Thin Lens Equation Calculator
The thin lens equation calculator helps you analyze the optical properties of a simple lens. Enter the object distance and focal length to instantly find the image distance and magnification — with full support for metric (mm, cm, m) and imperial (in, ft) units.
Thin Lens Equation
When you place an object near a lens, its image forms somewhere on the other side (or on the same side for virtual images). The position, orientation, and size of this image depend on two things: the focal length of the lens and the object distance. The thin lens equation relates all three:
1/x + 1/y = 1/f
Where:
- x — Distance between the object and the center of the lens (object distance)
- y — Distance between the image and the center of the lens (image distance)
- f — Focal length of the lens
Solving for the image distance gives: y = f · x / (x − f)
Types of Lenses
There are two basic types of lenses:
- Converging lens (f > 0) — thicker in the middle; causes parallel light rays to converge toward a focal point.
- Diverging lens (f < 0) — thinner in the middle; causes parallel light rays to spread out as if coming from a virtual focal point.
When the image distance is positive (y > 0), the image appears on the opposite side of the lens — this is a real image. When y < 0, the image appears on the same side as the object — a virtual image.
Magnification Lens Equation
The magnification M tells you how much larger or smaller the image is compared with the original object:
M = |y| / x
Magnification is always a positive number (we take the absolute value of y). When M < 1 the image is diminished; when M = 1 it is the same size; when M > 1 the image is magnified.
Images in the Converging Lens
For a converging lens (f > 0), five distinct cases arise depending on where the object is placed relative to the focal point:
| Object position | Image type | Magnification |
|---|---|---|
| x > 2f | Real (y > 0) | Diminished (M < 1) |
| x = 2f | Real (y > 0) | Same size (M = 1) |
| f < x < 2f | Real (y > 0) | Magnified (M > 1) |
| x = f | No image formed | y → ∞ |
| x < f | Virtual (y < 0) | Magnified (M > 1) |
For the diverging lens (f < 0) the image is always virtual, upright, and diminished regardless of where the object is placed.
Unit Systems
The thin lens equation is purely geometric — it works in any consistent length unit. This calculator supports:
- Metric: millimeters (mm), centimeters (cm), meters (m)
- Imperial / US: inches (in), feet (ft)
Simply select your preferred unit system and unit before entering values.
FAQs
What is the thin lens approximation?
The thin lens equation assumes the lens thickness is negligible compared with the focal length. For most everyday lenses (eyeglasses, camera lenses, magnifying glasses) this is an excellent approximation.
Can the focal length be negative?
Yes. A negative focal length indicates a diverging (concave) lens. You can enter negative values for f directly in the calculator.
What happens when the object is exactly at the focal point?
When x = f the denominator (x − f) becomes zero, so the image distance y → ∞. In practice, the refracted rays emerge as a parallel beam with no converging point — no image forms.
Is magnification the same as zoom?
Magnification here refers to the ratio of image size to object size, not optical zoom. A magnification of 2 means the image is twice as large as the object.
💡 Tip: To calculate the angle of refraction when light passes between media, use our Snell's Law Calculator.