The power of a lens in dioptres is 1 divided by its focal length in metres, and thin lenses placed together add their powers.
A lens of focal length 0.5 m has a power of 2 dioptres (D); +4 D brings parallel light to a focus at 25 cm; a diverging lens has a negative power. Because 1/f is what a thin lens subtracts from a ray’s slope, two lenses in contact subtract the sum: +2 D and +3 D together are +5 D, a focal length of 20 cm. Spectacle prescriptions are written in dioptres for this reason, and a short-sighted eye whose far point is 50 cm needs −2 D.
The addition holds for thin lenses in contact. If they are separated by a distance d, the power of the pair is P1 + P2 − d P1 P2 (with d in metres), less than the sum. For a thick lens the focal length is measured from principal planes inside the glass, not from the surface, and a spectacle lens’s power depends on how far it sits from the eye (the vertex distance), which matters above about 4 D.
Estimate with the rule, then check it against the calculator that models it properly.
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