# How to Cover Up Polygon Circles

When I gave a presentation on building round shapes a year ago, someone asked me how to cover up the top of the polygon circles. I couldn’t give a good answer at the time. So I’m going to correct that today. First of all, using trigonometry we can deduce that the distance between one end of the polygon circle to the other is not going to be a clean number in terms of studs. The best we can do is to approximate. But how can we do this. The technique used in set 7041, Troll Battle Wheel (shown above) and set 4183, The Mill gave us a hint. Let’s take a look at the geometry of the wheel structure.

In the original decahexagon, the total length from one vertex (at the center of the pivoting clip) to the other can be calculated with some simple trigonometry. Each side is 3 studs in length. a=3sin(22.5) b=3sin(45) c=3sin(67.5)  To fit the same amount of bricks in this length, we need 8 bricks + 4 studs + (1 brick – 4mm). 4mm is the distance from the top of the brick to the vertex. If we calculate the difference, {3[sin(22.5) + sin(45) + sin(67.5)]*2*8mm + 3*8mm} – [(9.6-4mm)*2 + 4*2*9.6mm + 4*8mm] = 0.6561mm.

0.6561mm is a very small difference already. To relieve any stress caused by even that difference, the trick is to connect the center piece with technic bar in a technic hole. See below.

## Top Studs

Using the technic above, we got concentric studs. But we still don’t have concentric studs that sits flat with the rim of the cylinder. To achieve that, we need to replace the connection to the 4×4 round brick with technic holes with something else.

The structures above has the same length, but the axles sit at different heights. Once we have concentric studs that sits flat with the rim, all there’s left is to close the cylinder with a circle. You can use any of the techniques that we talked about previously to build the circle. See example below.

## Other Sizes

Using the same concept, we can come up with circles of other sizes. You can try different combination of bricks, studs, brackets and half studs (technic brushes) to minimize the difference. I’ll skip all of the math derivations and jump straight to the results.

3 studs per side hexagon, length error = 0.0548mm

4 studs per side decahexagon, length error = 0.4749mm

5 studs per side decahexagon, length error = 0.6936mm

3 studs per side hexagon, length error = 0.3411mm

This last one is a bit different. Since it’s so small, you can’t fit a technic brick structure inside. The only way to do it is with a 1×1 brick with studs on 4 sides. So there is some stress in this structure.