = 395.45in garland --> 33.9 feet of garland
Sorry, no cheat-y short cut for doing it this way. You want to be this persnickety, you can do the math.  Don't forget to round up and consider slack in the line.

If you're wondering what the wrap angle is in this example, we can find out
6in / sin(x) = 19.77 --> 6 = 19.77 * sin(x) --> 6 / 19.77 = sin(x) --> 0.30 = sin(x) --> x = 17.45 degrees.

That's kinda flat, I think. I prefer the first method. I think wrap angle is more important than wrap spacing, but now you know how to do both!

Swags:

These swags are parabolas. Because that's how gravity works. The link has way more information about parabolas than you likely ever wanted to know. We will calculate the arc length (garland length) of a parabola based on a known/desired width (space between the bows in the picture) and drop height (space from railing to lowest point of the swag). This time I'm going to spare you the background explanation, because it involves calculus. Here's the formula (h = drop height, w = width):


0.5√16h²+w² + [w²/(8h)][Ln(4h + √16h²+w²) - Ln(w)] = garland length per swag

So lets say that you have a railing you'd like to swag, just like this picture. The railing is 30 feet long and has a post every 5 feet. You want to put the bows on the posts, so you want 6 swags, each 5ft wide. You want them to drop 2 feet.

0.5√16h²+w² + [w²/(8h)][Ln(4h + √16h²+w²) - Ln(w)] = garland length per swag -->
0.5√16*2²+5² + [5²/(8*2)][Ln(4*2 + √16*2²+5²) - Ln(5)] = g / swag -->
0.5√16*4+25 + [25/(8*2)][Ln(4*2 + √16*4+25) - Ln(5)] = g / swag -->
0.5√64+25 + [25/(16)][Ln(8 + √64+25) - Ln(5)] = g / swag -->
0.5√89 + [25/(16)][Ln(8 + √89) - Ln(5)] = g / swag -->
0.5*9.4 + [25/(16)][Ln(8 + 9.4) - Ln(5)] = g / swag -->
4.71 + [1.56][Ln(17.4) - Ln(5)] = g / swag --> 
4.71 + [1.56][2.86 - 1.61] = g / swag -->
4.71 + [1.56][1.25] = g / swag -->
4.71 + 1.95 = g / swag -->
6.66 feet of garland per swag * 6 swags --> 39.96 feet of garland (no being superstitious)


If that's a little more arithmetic than you're feeling confident with, Had2Know.com has a calculator you can use. You'll just take the arc length they calculate for you and multiply by the number of swags you need.