lens and mirror equation formulas and interpretation
The thin-lens and mirror equation relates focal, object, and image distances using a consistent sign convention.
The calculator preserves all three unknown-variable modes and classifies the resulting image.
How to use the lens and mirror equation calculator
- Choose a model: Select the relationship matching the problem.
- Choose the unknown: Select the quantity to calculate.
- Enter values: Enter all known values with matching units and signs.
- Calculate: Review the result, formula, units, and direction.
Formula and variables
Reciprocal focal length equals the sum of reciprocal object and image distances; magnification follows their signed ratio.
1/f = 1/do + 1/di; m = −di/do- f — Focal length
- Signed focal distance (cm)
- do — Object distance
- Signed object position (cm)
- di — Image distance
- Signed image position (cm)
- m — Magnification
- Image-to-object height ratio (dimensionless)
Converging lens example
A 10 cm lens views an object 30 cm away.
- Focal length
- 10 cm
- Object distance
- 30 cm
- 1/di = 1/10 − 1/30
- di = 15 cm; m = −0.5
Result: The image is 15 cm away and half-size.
Negative magnification indicates an inverted real image.
Understanding your results
Interpreting the result
Use one sign convention consistently; ideal equations neglect aberrations and finite lens thickness.
Assumptions
- The selected equation represents the physical system.
- Inputs use a consistent reference direction.
- Values are converted through coherent SI units.
Limitations
- Vector components must be resolved along a common axis.
- External forces or energy losses are not added automatically.
- Results depend on the accuracy of entered measurements.
Common mistakes
- Mixing incompatible units.
- Dropping negative signs that represent direction.
- Using weight where mass is required.
- Entering a zero divisor.
Practical use cases
Physics problems
Check classroom, laboratory, and mechanics calculations.
Practical estimates
Estimate motion, forces, and energy for real systems.
Frequently asked questions
Can a result be negative?
Yes. For directional quantities, the sign indicates direction relative to the chosen positive axis.
Should I use SI units?
The interface can convert supported units, while the formulas are evaluated through coherent SI units.
Sources and review
- SI Brochure, 9th edition — BIPM. Accessed 2026-07-11.
- Special Publication 811 — NIST. Accessed 2026-07-11.
Reviewed 2026-07-11.