Lens Maker’s Formula connects the focal length of a thin lens to the refractive index of the material and to the curvatures of its two refracting surfaces.
For a thin lens in air, the formula is
\[
\frac{1}{f}=(n-1)\left(\frac{1}{R_1}-\frac{1}{R_2}\right).
\]
Here \(f\) is the focal length, \(n\) is the refractive index of the lens material relative to air, and \(R_1\) and \(R_2\) are the signed radii of curvature of the first and second surfaces.
The formula is one of the most useful bridges between geometry and optics because it shows exactly how surface shape changes the focusing power of a lens.
The most important source of confusion is the sign convention. In this calculator, a radius is taken as positive when the center of curvature lies to the right of the surface vertex,
and negative when the center lies to the left. This is a common convention in introductory optics. A plano surface has infinite radius, so its reciprocal is zero:
\[
\frac{1}{\infty}=0.
\]
That makes plano-convex and plano-concave lenses especially easy to analyze. For example, if a glass lens has \(n=1.5\), \(R_1=20\ \text{cm}\), and \(R_2=\infty\), then
\[
\frac{1}{f}=(1.5-1)\left(\frac{1}{20}-0\right).
\]
Since \(1.5-1=0.5\), we obtain
\[
\frac{1}{f}=0.5\cdot \frac{1}{20}=\frac{1}{40},
\]
so
\[
f=40\ \text{cm}.
\]
Because the focal length is positive, the lens acts as a converging lens. This matches the intuition that a plano-convex glass lens can focus parallel light.
The formula also explains why changing either curvature can strengthen or weaken the focusing power. If the two reciprocal terms combine to give a large positive value,
then \(1/f\) is positive and the lens is converging. If the difference becomes negative, then \(f\) is negative and the lens acts as a diverging lens.
A symmetric bi-convex lens often produces a positive focal length, while a bi-concave lens often produces a negative one, assuming the refractive index is greater than 1.
Notice that Lens Maker’s Formula gives the focal length of the lens itself. It does not directly tell you where an image forms for a given object.
Once you know \(f\), you would then use the thin lens equation
\[
\frac{1}{f}=\frac{1}{d_o}+\frac{1}{d_i}
\]
to compute the image position. In that sense, Lens Maker’s Formula is a design equation, while the thin lens equation is an imaging equation.
The animated diagram in this calculator focuses on the geometry of the two surfaces. Each surface is drawn from the entered signed radius, and optional center markers show where the corresponding circle center lies.
This makes the signs in the formula much easier to understand. If the center of the first surface is on the right, \(R_1\) is positive. If it is on the left, \(R_1\) is negative.
The same idea applies to \(R_2\). For a plano surface, the calculator treats the center as “at infinity,” so no finite center point is drawn and the reciprocal term becomes zero.
In real optical engineering, the full treatment can become more complex. The surrounding medium may not be air, the lens may not be thin, and the refractive index may vary with wavelength.
That leads to more advanced versions of the formula involving the external medium, thick-lens corrections, principal planes, and chromatic aberration.
Still, the thin-lens-in-air form remains the best starting point for understanding how lens curvature controls focal length.
This is why Lens Maker’s Formula appears naturally in topics such as eyeglass design, magnifiers, camera lenses, and basic instrument optics.
It lets you see that focal length is not a mysterious property assigned to a lens after the fact; it comes directly from the material index and the shapes of the two refracting surfaces.
