PhysicsCalcs
Optics

Thin Lens Equation Calculator

Calculate image distance, focal length, and magnification for converging and diverging lenses.

Interactive calculator

Thin Lens Equation Calculator

Calculate image distance, object distance, focal length, magnification, and image height for converging and diverging lenses using 1/f = 1/dₒ + 1/dᵢ.

Try an example

Your result will appear here.

Choose a calculation mode, fill in the known values, and click Calculate.

Quick Guide

  • Converging lens: enter positive f. Diverging: enter negative f.
  • Object distance dₒ is always positive for real objects.
  • Results show real/virtual, upright/inverted, and size.

Table of Contents

Key Takeaways

  • Thin lens equation: 1/f = 1/dₒ + 1/dᵢ (same form as mirror equation).
  • Converging (convex) lens: f > 0; diverging (concave) lens: f < 0.
  • dᵢ > 0 = real image (opposite side); dᵢ < 0 = virtual image (same side).
  • Magnification m = −dᵢ/dₒ: positive = upright, negative = inverted.
  • A magnifying glass works when dₒ < f (virtual, upright, magnified image).

The Thin Lens Equation

The thin lens equation relates focal length, object distance, and image distance for an ideal thin lens. It applies to both converging (convex, f > 0) and diverging (concave, f < 0) lenses, and is one of the most fundamental relationships in geometric optics.

Formula & Sign Convention

1f=1do+1di\frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i}
m=−dido=hihom = -\frac{d_i}{d_o} = \frac{h_i}{h_o}
QuantityPositiveNegative
fConverging (convex) lensDiverging (concave) lens
dₒReal object (in front)Virtual object
dᵢReal image (opposite side)Virtual image (same side)
mUpright imageInverted image

Image Formation Cases (Converging Lens)

Object PositionImageSize
dₒ > 2fReal, invertedDiminished
dₒ = 2fReal, invertedSame size
f < dₒ < 2fReal, invertedEnlarged
dₒ = fAt infinityN/A
dₒ < fVirtual, uprightEnlarged (magnifying glass)

Lens vs. Mirror

The thin lens equation has the same form as the mirror equation (1/f = 1/u + 1/v), but the sign convention differs. For lenses, real images form on the opposite side from the object. For mirrors, real images form on the same side. Use the Mirror Equation Calculator for reflective optics.

How to Use

  1. Select what to solve: image distance, object distance, focal length, magnification, or image height.
  2. Enter values with correct signs (converging f > 0, diverging f < 0).
  3. Click Calculate for results with image characterization.

Examples

Magnifying glass (f=10cm, dₒ=8cm)

1/dᵢ = 1/10 − 1/8 = −1/40 → dᵢ = −40 cm (virtual, upright, 5× magnified)

Camera lens (f=50mm, dₒ=2m)

1/dᵢ = 1/0.05 − 1/2 = 19.5 → dᵢ ≈ 51.3 mm (real, inverted, 0.026× reduced)

FAQ

What is the difference between the thin lens and mirror equations?›

They use the same formula (1/f = 1/dₒ + 1/dᵢ), but the sign convention differs. For lenses, real images form on the opposite side from the object (dᵢ > 0). For mirrors, real images form on the same side as the object. The physics is the same — only the geometry changes.

When does a converging lens form a virtual image?›

When the object is inside the focal point (dₒ < f), a converging lens produces a virtual, upright, magnified image. This is the magnifying glass effect. The image appears on the same side as the object and cannot be projected on a screen.

What does the thin lens equation not account for?›

The thin lens equation assumes zero lens thickness, no aberrations, and paraxial rays (close to the optical axis). Real lenses have spherical aberration, chromatic aberration, coma, astigmatism, and distortion that this equation ignores.

How is focal length related to lens power?›

Lens power P (in diopters, D) is the reciprocal of focal length in metres: P = 1/f. A +2 D lens has f = 0.5 m (converging). A −4 D lens has f = −0.25 m (diverging). Optometrists use diopters for eyeglass prescriptions.

Sources

  1. Hecht, E. – Optics (5th ed.)
  2. OpenStax University Physics Vol. 3
  3. The Physics Classroom – Thin Lens Equation
Manish Kumar

Author & technical reviewer

Manish Kumar

PhysicsCalcs tools are reviewed with an educational focus: clear formulas, transparent assumptions, and practical context for students and science learners.

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