Magnification Calculator
Calculate linear magnification, image size, or object size using m = −dᵢ/dₒ = hᵢ/hₒ.
What Is Magnification?
Linear magnification (m) for a single lens or mirror is: m = −dᵢ/dₒ = hᵢ/hₒ, where dᵢ is image distance, dₒ is object distance, hᵢ is image height, and hₒ is object height. A negative m means the image is inverted; positive m means upright. |m| > 1 means enlarged; |m| < 1 means reduced.
The sign conventions: m = +2 means upright and twice the size (virtual image from concave mirror or magnifying glass). m = −0.5 means inverted and half the size (real image from converging lens or concave mirror). The magnitude |m| tells you the size ratio; the sign tells you the orientation.
For compound optical systems (microscopes, telescopes), total magnification = m₁ × m₂ × ... × mₙ. A compound microscope: m_objective × m_eyepiece ≈ 100 × 10 = 1,000×. Angular magnification for simple magnifiers: M = 25 cm/f (the ratio of near-point to focal length), giving a 'virtual magnifier' comparison.
Magnification also applies in electronics: voltage gain in amplifiers (Vout/Vin), current gain (Iout/Iin), and power gain (Pout/Pin) are all forms of 'magnification' in the signal domain. The concept extends to any ratio of output to input in a system.
Formula Reference Table
| Solve For | Formula | Notes |
|---|---|---|
| Linear magnification | m = −dᵢ/dₒ = hᵢ/hₒ | For lenses and mirrors |
| Image height | hᵢ = m · hₒ | Sign gives orientation |
| Object height | hₒ = hᵢ / m | From known m and hᵢ |
| Simple magnifier | M = 25/f (cm) | f = focal length; virtual at ∞ |
| Compound microscope | M_total = m_obj × M_eye | m_obj = dᵢ/f_obj typically |
| Telescope | M = −f_obj/f_eye | Negative = inverted |
3 Worked Examples
Projector slide (hₒ = 36 mm), dₒ = 105 mm, dᵢ = 3.5 m = 3,500 mm.
- m = −dᵢ/dₒ = −3500/105 = −33.3
- hᵢ = m × hₒ = −33.3 × 36 = −1,200 mm = 1.2 m
- Negative → inverted (projectors invert the slide to project correctly)
A 5 cm focal length magnifying glass used at 25 cm near point.
- Angular magnification M = 25/f = 25/5 = 5×
- Object 4.5 cm from lens: dᵢ = 1/(1/5 − 1/4.5) = −45 cm (virtual)
- Linear m = −(−45)/4.5 = +10 (upright, 10× linear)
Camera: f = 50 mm, object 2 m away. Photo is 35 mm frame. Object height 0.7 m in frame.
- dᵢ = 1/(1/50 − 1/2000)= 51.28 mm
- m = −51.28/2000 = −0.02564
- hₒ = hᵢ/m = 35/(−0.02564) = −1,365 mm — but that's the entire frame
Real-World Applications
Common Mistakes to Avoid
A converging lens forming a real image has both dᵢ and dₒ positive, so m = −dᵢ/dₒ is negative (inverted). Omitting the minus gives a wrong orientation.
Camera 'zoom' changes focal length f (optical zoom) or interpolates pixels (digital zoom). Magnification m = f/(dₒ−f) changes with both f and subject distance.
For a system: m_total = m₁ × m₂. Each m has its sign. If m₁ = −3 and m₂ = −0.5, total = +1.5 (upright overall).
Angular magnification M = angle subtended with optic / angle without = 25/f for simple magnifier. Linear m = hᵢ/hₒ. They are different for the same optical system.
Higher magnification reduces field of view and often requires shorter working distance. Microscopes balance magnification with resolution (diffraction-limited) — beyond 1,000× light microscopy adds no useful detail.
Frequently Asked Questions
Related Physics Calculators
Formula Explorer connections
Interpretation: This relationship connects light propagation, geometry, wavelength, refraction, interference or image formation. Assumption: Use a consistent sign convention and units. Paraxial rays, thin elements, coherent light, vacuum wavelength or ideal optical components may be assumed.