Lens Calculator - Thin Lens Equation 1/f = 1/do + 1/di

How to use this tool

1. Select which value to solve for (focal length, object distance, or image distance).
2. Enter the two known distances in metres.
3. View the result, magnification, and image properties.

Thin Lens Equation

1/f = 1/d_o + 1/d_i - relates the focal length of a thin lens to the object and image distances. Converging (convex) lenses have a positive focal length; diverging (concave) lenses have a negative focal length.

Sign convention

The standard sign convention for thin lenses (real-is-positive or Cartesian):

• d_o > 0: object is on the incoming-light side of the lens (real object - the usual case).
• d_i > 0: image forms on the outgoing-light side - a real image where light rays actually converge (can be projected onto a screen).
• d_i < 0: image appears on the same side as the object - a virtual image where rays only appear to diverge from a point (cannot be projected). A magnifying glass always produces a virtual image.

Magnification formula

m = −d_i / d_o

• m < 0 (negative): image is inverted - typical of real images formed by converging lenses (cameras, projectors).
• m > 0 (positive): image is upright - typical of virtual images (magnifying glass, eyepiece lenses).
• |m| > 1: image is enlarged; |m| < 1: image is reduced.

Common applications

Lensmaker's equation

For a lens made of material with refractive index n and radii of curvature R₁ and R₂:

1/f = (n − 1)(1/R₁ − 1/R₂)

R₁ is the radius of the surface facing the incoming light; R₂ is the radius of the exit surface. By convention, a surface curving toward the incoming light has a positive radius. This equation explains why a biconvex lens (R₁ > 0, R₂ < 0) always has a positive focal length.