How to use it
- Choose the Lens, converging or diverging.
- Type the Focal length, cm, the Object distance, cm and the Object height, cm, for example 10, 50 and 3.
- Read the Image distance, cm, the Magnification, the image height and whether the image is real or virtual, then the steps.
The thin-lens equation
For a lens that is thin compared with the distances involved, the object distance d_o, the image distance d_i and the focal length f are related by the thin-lens equation. The magnification is the ratio of the image height to the object height (OpenStax, University Physics Volume 3, Section 2.4).
1 ÷ d_o + 1 ÷ d_i = 1 ÷ f
d_i = f × d_o ÷ (d_o − f)
m = h_i ÷ h_o = −d_i ÷ d_o
f = focal length: positive for a converging lens, negative for a diverging lens
d_o = object distance, positive
d_i = image distance: positive for a real image, negative for a virtual image
m = magnification: positive upright, negative inverted- A diverging lens has a negative focal length, so for every object distance the image is virtual and upright. It is also smaller than the object, because the magnification is less than one.
- An object at the focal point gives no image at a finite distance: the image is at infinity.
- The source gives the same sign rules for the image distance, the focal length and the radius of the surfaces.
Three cases of a converging lens
OpenStax, Example 2.4, takes a converging lens with a focal length of +10.0 cm and an object 3.0 cm high. The table gives the three cases of the example, which the page reproduces.
| Object distance, cm | Image distance, cm | Magnification | Image |
|---|---|---|---|
| 50.0 | 12.5 | -0.250 | Real, inverted, smaller |
| 5.00 | -10.0 | +2.00 | Virtual, upright, larger |
| 20.0 | 20.0 | -1.00 | Real, inverted, same size |
What the calculator does not include
- Thick lenses, and systems of several lenses, which need the lens-maker's data or the combination of the lenses.
- Aberrations, the blur at the edge of a lens and the colours of the light.
- The eye and other optical instruments. The page models one thin lens in air.
Worked example: a converging lens, f = 10 cm, object 50 cm away
With 1 ÷ 50 + 1 ÷ d_i = 1 ÷ 10, the image distance is d_i = 10 × 50 ÷ (50 − 10) = 12.5 cm. The magnification is m = −12.5 ÷ 50 = −0.250, so the image is 0.75 cm high, real, and inverted.
| Figure | Value |
|---|---|
| Image distance | 12.50 cm |
| Magnification | -0.250 |
| Image height | 0.75 cm |
| Image | Real, inverted |
Frequently asked questions
Why is the image distance negative for some images?
The source's sign convention makes d_i negative when the image is on the same side of the lens as the object. That is the virtual, upright and enlarged image of case (b) in its Example 2.4.
What does a negative focal length mean?
The lens is diverging. The source takes f as negative for a diverging lens, and then the equation gives a virtual, upright and smaller image for any object distance.
Why is there no image when the object is at the focal point?
The rays leave the lens parallel, so they never meet at a finite distance. The image is at infinity, which the calculator reports as an error.
Does the calculator allow for the thickness of the lens?
No. It uses the thin-lens equation, which assumes that the lens is thin compared with the distances in the problem, as the source does throughout the section.
Is my data sent to a server?
No. The calculation runs in your browser, and nothing you type is sent to us or stored.
Sources
Figures and text last checked on Wednesday, November 11, 2026.