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3. Everything / Maths, Science & Technology / Mathematics

How to Draw a Pentagram

A schoolboy is caught in the act of spraying a pentagram onto a school wall. But at least he used a compass and protractor to do it.

Have you ever tried to draw a pentagram? You know, the five-pointed star you see Christopher Lee drawing on the floor in old Hammer Horror films to summon up the devil? Or perhaps the more gentle one used by Wiccans for invoking the five elements in their rites? (Same pentagram: different use.) Ignoring magic all together, you might just like to draw a five-pointed star because it is a pleasing pattern. You've probably never drawn an accurate pentagram, with all sides equal and all angles equal. It's not as easy as you may think!

So, here are a few methods for accomplishing this task.

Basic Equipment

Other Equipment

  • Protractor (for Angles Method)

  • Calculator (for Measurement Method)

  • Compass1 of the circle-drawing variety (for other methods)

The Angles Method

For this method you'll need a special tool - a protractor. All you need to do is draw a line of the desired length, then use the protractor to measure 36 degrees from its tip and draw another line of the same length. Repeat with the rest of the tips until you close the pentagram.

This method is nice and simple, but it's not very accurate. If you are a tiny bit out with the 36° angles, the whole thing will look rather crooked by the time you get to the last line. Who walks around with a protractor, anyway? (or even knows what this word means?)

The Measurement Method

The following method is based on a specific ratio that exists between the sides of the regular pentagon that form the centre of the pentagram and the sides of the triangles that make its points. This ratio is (1+√5)/2 to 1, which is approximately 1.618 to 1. This ratio, often called Φ or the golden ratio, is referred to here as P. You're probably going to need a calculator to do some of the calculations.

Method

  1. Draw a horizontal line of desired length (X will be used here for demonstration) and mark its ends. This line will be one side of the little pentagon in the middle of the pentagram, so it should be quite small.

  2. Extend this line at each end by a distance of P*X (no need to be extremely accurate), so if you had a line of length 1cm, you now have a line of length 4.236cm (2*P*1 + 1).

  3. Mark the centre of the first line and draw a weak guideline upwards at right angles to it. This guideline should be nearly at length P*X.

  4. Draw lines from the marks you made at step 1 to the guideline. These lines should be at length P*X and should meet at the same point. You can now erase the guideline.

  5. Extend each of the lines you made in step 4 downwards for a distance of X + P*X from the point where they meet the first line.

  6. Connect the end of each downward-pointing line to the opposite end of the horizontal line. There you have it! A pentagram!

The Improve-Your-Guess Method

This is probably the best method for general use. It doesn't take too long, is easy to do and is easy to remember, too! It can also be used to draw stars with other numbers of points. You'll need a compass.

Method

We're going to draw a circle and mark off five points on it which are evenly spaced around the circle. These points can then be joined to make the pentagram.

  1. Draw a circle and mark the first point at the top of the circle.

  2. Look at the length of the circle and guess what a fifth of that would be. Set your compass to this length.

  3. With the first point as centre, draw a short arc which cuts the circle. Where the arc and circle intersect is the second point.

  4. With the second point as centre, draw another arc to cut the circle at the third point. Proceed around the circle, marking the fourth and fifth point. With the fifth point as centre, draw the fifth arc. You should be roughly back to your original drawn point.

  5. Now, if you overshot the original point, your compass is too wide. If you undershot, it is too narrow. Estimate one fifth of the distance between the first point and where you ended up. Adjust the width of the compass in or out by this distance.

  6. Repeat the process with the new compass width. This time you'll be much closer to the mark.

  7. If necessary, adjust the width of the compass again and repeat.

  8. At this stage, your final point should be within a millimetre of the original point, which is close enough. Join the five points you've marked around the circle and you've got a pentagram.

The Method of Many Circles

The following method is the most accurate, and doesn't involve any guessing, but it is quite tricky to do, and the reason why it works is far beyond the scope of this Entry. Again, you will need a compass.

Method

  1. Draw a circle with two lines at 90 degrees to each other crossing in the centre. We'll use clock face directions here. 12 means towards the top, 3 means to the right, 6 means to the bottom, and, guess what, 9 means to the left. The lines you've drawn are from 12 to 6 and from 9 to 3.

  2. Divide the line from centre to 3 in half. This can be easily done by setting the compass at over half the distance and then drawing an arc from both the centre and 3 points. A line between the two crossings of the arcs, above and below the centre-to-3 line, will divide it in half.

  3. Place the point of the compass on the half way point of the centre-to-3 line and set the radius to reach the 12 point. Draw an arc through the 9-to-centre line.

  4. Draw one more arc with the centre at 12 and the radius set at the point were the last arc intersects the 9-to-centre line. Draw this arc through the initial circle. This radius is 1/5 of the circle.

  5. Use the above intersections as centre to find the other two points on the initial circle (the fifth is at 12).

  6. Connect the five points, each one with a point not neighbouring it, and there you're done!

Historical Note

An Ancient Greek philosopher by the name of Pythagoras believed that mathematics was at the root of all philosophy. He formed a band of followers who were sworn to secrecy and who investigated the properties of geometry. Legend has it that they adopted a pentagram inside a standard pentagon as their symbol. If you knew enough geometry to draw the symbol accurately, then you were in. Of course, only the method of many circles would have been good enough for such perfectionists.


1 Often known as a 'pair of compasses'.

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Simple method using a compass
(Last Posting: Aug 14, 2006)




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Entry Data
Entry ID: A13083202 (Edited)

Written and Researched by:
U151503 - Gnomon - Future Guide Editor - towels at the ready
Fire Bat
U235886

Edited by:
U205101


Date: 14   August   2006


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Referenced Guide Entries
The Five Elements in Relation to the Wiccan Religion
Trigonometry
Pencils
Rulers
How to Be a Philosopher
Hammer Horror
Quill Pens
The Ballpoint Pen
Phi


Related BBC Pages
BBC GCSE Bitesize Maths


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