Thin Lens Equation Calculator

Thin Lens Equation

The thin lens equation links object distance, image distance, and focal length for paraxial rays passing through a lens thin enough that refraction happens at one plane. Photographers, microscope designers, and physics students use it to predict where an image forms and how large it appears.

Lens and Magnification Formulas

$$\frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i}$$ $$M = -\frac{d_i}{d_o}$$

Real objects use positive $d_o$. Real images have positive $d_i$; virtual images have negative $d_i$. Magnification magnitude tells image size relative to the object; the minus sign indicates inversion.

Example

A 45 mm lens with an object 180 mm away gives $1/d_i = 1/45 - 1/180 = 0.01667\,\text{mm}^{-1}$, so $d_i = 60\,\text{mm}$. Magnification $M = -60/180 = -0.33$: a smaller inverted real image.

Frequently Asked Questions

What sign convention does this tool use?

Object distances are positive to the left of the lens. Real images are positive on the opposite side; virtual images are negative. Magnification follows M = −di/do.

When is the image virtual?

When the calculated image distance is negative. That happens for objects inside the focal length of a converging lens.

Can focal length be negative?

Yes for diverging lenses. Enter a negative focal length and the same equations apply.

What if the object sits at the focal point?

1/di goes to zero and the image moves to infinity. The calculator reports that condition when it occurs.

Does lens thickness matter?

Only for thick lenses. This thin-lens model ignores thickness and assumes paraxial rays.

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