Solve thin lens problems by finding the missing value (f, do, di, or magnification) and getting a plain-language image description (real/virtual, upright/inverted, magnified/reduced). Choose beginner-friendly positive distances or enter signed values if your class uses a sign convention.
Advanced options
How to use our Thin Lens Equation Calculator
- Choose Solve for (the calculator hides the value you are solving for).
- Choose Input mode: use Lens type + positive distances if you want to enter normal positive distances, or choose the manual signed option if your homework requires + and - signs.
- Pick a Distance unit (mm, cm, or m). Use that same unit for every distance and height you enter.
- If you are in Lens type mode, choose Lens type (only for Lens type mode), then enter Focal length f (selected unit) as a positive magnitude.
- Enter Object distance do (selected unit). (In manual signed mode, do is usually positive for a real object.)
- If you are not solving for it, open Advanced options and enter Image distance di (selected unit). In manual signed mode, di > 0 means a real image and di < 0 means a virtual image.
- Optional: Enter Object height ho (selected unit) to compute Image height hi (selected unit). Expect hi to be negative when the image is inverted.
- Optional: In Advanced options, set Number display format if you want to force Plain numbers or Scientific notation.
- Click Calculate.
- Sanity-check: look at Equation check (1/f and 1/do + 1/di). The two sides should match closely (small differences are just rounding). If they do not, re-check units, which value you solved for, and whether you used the right sign mode.
Common mistakes and quick fixes
Mistake: Mixing units (for example, f in cm but do in mm).
Fix: Pick one Distance unit and enter all distances and heights in that same unit.
Mistake: Typing negative values while still in Lens type + positive distances mode.
Fix: If you want to enter a negative f or di, switch Input mode to the manual signed option first.
Mistake: Solving for di when Object distance do (selected unit) equals Focal length f (selected unit).
Fix: Change do so it is not equal to f. At do = f, the model says the image is at infinity, so di is undefined and the calculator will show N/A with a warning.
Mistake: Using only Magnification magnitude |m| and ignoring the sign of Magnification m (signed).
Fix: Use signed m for orientation (m > 0 upright, m < 0 inverted). Use |m| only to tell how much bigger or smaller the image is.
Mistake: Being surprised when Image height hi (selected unit) is negative after entering a positive Object height ho (selected unit).
Fix: Negative hi is expected for an inverted image because hi = m times ho. Check the sign of Magnification m (signed).
Mistake: Leaving a needed field blank for your chosen Solve for option (for example, di is blank when you are solving for f).
Fix: Enter the hidden-needed value in Advanced options (or change Solve for to match what you actually know).
Definitions
Thin lens equation: A relationship for a thin lens that connects focal length, object distance, and image distance. In one common form: 1/f = 1/do + 1/di. [1]
Focal length f: How strongly a lens bends light. With the sign convention used in many textbooks, converging (convex) lenses have f > 0 and diverging (concave) lenses have f < 0. [1]
Object distance do: The distance from the lens to the object. In the manual signed convention used here, a real object is usually do > 0. [4]
Image distance di: The distance from the lens to the image location. In the manual signed convention used here, di > 0 means a real image (on the opposite side of the lens from the object) and di < 0 means a virtual image (on the same side as the object). [4]
Magnification m (signed): How the image size and orientation compare to the object, computed as m = -di/do. m > 0 is upright, m < 0 is inverted. [1]
Image height hi: The image height computed from hi = m times ho when you enter an object height ho. A negative hi means the image is inverted relative to a positive ho. [1]
Methodology
Sign convention options (what the calculator means by + and -): You can use either mode. In Lens type + positive distances mode, you enter positive magnitudes for f, do, and di, then the calculator assigns f > 0 for a converging lens and f < 0 for a diverging lens; do is treated as positive. In manual signed mode, you enter signed values using a common convention: f > 0 converging, f < 0 diverging; do > 0 for a real object; di > 0 for a real image and di < 0 for a virtual image. [1] [4]
Core equations used:
1/f = 1/do + 1/di
di = 1 / (1/f - 1/do)
do = 1 / (1/f - 1/di)
f = 1 / (1/do + 1/di)
m = -di/do
hi = m * ho
These relationships are standard for thin lenses in intro optics. [1] [2]
How outputs are interpreted: Real vs virtual comes from the sign of di (in manual signed mode): di > 0 real, di < 0 virtual. Upright vs inverted comes from the sign of m: m > 0 upright, m < 0 inverted. Magnified vs reduced comes from |m|: |m| > 1 larger, |m| < 1 smaller. The calculator keeps m signed (it does not force an absolute value) so you do not lose the orientation info. [1]
Equation check output: The tool also calculates both sides of the thin lens equation as values in 1 per selected unit: left side is 1/f and right side is 1/do + 1/di. If they are far apart, it usually means a wrong sign mode, a missing required input, or a rounding-heavy value was typed.
Worked mini-example (cm): Converging lens, f = 10 cm and do = 25 cm. Solve for di.
di = 1 / (1/10 - 1/25) = 1 / (0.1 - 0.04) = 1 / 0.06 = 16.6667 cm
m = -di/do = -16.6667/25 = -0.6667
Interpretation: di is positive so the image is real. m is negative so the image is inverted. |m| is about 0.667, so the image is smaller than the object. [1]
Assumptions and limits: This is the thin lens, paraxial (small-angle) model, so real lenses can differ due to thickness, aberrations, and large angles. Special case: when solving for di, if do equals f then (1/f - 1/do) becomes 0 and the equation predicts an image at infinity; the calculator reports N/A and shows a warning. [1]