Use the thin lens equation to find focal length, object distance, or image distance. Determines magnification and whether the image is real or virtual, upright or inverted.
The thin lens equation is the foundation of geometrical optics: a single relationship that ties focal length, object distance, and image distance for any "thin" lens (one whose thickness is small compared to focal length and object distances). It works for cameras, microscopes, telescopes, eyeglasses, magnifiers, and any other simple lens system.
This calculator solves the thin lens equation for any one of the three variables given the other two, and reports the magnification and whether the resulting image is real or virtual, upright or inverted. Use it for physics homework, optics design intuition, photography depth-of-field reasoning, or to understand how a magnifier or telescope produces its image.
The thin lens equation uses a sign convention. Distances on the same side as the incoming light (object side) are positive for objects; distances on the opposite side are positive for real images. Focal length is positive for converging (convex) lenses, negative for diverging (concave) lenses. Different textbooks vary in convention, but the formula and sign rules used here are the standard physics-textbook convention.
A 10 cm focal length convex lens used as a magnifier. Object placed at do = 7 cm. 1/di = 1/10 − 1/7 = (7 − 10)/70 = −3/70 di = −23.3 cm (virtual) m = −(−23.3)/7 ≈ +3.3 (upright, 3.3× enlarged) Virtual upright magnified image — exactly how a hand magnifier works.
Subject at do = 1000 cm (10 m) photographed with f = 50 mm = 5 cm lens. 1/di = 1/5 − 1/1000 ≈ 0.199 di ≈ 5.025 cm m = −5.025/1000 = −0.005 (inverted, ~1/200 size) The image forms just past the focal length, very small, inverted — exactly what hits the camera sensor.
Use this for any single-lens optics problem in introductory physics or applied imaging contexts: cameras, simple microscopes, telescopes, eyeglasses, projectors, and magnifiers.
Limitations: - "Thin lens" assumption: lens thickness much less than focal length. Real photographic lenses are systems of multiple elements; their effective focal length still works in this equation, but aberrations and field of view aren't captured here. - Paraxial approximation: rays close to the optical axis. For wide-angle lenses or off-axis points, more elaborate ray tracing is needed. - Single wavelength: chromatic aberration (different focal length per wavelength) requires separate analysis.
For real lens systems with multiple elements, use combinations of thin lens equations or full optical-design software. For mirrors instead of lenses, use the mirror equation (same form, but with sign convention adjustments for reflection).
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Positive for converging, negative for diverging
Image Distance
15.00 cm
Magnification
-0.500×
Image
Real, Inverted
| Parameter | Value |
|---|---|
| Focal Length f | 10.0000 cm |
| Object Distance do | 30.0000 cm |
| Image Distance di | 15.0000 cm |
| Magnification M | -0.5000× |
| Image Size | 0.5000× object size |
| Image Type | Real |
| Orientation | Inverted |
| Lens Type | Converging (convex) |
| Formula | 1/f = 1/do + 1/di |