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## 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