About the Thin Lens Equation Calculator
This thin lens equation calculator solves 1/f = 1/dₒ + 1/dᵢ for whichever quantity you are missing — the image distance, the object distance or the focal length — and then tells you the magnification, the image height and whether the image is real or virtual, inverted or upright, magnified or reduced. It works for converging (convex) lenses with a positive focal length and diverging (concave) lenses with a negative one.
Use it for physics and optics homework, to check a ray diagram, to plan a simple magnifier or projector, or to understand why a camera must refocus for close subjects. The table below the result shows where this lens forms images for a range of object positions, which makes the classic “object beyond 2f, between f and 2f, inside f” cases easy to see.
Distances use the real-is-positive sign convention: object distance is positive for a real object, image distance is positive for a real image on the far side of the lens and negative for a virtual image on the same side as the object. The thin-lens model ignores lens thickness and aberrations; the same equation also works for spherical mirrors, except that a mirror’s real image forms in front of it, on the object side.
How to use the thin lens equation calculator
- 1Choose which quantity you want to find.
- 2Enter the focal length — negative for a diverging lens.
- 3Enter the object distance or image distance in centimetres.
- 4Optionally enter the object height to get the image height.
- 5Read the result, magnification and image type, then check the table for other object positions.
Formula and method
The thin lens equation links the focal length f to the distances of the object (dₒ) and image (dᵢ) from the centre of the lens. Rearranging gives dᵢ = 1 ÷ (1/f − 1/dₒ), dₒ = 1 ÷ (1/f − 1/dᵢ) and f = 1 ÷ (1/dₒ + 1/dᵢ). With the real-is-positive convention, a positive dᵢ is a real image on the far side of the lens and a negative dᵢ is a virtual image on the object side.
Linear magnification is m = −dᵢ ÷ dₒ. A negative m means the image is inverted, a positive m means upright, and |m| > 1 means it is larger than the object. The image height is m × hₒ. Lens power in dioptres is the reciprocal of the focal length in metres, so a 10 cm lens has a power of +10 D.
- f
- Focal length (+ converging, − diverging)
- dₒ
- Object distance from the lens
- dᵢ
- Image distance (+ real, − virtual)
- m
- Magnification (− inverted, + upright)
- P
- Lens power in dioptres (1/m)
Worked examples
Object at 30 cm from a 10 cm convex lens
1/dᵢ = 1/10 − 1/30 = 2/30, so dᵢ = 15 cm. The magnification is −15 ÷ 30 = −0.5: a real, inverted image half the size, so a 5 cm object gives a 2.5 cm image.
Magnifying glass: object inside the focal length
1/dᵢ = 1/10 − 1/5 = −1/10, so dᵢ = −10 cm. The negative sign means a virtual image on the same side as the object, upright and magnified 2×.
Finding the focal length from a sharp image
An object 20 cm away focuses 60 cm behind the lens. 1/f = 1/20 + 1/60 = 4/60, so f = 15 cm (+6.67 D) and the image is inverted and 3× larger.
Frequently asked questions
What is the thin lens equation?+
The thin lens equation is 1/f = 1/dₒ + 1/dᵢ, relating a lens’s focal length to the object and image distances. It assumes the lens is thin compared with those distances.
What does a negative image distance mean?+
A negative image distance means the image is virtual: it forms on the same side of the lens as the object and cannot be projected onto a screen. Magnifying glasses and diverging lenses produce virtual images.
How do I calculate magnification?+
Magnification is m = −dᵢ/dₒ, which also equals image height ÷ object height. A negative value means the image is inverted; a magnitude above 1 means it is larger than the object.
What sign should the focal length have?+
Use a positive focal length for converging (convex) lenses and concave mirrors, and a negative focal length for diverging (concave) lenses and convex mirrors.
What happens when the object is at the focal point?+
When dₒ equals f, the refracted rays leave parallel and never meet, so no image forms (it is “at infinity”). This is how a flashlight or projector collimates light.
What is lens power in dioptres?+
Lens power is 1 ÷ focal length in metres. A +2.5 D reading-glass lens has a focal length of 40 cm; eyeglass prescriptions list power in dioptres.