Lens sign convention and common lens shapes
How the radius sign rule works and how biconvex, plano-convex and meniscus shapes change focal length in the lens maker's equation.
The equation and where power comes from
The thin lens maker's equation is 1/f = (n - 1)(1/R1 - 1/R2). Two things set the focusing power: how much the glass bends light, captured by n minus 1, and how sharply the two surfaces curve, captured by the difference of their curvatures 1/R1 and 1/R2. A higher index or tighter radii both raise the power. The result, 1/f, is the optical power in diopters, and its reciprocal is the focal length in meters.
Making sense of the sign rule
This calculator treats a surface that bulges toward the incoming light as having a positive radius, and one that curves away as negative. For a biconvex lens the front surface bulges toward the light so R1 is positive, while the back surface curves away so R2 is negative. Subtracting a negative R2 makes the two curvatures add, which is why a symmetric biconvex lens of n = 1.5 with radii of 0.1 m and -0.1 m gives 1/f = 0.5 times (10 minus -10) = 10, a focal length of 0.1 m and a power of 10 D.
Common shapes and their behavior
A plano-convex lens has one curved side and one flat side, so one 1/R term drops to zero. With n = 1.5 and R1 = 0.2 m against a flat back, 1/f = 0.5 times (5 minus 0) = 2.5, giving a 0.4 m focal length and 2.5 D. A biconcave lens has both surfaces curving inward, producing a negative focal length and a diverging beam used to spread light. A meniscus lens has both surfaces curving the same way, one convex and one concave, which lets designers fine tune power while controlling aberrations.
Limits of the thin lens model
The equation assumes the lens is thin enough that its center thickness can be ignored and that rays stay close to the axis, the paraxial approximation. Real lenses have thickness, so a full design adds a term involving the thickness divided by n times R1 times R2, and wide beams introduce spherical aberration that a single formula cannot capture. The refractive index also varies with wavelength, so blue and red light focus at slightly different points, an effect called chromatic aberration. For quick estimates and teaching, though, 1/f = (n - 1)(1/R1 - 1/R2) is accurate and fast.