Thin Lens Equation Calculator
Calculate object distance, image distance, or focal length using the thin lens equation (1/f = 1/dₒ + 1/dᵢ). Also computes magnification and image characteristics. See also our Snell's Law Calculator and Distance Calculator.
How to Use the Thin Lens Equation
The thin lens equation relates three fundamental quantities in optics: the focal length of a lens, the distance from the lens to the object, and the distance from the lens to the image. This equation applies to any thin lens — one whose thickness is negligible compared to the focal length. It works for both converging (convex) lenses with positive focal lengths and diverging (concave) lenses with negative focal lengths.
To use this calculator, select which quantity you want to solve for, enter the known values, and click Calculate. The sign convention is important: positive distances are on the side where light exits the lens (real images), and negative distances indicate virtual images on the same side as the object. A positive focal length indicates a converging lens, while a negative focal length indicates a diverging lens.
The magnification tells you about the image size and orientation. A magnification of -2 means the image is twice as large as the object and inverted. A magnification of +0.5 means the image is half the size and upright (virtual). The magnitude gives the size ratio, while the sign indicates orientation — negative means inverted, positive means upright.
Thin Lens Equation Formula
Thin Lens Equation:
1/f = 1/dₒ + 1/dᵢ
Magnification:
M = -dᵢ/dₒ = hᵢ/hₒ
Lens Power (Diopters):
P = 1/f (f in meters)
Lensmaker's Equation:
1/f = (n-1)[1/R₁ - 1/R₂]
Two Thin Lenses in Contact:
1/f_total = 1/f₁ + 1/f₂
Sign Convention:
dₒ > 0: real object (left of lens)
dᵢ > 0: real image (right of lens)
dᵢ < 0: virtual image (left of lens)
f > 0: converging lens
f < 0: diverging lens
Example Calculation
An object is placed 30 cm from a converging lens with focal length 10 cm. Find the image distance and magnification:
Given: dₒ = 30 cm, f = 10 cm
1/f = 1/dₒ + 1/dᵢ
1/10 = 1/30 + 1/dᵢ
1/dᵢ = 1/10 - 1/30 = 3/30 - 1/30 = 2/30
dᵢ = 30/2 = 15 cm
Magnification: M = -dᵢ/dₒ = -15/30 = -0.5
The image is:
• Real (dᵢ positive, on opposite side of lens)
• Inverted (M negative)
• Diminished (|M| < 1, half the object size)
• Located 15 cm from the lens
Thin Lens Reference Table
| Object Dist (dₒ) | Image Dist (dᵢ) | Magnification | Image Type |
|---|---|---|---|
| ∞ | f | 0 | Point at focus |
| 2f | 2f | -1 | Real, inverted, same size |
| 3f | 1.5f | -0.5 | Real, inverted, diminished |
| 1.5f | 3f | -2 | Real, inverted, magnified |
| f | ∞ | ∞ | No image (parallel rays) |
| 0.5f | -f | +2 | Virtual, upright, magnified |
| 30 | 15 | -0.5 | Real, inverted (f=10) |
| 20 | 20 | -1 | Real, inverted (f=10) |
| 15 | 30 | -2 | Real, inverted (f=10) |
| 5 | -10 | +2 | Virtual, upright (f=10) |
Frequently Asked Questions
What is the thin lens equation?
The thin lens equation (1/f = 1/dₒ + 1/dᵢ) relates the focal length of a thin lens to the object and image distances. It applies to lenses whose thickness is much smaller than the focal length. The equation works for both converging (convex) and diverging (concave) lenses using the appropriate sign convention. It is derived from geometric optics by tracing rays through the lens and is fundamental to understanding cameras, telescopes, microscopes, and the human eye.
What is the difference between real and virtual images?
A real image forms where light rays actually converge — it can be projected onto a screen. It has a positive image distance (dᵢ > 0) and is always inverted. A virtual image forms where light rays appear to diverge from — it cannot be projected onto a screen but can be seen by looking through the lens. It has a negative image distance (dᵢ < 0) and is always upright. Magnifying glasses produce virtual images; projectors produce real images.
What does magnification tell us?
Magnification (M = -dᵢ/dₒ) tells you two things: the size ratio and the orientation of the image. The magnitude |M| gives the ratio of image height to object height. If |M| > 1, the image is larger; if |M| < 1, it is smaller. The sign tells orientation: negative M means the image is inverted (flipped upside down), positive M means it is upright. For example, M = -2 means the image is twice as tall and inverted.
What happens when the object is at the focal point?
When the object is placed exactly at the focal point (dₒ = f), the thin lens equation gives 1/dᵢ = 1/f - 1/f = 0, meaning dᵢ = infinity. Physically, the refracted rays emerge parallel and never converge — no image is formed at a finite distance. This is the principle behind collimated light sources and searchlights, where a light source at the focal point produces a parallel beam.
How do converging and diverging lenses differ?
A converging (convex) lens has a positive focal length and bends parallel rays inward to a focal point. It can produce both real and virtual images depending on object placement. A diverging (concave) lens has a negative focal length and spreads parallel rays outward — it always produces virtual, upright, diminished images regardless of object position. Eyeglasses for nearsightedness use diverging lenses; those for farsightedness use converging lenses.
What is lens power in diopters?
Lens power is the reciprocal of focal length measured in meters: P = 1/f (diopters). A lens with f = 0.5 m has power +2 diopters. Positive power means converging, negative means diverging. Optometrists use diopters to prescribe eyeglasses — a prescription of -3.0 D means a diverging lens with f = -0.33 m. The advantage of using diopters is that powers of thin lenses in contact simply add: P_total = P₁ + P₂.
Applications of the Thin Lens Equation
The thin lens equation is essential in designing optical instruments. Cameras use it to determine the sensor-to-lens distance for focusing at different object distances. Telescopes and microscopes combine multiple lenses, each analyzed with this equation. The human eye is modeled as a thin lens system where the cornea and crystalline lens focus light onto the retina. Ophthalmologists use the equation to calculate corrective lens prescriptions.
In photography, the thin lens equation explains depth of field, bokeh, and the relationship between aperture and focus. In projector design, it determines the throw distance needed for a given screen size. Magnifying glasses, reading glasses, and microscope eyepieces all rely on placing the object inside the focal length to produce a magnified virtual image that the eye can comfortably view.