What is the thin lens equation?

The thin lens equation is 1/f = 1/d₀ + 1/dᵢ, where f is focal length, d₀ is object distance (object to lens), and dᵢ is image distance (lens to image). It holds when the lens thickness is negligible compared to the distances.

What units should I use?

Any consistent unit works — millimeters, centimeters, or meters. All three values must be in the same unit. The result will be in that same unit.

What does a negative focal length mean?

A negative focal length indicates a diverging (concave) lens. Diverging lenses spread light rays apart and always produce virtual, upright, diminished images regardless of where the object is placed.

What does a negative image distance mean?

A negative dᵢ means the image is virtual — it forms on the same side as the incoming light (the same side as the object). Virtual images cannot be projected on a screen but can be seen by looking through the lens.

How is magnification calculated?

Linear magnification is m = −dᵢ/d₀. A positive value means the image is upright; negative means inverted. |m| > 1 is magnified; |m| < 1 is diminished.

What happens when the object is at the focal point?

When d₀ = f, the equation gives 1/dᵢ = 0, meaning the image forms at infinity. In practice, refracted rays emerge parallel and never converge to a real image.

Focal Length Calculator

Name: Focal Length Calculator
Availability: InStock
Author: Nham Vu

Enter any two of focal length, object distance, or image distance to solve the thin lens equation (1/f = 1/d₀ + 1/dᵢ) and find the unknown.

Lens Parameters

Use the same unit for all values (mm, cm, or m).

Solve for

Focal Length (f)

Positive = converging; negative = diverging.

Object Distance (d_o)

Distance from object to lens center.

Image Distance (d_i)

Positive = real image; negative = virtual image.

Select what to solve for, enter the two known values, then click Calculate.

Summary

Enter any two of focal length, object distance, or image distance to solve the thin lens equation (1/f = 1/d₀ + 1/dᵢ) and find the unknown.

How it works

Choose which variable to solve for: focal length (f), object distance (d₀), or image distance (dᵢ).
Enter the two known values in the input fields (any consistent unit — mm, cm, m).
Click Calculate. The tool applies 1/f = 1/d₀ + 1/dᵢ and returns the unknown.
Review the magnification (m = −dᵢ/d₀) and the image type (real vs. virtual, upright vs. inverted).
Adjust any input and recalculate to explore how changing one variable affects the others.

Use cases

Determine the focal length of an unknown lens by measuring where an image forms.
Find the required object distance to place a specimen for a desired image position.
Calculate image distance for a projector or camera with a known focal length.
Verify optics homework or lab results quickly.
Explore how converging and diverging lenses behave under different setups.
Cross-check magnification for microscopes, telescopes, or camera macro setups.

Frequently Asked Questions

Last updated: 2026-07-23 · Reviewed by Nham Vu