Thin Lens Equation Calculator — 1/f = 1/dₒ + 1/dᵢ
Enter the focal length and object distance to find image distance, magnification and whether the image is real or virtual, inverted or upright.
Lens type
cm
cm
Real image — forms on the far side of the lens
- 1
Signed focal length
f = +10 cm = 10 - 2
Denominator (dₒ − f)
30 − (10) = 20The denominator changes sign when the object crosses the focal point. - 3
Image distance dᵢ = f × dₒ ÷ (dₒ − f)
10 × 30 ÷ 20 = 15
How does this calculator work?
Solve 1/f = 1/dₒ + 1/dᵢ to get dᵢ = f·dₒ/(dₒ−f). Magnification m = −dᵢ/dₒ: negative means inverted, |m| > 1 means enlarged. A converging lens (f > 0) forms a real image when dₒ > f; a diverging lens (f < 0) always gives a virtual, upright, reduced image.
Formula
How this is calculated
The thin lens equation 1/f = 1/dₒ + 1/dᵢ links three distances measured from the centre of a thin lens: the focal length f, the object distance dₒ (always positive for a real object placed in front of the lens), and the image distance dᵢ. Rearranging gives dᵢ = f × dₒ / (dₒ − f). A converging (convex) lens has f > 0; a diverging (concave) lens has f < 0.
The lateral magnification m = −dᵢ/dₒ gives the ratio of image height to object height. A negative magnification means the image is inverted; |m| > 1 means it is enlarged. For a converging lens with the object beyond the focal point (dₒ > f) the image is real (dᵢ > 0) and forms on the far side of the lens — it can be projected onto a screen. When the object is inside the focal point (dₒ < f), dᵢ becomes negative: a virtual, upright, magnified image appears on the same side as the object (like a magnifying glass). Diverging lenses always produce virtual, upright, reduced images regardless of object position.
This model assumes an ideal thin lens with no aberrations. Real thick lenses and imperfect optics deviate from the formula, particularly near the edges (spherical aberration) and for different wavelengths (chromatic aberration). The equation does not account for the lens's physical thickness, aperture effects, or absorption.
Frequently asked questions
A real image (dᵢ > 0) forms on the far side of the lens where refracted rays actually converge — it can be projected onto a screen. A virtual image (dᵢ < 0) forms where diverging rays appear to originate when extended back — it cannot be projected and is only visible by looking through the lens.
With dₒ = f the denominator becomes zero and dᵢ → ∞: rays from the object emerge parallel on the other side. This is the principle behind collimators and spotlights — placing a point source at the focal point produces a parallel beam.
Yes — the same formula works for spherical mirrors using the mirror sign convention (f = R/2 with f > 0 for a concave mirror). The sign conventions differ slightly from those for lenses, so always check which convention your textbook uses when applying it to mirrors.
Also known as
TG we-Calculate Editorial Team. (2026). Thin Lens Equation Calculator — 1/f = 1/dₒ + 1/dᵢ [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/thin-lens-equation-calculator
TG we-Calculate Editorial Team. "Thin Lens Equation Calculator — 1/f = 1/dₒ + 1/dᵢ." TG we-Calculate. 2026. https://we-calculate.com/calculator/thin-lens-equation-calculator.
TG we-Calculate Editorial Team, "Thin Lens Equation Calculator — 1/f = 1/dₒ + 1/dᵢ," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/thin-lens-equation-calculator
@misc{wecalculate_thin_lens_equation_calculator, title = {Thin Lens Equation Calculator — 1/f = 1/dₒ + 1/dᵢ}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/thin-lens-equation-calculator}}, year = {2026}, note = {TG we-Calculate} }
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