I Created a Perfect Star (Again)

I woke up early in the morning and sat at my computer. I was about to study Japanese when I suddenly remembered how I made a perfect star when I was in high school. At that time, I mastered elementary geometry, so I was able to easily make it with a ruler and a protractor.

With a ruler to measure length and a protractor to measure angles, it was easy to make a star.

A star is a polygon with ten sides. Technically, it is a decagon. But a star can be created by laying out only five lines from its center as its frame.

To make a perfect star, all five lines must have the same length and must have equal angles between them.

A complete rotation has 360°. To determine the angle from one line to the other, 360° is divided by the number of lines, which is 5.

360° / 5 = 72°

From the angle alone, and with a ruler and a protractor, a perfect star can be drawn like so:

Draw a line with length x from the center upward.

Then, measure 72° from the center to its left and right, and draw the lines with the same length x from the center.

From there, measure 72° again from the two new lines and draw the remaining two lines.

When the angle between the last two lines is measured, they must also be 72° apart. If not, a mistake was made in the earlier steps.

When done correctly, all lines should be 72° apart. From here, the perfect star can be made by connecting the ends of the five lines with straight lines.

I did this method almost 15 years ago, and it was easy.

This time, I thought of creating a perfect star again, but without measuring angles with a protractor or with any app.

I thought of doing this to check if my brain still works. I was happy to find out that it still does. (lol)

A star can be circumscribed by a pentagon, like so:

To make a new star from scratch without an angle-measuring device, I started with the base of the pentagon. The exact length of that line could be anything, and I assigned it to be 1 unit.

I knew that both ends of this base must connect to the center of the pentagon (and of the star), so creating that triangle was my first goal.

To do that, I needed to know the exact height of the center of the pentagon from its base. If I drew a line from the center of the pentagon perpendicular to the base, that would be the altitude l of the red triangle.

I knew that the angle between the two legs of the red triangle must be 72°. From that, the angle opposite to the altitude can be deduced. There is more than one way to move forward from here, but I proceeded with my calculations using just half of the red triangle. I used the right triangle (no pun intended).

Note that the red triangle was an isosceles triangle. Therefore, when it was halved, the result would be two identical and symmetric right triangles. The red isosceles triangle's base was halved, and the angle opposite to it was also halved (0.5 units and 36°, respectively).

From there, the angle opposite to the altitude l was calculated. Any triangle has all its three angles equal to 180°. Therefore, 180°-90°-36°=54°. (For those who do not know yet, and for those who hated math so much that they intentionally unlearned it, the angle opposite to the hypotenuse c is a right angle, and right angles are 90°.)

Now that I had the angle opposite to the altitude l, I could calculate its length using the Law of Sines.

The solution would be:

l = 0.68819 units

After I had obtained the length of the altitude l, I now knew exactly where the center of the pentagon was located, which was 0.68819 units above the midpoint of the pentagon's base.

Next, I could now draw the legs of the triangle with complete certainty that their lengths would be correct.

It was not enough to draw the legs of the triangle and to know that the length was correct. It was also important to identify its exact length because these two lines were already the length from the center of the star to the vertices of its two legs.

Therefore, whatever its length was, that would also be the length of the other three lines from the center of the pentagon.

There are two ways to identify the exact length of the legs: measure it with a ruler and calculate it. I did both to verify that they would be the same.

To calculate it, I could use the sine law again, but I used the Pythagorean theorem.

The solution would be:

c = 0.85065 units

I got 0.85 from my physical measurement, indicating a (very) high chance that my calculation and drawing were correct.

So far, I have had this red triangle:

(Remember that the gray lines were just for imagination and guidance, and they didn't yet exist at this point.)

The easiest line to draw next from here was the vertical line from the center of the star to the vertex of the star's head. The reason was that I knew its exact length—the same length as the triangle's legs—0.85065 units, and the angle from the horizon, which was just 90° because it was perpendicular. Even still, I didn't need to measure 90° to draw it. I just used a rectangular object as a rough guide for 90°.

If I could just draw the other two lines from the center of the star, the job would be almost done. But it was not possible because even though I knew their length, I did not know their angle from the horizon. And even if I calculated the angle from the horizon, I still could not draw it because the point of this method was to draw the perfect star without measuring the angles.

The only possible move from here was to draw the horizontal line that was across the vertical line.

To do it, I needed to know the distance from its midpoint to the center of the pentagon. It would be y.

To identify the length y, I needed to summon all relevant imaginary lines and create imaginary triangles.

With these right triangles, I could use the sine law again to find the length of y.

In this right triangle, I knew that the length of the hypotenuse was 0.85065 and that the angle opposite to it was 90°. I also knew that the angle between the hypotenuse and the line y was 72°. To calculate the length of y using the sine law, I needed to know the angle opposite to it, which was 180°-90°-72°=18°.

Calculating using the sine law, length y (a) was 0.26287 units

y = 0.26287 units

Now that I had the length y, I could identify where to start drawing the horizontal line z from.

To calculate the length of the line z, I used the Pythagorean theorem.

z = 0.80901 units

I could now draw the line z from the vertical line to the right and know where to stop. I did the same on the left side.

Now that I had drawn the horizontal line, I could draw the other two lines of the star from each end of the horizontal line to the center of the pentagon. Their lengths must be 0.85065 units as well, and I was able to verify by getting a physical measurement of 0.85 units.