Reference Materials
Tables, conversions, and the manual.
Shadow Square Calculations
Using the shadow square makes it possible to convert a measured angular size to 1/tan(α), according to the formula:
distance = known size × [shadow square length]/reading on the shadow square
where [shadow square length]/reading = 1/tan(α)
Calculations for a 24 × 24 shadow square
Note that Umbra Versa = distance and Umbra Recta = height.
| Distance (Umbra Versa) | Height (Umbra Recta) | Angle, Degrees | 1/tan(α) |
|---|---|---|---|
| 24 | 1 | 2.39 | 24.000 |
| 24 | 2 | 4.76 | 12.000 |
| 24 | 3 | 7.13 | 8.000 |
| 24 | 4 | 9.46 | 6.000 |
| 24 | 5 | 11.77 | 4.800 |
| 24 | 6 | 14.04 | 4.000 |
| 24 | 7 | 16.26 | 3.429 |
| 24 | 8 | 18.43 | 3.000 |
| 24 | 9 | 20.56 | 2.667 |
| 24 | 10 | 22.62 | 2.400 |
| 24 | 11 | 24.62 | 2.182 |
| 24 | 12 | 26.57 | 2.000 |
| 24 | 13 | 28.44 | 1.846 |
| 24 | 14 | 30.26 | 1.714 |
| 24 | 15 | 32.01 | 1.600 |
| 24 | 16 | 33.69 | 1.500 |
| 24 | 17 | 35.31 | 1.412 |
| 24 | 18 | 36.87 | 1.333 |
| 24 | 19 | 38.37 | 1.263 |
| 24 | 20 | 39.81 | 1.200 |
| 24 | 21 | 41.19 | 1.143 |
| 24 | 22 | 42.51 | 1.091 |
| 24 | 23 | 43.78 | 1.043 |
| 24 | 24 | 45.00 | 1.000 |
| 23 | 24 | 46.22 | 0.958 |
| 22 | 24 | 47.49 | 0.917 |
| 21 | 24 | 48.81 | 0.875 |
| 20 | 24 | 50.19 | 0.833 |
| 19 | 24 | 51.63 | 0.792 |
| 18 | 24 | 53.13 | 0.750 |
| 17 | 24 | 54.69 | 0.708 |
| 16 | 24 | 56.31 | 0.667 |
| 15 | 24 | 57.99 | 0.625 |
| 14 | 24 | 59.74 | 0.583 |
| 13 | 24 | 61.56 | 0.542 |
| 12 | 24 | 63.43 | 0.500 |
| 11 | 24 | 65.38 | 0.458 |
| 10 | 24 | 67.38 | 0.417 |
| 9 | 24 | 69.44 | 0.375 |
| 8 | 24 | 71.57 | 0.333 |
| 7 | 24 | 73.74 | 0.292 |
| 6 | 24 | 75.96 | 0.250 |
| 5 | 24 | 78.23 | 0.208 |
| 4 | 24 | 80.54 | 0.167 |
| 3 | 24 | 82.87 | 0.125 |
| 2 | 24 | 85.24 | 0.083 |
| 1 | 24 | 87.61 | 0.042 |
Calculations for a 20 × 20 shadow square
Note that Umbra Versa = distance and Umbra Recta = height.
| Distance (Umbra Versa) | Height (Umbra Recta) | Angle, Degrees | 1/tan(α) |
|---|---|---|---|
| 20 | 1 | 2.86 | 20.000 |
| 20 | 2 | 5.71 | 10.000 |
| 20 | 3 | 8.53 | 6.667 |
| 20 | 4 | 11.31 | 5.000 |
| 20 | 5 | 14.04 | 4.000 |
| 20 | 6 | 16.70 | 3.333 |
| 20 | 7 | 19.29 | 2.857 |
| 20 | 8 | 21.80 | 2.500 |
| 20 | 9 | 24.23 | 2.222 |
| 20 | 10 | 26.57 | 2.000 |
| 20 | 11 | 28.81 | 1.818 |
| 20 | 12 | 30.96 | 1.667 |
| 20 | 13 | 33.02 | 1.538 |
| 20 | 14 | 34.99 | 1.429 |
| 20 | 15 | 36.87 | 1.333 |
| 20 | 16 | 38.66 | 1.250 |
| 20 | 17 | 40.36 | 1.176 |
| 20 | 18 | 41.99 | 1.111 |
| 20 | 19 | 43.53 | 1.053 |
| 20 | 20 | 45.00 | 1.000 |
| 19 | 20 | 46.47 | 0.950 |
| 18 | 20 | 48.01 | 0.900 |
| 17 | 20 | 49.64 | 0.850 |
| 16 | 20 | 51.34 | 0.800 |
| 15 | 20 | 53.13 | 0.750 |
| 14 | 20 | 55.01 | 0.700 |
| 13 | 20 | 56.98 | 0.650 |
| 12 | 20 | 59.04 | 0.600 |
| 11 | 20 | 61.19 | 0.550 |
| 10 | 20 | 63.43 | 0.500 |
| 9 | 20 | 65.77 | 0.450 |
| 8 | 20 | 68.20 | 0.400 |
| 7 | 20 | 70.71 | 0.350 |
| 6 | 20 | 73.30 | 0.300 |
| 5 | 20 | 75.96 | 0.250 |
| 4 | 20 | 78.69 | 0.200 |
| 3 | 20 | 81.47 | 0.150 |
| 2 | 20 | 84.29 | 0.100 |
| 1 | 20 | 87.14 | 0.050 |