Figuring Measurements of a 5-pointed Symmetrical Lighted Star

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Have a frame and want to know how many feet of light strings you'll want to go around the perimeter or across the points?

Want to form a star of a particular height? Take a look at the photos.

Note, this is for a Symmetrical Star -- all 5 points have the same angle, and all 10 outer sides along the perimeter are equal.

QUICK FACTS:

  • The angle in any point (or tip) of the star is 36 degrees.
  • If you bisect the angle in any point (split it in half), each angle is 18 degrees.

Because this angle is always the same, given only the measurement along one of the outer sides of the star, as long as it is a Symmetrical Star, you can calculate the other dimensions. Or given one of the other dimensions, you can calculate the side.

The parts of this Symmetrical Star are always in the same ratios or proportions to each other. So they're easy to calculate.

Step 1: Sides, Base, Inner Chord Measurements

If you know the measurement along one side of a star point -- for instance, the distance between A and B in the drawing, then the following calculations are possible.

( Note that for a "Side," any two points along the perimeter would do, as long as they're neighbors. )

QUICK RESULTS:

  • The "Base" (see drawing) will be always be 0.309 times the measurement of any one of the sides of a StarPoint. Another way of saying this is that the Base is 30.9% of the size of any Side -- when it's a Symmetrical Star. If the Side of the star were 100 miles long, the Base would be 30.9 miles long.
  • Because the "Inner Chord" (see drawing) is just two Bases added together, the length of the Inner Cord will be (2 x 0.309 ) = ( 0.618) times the measurement of any one of the Sides of a star's point. (Remember the sides are all the same measurements for any particular Symmetrical Star.)

Step 2: Measuring Around the Perimeter (10 Sides) or Across From Point to Opposite Point in 5-Segments

If you know the measurement along one side of a star point, then:

QUICK RESULTS:

  • The Perimeter will be the sum of ten of the sides of a star point.

(Obviously. Just go around the star on the outside, starting at any point and ending up back there.)

  • The"Length Across" from any one of the 5 points to the point opposite it will be 2.618 times the measurement of the Side of a Star Point.

(For instance, in the drawing, the distance from A to E will be 2.618 times the measurement from A to B. This is because you're adding together AB, an Inner Chord, and BE. AB and BE are the same length because it's a Symmetrical Star, so that adds up to 2 Sides, and the Inner Chord is 0.618 of any Side. The sum across from A to E is 2.618 x AB or 2.618 times a Side. )

  • The total measurement for forming the star by going across from point to opposite point in succession will be 5 times the length between two opposite points. Or just multiply your known measurement along one side of a StarPoint by (5 x 2.618) = 13.09.

We're talking about starting at the bottom left point C, going across to the opposite point I, then
down to E, back across to A, horizontally across to G, then down and across -- until you get back to your starting point C.

Step 3: From Knowing a Side to Height -- and Back Again

So if you know the measurement of a Side, then you can add up the measurement from one point across to it's opposite point. (See previous steps.)

From that number you can calculate the actual height of the star. (The red line in the drawings.)

QUICK RESULTS:

  • The height of the star will be 0.9511 times that length across between opposite points.

Working backwards, if you know the height of the star, then:

QUICK RESULTS:

  • Divide the height by 0.9511 to get the length across between two opposite points.
  • Divide the height by 0.9511 and then divide that result by 2.618 to get the measurement along one Side of one StarPoint.

Step 4: Example 30" Star

If I want a star that is 2.5 feet (30 inches) high, then...

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How much wire do I need to go around the perimeter? I need to figure out the measurement of one Side of a "StarPoint" then I know I'll need TEN of those to complete the perimeter.

Height = 30"

One Side of a StarPoint = Height / (.9511)x(2.618)

For my 30" high star, one side will be 30" divided by those 2 numbers... or 12.0483 inches, just barely over 12 inches.

Ten times that number should be about ten feet for the perimeter, of course, almost a half inch over. Plus whatever overlap or losses I'd have in real life, of course.

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If I choose to traverse the star by going from one point to another across the star to opposite points, how much wire would I need to complete the star? I need to figure out the length from one point to the opposite point, then multiply that by FIVE.

Height = 30"

Length Across to Opposite Point = Height / (.9511)

For my 30" high star, 30"/.9511 is 28.533 inches, and five of those segments will complete a star. I need 142.665 inches or 11 feet plus 10.665 inches. (Or 11 feet and a bit under10 and11/16" -- how accurate do you need to be for a 30" star?)

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If I only want to know the "Inner Chord" for a 30" high star, that's two Base measurements.

Height = 30"

One Side of a StarPoint = Height divided by [ (.9511)x(2.618) ] = 12.0483inches

Inner Chord = Base + Base = 2 x Base = 2 x (0.309) x One Side of a StarPoint

= 2 x 0.309 x 12.0483inches = 7.4458 inches

So the inner chord is a bit over 7 and 7/16 inches.

Step 5: A Real Life Star

So check out my real life star.

https://www.instructables.com/id/Intro-DIY-Lighted-Star-From-5-Coat-Hangers-a-Rope-

I started with the sides of the star, about 8 5/8" long. (That was determined by the size of my coat hangers that I built the frame with before laying rope light over it.)

How tall did it turn out to be? How much rope light did I use going across from one point to the other? (You can see we didn't just go around the perimeter, which gave it more strength.)

Answers:

  • I guesstimated about a little under 2 feet tall. What does the Math say?

Height = (One side of the star point) x (0.9511) x (2.618)

= 8.625" x 0.9511 x 2.618 = 21.476" tall

When we pull down the Christmas outside lights in January, I guess I'll find out how close I was.

It's just the right size for our outside lighted "tree."

  • I had an 18 ft rope light. It went across all the legs and then I knew I'd have some left over, plus some of it was the plug end without lights. What does the math say?

Length across to opposite point = 2.618 x 8.625" side

= 22.58 inches

And five times that to complete one pass around the star should have been 112.9 inches.

Again, it's 15 ft up in the air for now. Have to wait to see if the math works. But I knew from doing the math that I had enough rope light to go around once, but not twice.

If you want to know how to build this lighted star, I'm publishing the Instructable. I'll include a link when it's up. Making my own star is why I had to jot down the geometry of the star.

Hope it's useful to someone else.

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    9 Discussions

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    Soosekendallwilliams

    Reply 6 months ago

    Thanks, Kendall Williams, for providing the link to the little calculator. Very helpful. I didn't think of looking for a calculator online, last holiday when I was making my own star. No matter, I like pulling out a few trig skills and exercising them. :)

    I like your PVC frame, too. You're getting a good start on the holiday! I doubt I'll be making more stars, though. The one from last year is still in great shape, very sturdy with only box tape holding it. Unless we decide to mount another lighted tree on another bird house... I guess I could cover the whole field with stars and trees!

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    kendallwilliamsSoose

    Reply 6 months ago

    Thanks, Soose, for your comments. I liked your rigorous geometry effort for instructing other users on star geometry; and it made me recall Sine and Cosine functions. In high-school and college, I still had a few of those skills, but at my present age, they are very rusty! I fell back on the "calculator" as it was easier on my head, shameful for a retired engineer!
    I've completed the basic PVC star, consisting of 5 X 5' of PVC tubing and 10X 10-24 X 2-1/2 screws with nuts. Next will be 5X strings of 50 LED lights. Good thing that they are low power! We have two other large PVC tube stars, mounted on large redwood trees in front of our house in Sebastopol. They were designed/built by a late brother-in-law, using PVC plumbing fittings; and their geometry is not accurate, as such fittings cannot achieve that. Your tutorial pushed me forward to a more elegant design, a true pentagon. Thanks very much. Ken

    Five Point Star Assembled.JPG
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    Soosekendallwilliams

    Reply 6 months ago

    I had to look for Sebastopol... with Redwoods, that's gotta be in California.

    I enjoyed figuring out the math again; like you say, if you don't use it you get rusty. Figuring out how to explain it to someone else -- well, that's difficult. Reading this almost a year later, I probably made it overly complex to follow. Real skill is needed to make something understandable.

    Maybe I should add a step at the beginning with simple resulting equations or methods -- maybe using the calculator you found -- allowing someone to derive actual measurements. Just for clarity. We're not trying to teach math in this Instructable, imo-- we're trying to build a lighted star!

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    kendallwilliams

    6 months ago

    Attached is an image of the assembled PVC tube 5-Pointed-Star, still without lights. It is constructed of 5X 5 foot, 3/4" PVC tubing and 10X 10-24 X 2-1/2 in. bolts with nuts. It would be better to use 2-1/4 in. long bolts; but they're not easily available. So, must watch your hands and arms while handling this large structure. Next: Adding LED Lights. Thanks, again, to Soose for his geometry discussions and guidance.

    Five Point Star Assembled.JPG
    1 reply
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    Soosekendallwilliams

    Reply 6 months ago

    Kendall, do show us when you get your LED lights attached! (And don't forget, a photo of how you get that up to the top of a Redwood.)

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    Soose

    6 months ago

    If I had oodles of time, I'd like to create my own LED circuit for such a star. Have you thought of making a star out of the actual hurricane strips? (Are they sturdy enough to be the frame for a light structure?) You could put an LED thru every hole, or at intervals, directly in the hurricane strips? Maybe some other member here will get inspired and try that out. :)

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    kendallwilliams

    6 months ago on Step 4

    A large star can be built using 3/4" PVC tubing, for example 5 feet long, drilled and bolted at the ends. After initial assembly, aligning star points is difficult, as every tube moves! The star's point-to-point distance calculates, for this example, close to 36 inches. Steel "earthquake" straps, 48" long with holes every inch, solve this problem: The five bolt ends fit nicely into the strap holes 36" apart. Holes in the PVC tubing can then be drilled, and bolts provided, for the inner pentagon, securing the star's geometry. Add lights after that. Other star sizes can be determined from the excellent method described by Soose or from the following site: https://rechneronline.de/pi/pentagon.php

    Five Point Star Alignment.jpg
    1 reply
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    Soosekendallwilliams

    Reply 6 months ago

    That is one huge star! I honestly had to go back and read my own Instructable above, as it's been 10 months or more.

    Great to have another example in case I've really confused someone above...

    So if your "sides of a star" are 5 feet long or 60 inches (!) then for the height, about 95% of any side means (.9511 x 60 = 57.066 inches ), wow, 57 and about 1/16th tall!

    How BIG are your redwoods and how do you get the stars up there!? Do tell! That sounds like an Instructable all on its own!

    Our purple martin house is 18ft high, I think? And the pole has a hinge at the bottom, about waist height. Plus, the metal house itself is on a pulley and string, so we just untie and lower it to attach both the light strings of the tree and last year we just attached the lightweight star with twist ties. Too easy. I've pondered erecting a similar light-string tree on some of our trees, but getting the strings or a star up to the top just seemed too difficult without erecting scaffolding and I am not going up on a ladder against a tree, lol.

    BTW, on my cell phone, I couldn't see what the "hurricane straps" really were, and didn't recognize them. Glad I took a second look on the desktop. I think, if I were making my version of your star, I'd want to spray paint it all dark, maybe dark green.

    It's certainly a better frame than mine of 5 coat hangers. :) Built to LAST! And you can unbolt it for storage -- a great advantage!