It can sometimes be useful to estimate the Snellen or logMAR acuity of a patient based on the size of the font they read at near. While there are tables available showing the conversions, this can not cover all font sizes and reading distances. With some basic trigonometry, you can do the calculations yourself.
First, let’s draw out the problem:
Here, d is the reading distance and h is the height of the letter read by the patient. If the chart uses the N scale, the letter size is measure in points where N1 = 1 point = 1/72 inches.
Let’s assume the patient reads N5 (h) at a distance of 40cm (d). We need d and h to be in the same units, so we convert N5 to metric like so:
N5 = 5/72 inches = 0.0694 inches
1 inch = 2.54cm
Therefore N5 = 0.0694 x 2.54 = 0.176cm
From here we have a small adjustment to make; font sizes are measured as the height from the lowest tail to the highest stem or accent of any letter in that font. This is known as the em height – This wikipedia article explains this concept in detail. The character | is a rough estimation of em height, but most letters are much smaller than that – around the same hight as the letter x. This x-height is what we want to use in our calculation. While this differs between fonts, it is usually around half of the em height, so we multiply our value by 0.5:
h = 0.176 x 0.5 = 0.088cm.
So we have d = 40cm and h = 0.088cm. From here we can calculate Θ:
Θ = atan(h/d) = atan(0.088/40) = 0.126° = 7.6 minutes of arc
Finally, we need to convert Θ to our Snellen fraction. We know than 6/6 (or 20/20) corresponds to an optotype subtending 5 minutes of arc. We can therefore calculate MAR (minimum angle of resolution) as:
MAR = Θ/5 = 7.6/5 = 1.5
and from here, the logMAR and Snellen equivalents:
logMAR = log(1.5) = 0.18
Snellen = 6/(6xMAR) = 6/(6×1.5) = 6/9
Therefore, N5 is approximately equivalent to 6/9 at a distance of 40cm.