How do I calculate the position of 12 squares aligned around the edges of a do-decagon?

Answer 1

To calculate the position of the 12 squares aligned around the edges of a perfect dodecagon, you can use the following steps:

1. Calculate the radius of the dodecagon:

   • The edges of the dodecagon are 576 units long, and the dodecagon is a regular polygon, so the radius can be calculated as:

   • Radius = (Edge Length) / (2 × tan(π/12)) = 576 / (2 × tan(π/12)) ≈ 500.00 units

2. Calculate the position of the centers of the squares:

   • The squares are 576 × 576 units, so their centers will be 288 units away from the edge of the dodecagon.

   • The centers of the squares will be evenly spaced around the dodecagon, with an angular separation of 30 degrees (360 degrees / 12 squares).

   • The coordinates of the centers of the squares can be calculated using the following formulas:

     • x = (Radius + 288) × cos(θ)

     • y = (Radius + 288) × sin(θ)

   • Where θ is the angle of the square, starting from the positive x-axis and increasing counterclockwise.

Applying these formulas, the coordinates of the centers of the 12 squares are:

1. (681.40, 1801.21)

2. (1365.45, 1117.77)

3. (1365.45, -1117.77)

4. (681.40, -1801.21)

5. (-681.40, -1801.21)

6. (-1365.45, -1117.77)

7. (-1365.45, 1117.77)

8. (-681.40, 1801.21)

9. (0.00, 2000.00)

10. (1000.00, 1732.05)

11. (1000.00, -1732.05)

12. (0.00, -2000.00)

Okay, let's solve this step-by-step:

1. We have a regular dodecagon with edge length of 576 units.

2. The dodecagon is centered at (0, 0).

3. You want to place 12 squares of size 576 x 576 units around the edges of the dodecagon.

4. The squares should be rotated 30 degrees more than the previous one.

To calculate the position of the centers of the 12 squares:

1. The angle between each square is 360/12 = 30 degrees.

2. The radius of the dodecagon is 576 / (2 * tan(π/12)) = 500 units.

3. The center of the first square is at:

   • x = 500 * cos(0) = 500

   • y = 500 * sin(0) = 0

4. The center of the second square is at:

   • x = 500 * cos(30 * π/180) = 433.01

   • y = 500 * sin(30 * π/180) = 250.00

5. Continuing this pattern, the coordinates of the 12 square centers are:

| Square | X | Y |

| --- | --- | --- |

| 1 | 500.00 | 0.00 |

| 2 | 433.01 | 250.00 |

| 3 | 250.00 | 433.01 |

| 4 | 0.00 | 500.00 |

| 5 | -250.00 | 433.01 |

| 6 | -433.01 | 250.00 |

| 7 | -500.00 | 0.00 |

| 8 | -433.01 | -250.00 |

| 9 | -250.00 | -433.01 |

| 10 | 0.00 | -500.00 |

| 11 | 250.00 | -433.01 |

| 12 | 433.01 | -355.o1

The coordinates are rounded to two decimal places, as requested.

Answer 2

All answers are positive, negative and reverse (X,Y) of the following 2 sets

(681.4, 1180.22)

(0, 1362.8)