What is the thin lens equation?

1/f = 1/do + 1/di, where f is focal length, do is object distance, and di is image distance. Distances are positive for real objects/images (in front of lens for objects, behind for images).

What determines if an image is real or virtual?

A real image forms where light rays converge (positive di) and can be projected on a screen. A virtual image forms where rays appear to diverge from (negative di) and cannot be projected.

Thin Lens Calculator

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.

Inputs

Solve For

Focal Length (cm)

Positive for converging, negative for diverging

Object Distance do (cm)

Image Distance di (cm)

Results

Image Distance

15.00 cm

Magnification

-0.500×

Image

Real, Inverted

Thin Lens Results

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

Last updated: April 16, 2026

Formula

Thin lens equation: 1/f = 1/do + 1/di Magnification: m = −di / do = h_image / h_object Where: f = Focal length (cm; positive for converging lens) do = Object distance (cm; positive when object is on incoming-light side) di = Image distance (cm; positive for real image on opposite side, negative for virtual image) m = Magnification (positive = upright, negative = inverted; |m|>1 enlarged, <1 reduced) Image characteristics: di > 0 → Real image (can be projected on a screen), inverted di < 0 → Virtual image (apparent only, can't be projected), upright |m| > 1 → Magnified |m| < 1 → Reduced Example: f = 10 cm (converging), do = 30 cm 1/di = 1/10 − 1/30 = 2/30 → di = 15 cm m = −15/30 = −0.5 Real image, inverted, half size — typical of a camera lens imaging a distant subject.

How to use this calculator

Choose which variable you want to solve for: image distance, object distance, or focal length.
Enter the two known variables. For focal length, positive = converging (convex) lens, negative = diverging (concave).
Read the result and the magnification. Sign tells you image type and orientation.
For real-world photography: object distances are usually large (meters); image distances are within the camera (centimeters). For magnifiers: object distance is just inside the focal length; image distance is virtual.

Worked examples

Magnifying glass

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.

Camera lens of distant subject

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.

When to use this calculator

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).

Common mistakes to avoid

Mixing sign conventions across textbooks. Always check whether a given source uses real-positive or Cartesian sign convention.
Forgetting that diverging lenses always have negative focal length. A −10 cm lens always produces a virtual, reduced, upright image.
Confusing magnification sign. Negative magnification = inverted; magnitude < 1 = reduced. A real-camera image is m ≈ −0.005, both inverted and reduced.
Using the thin lens equation for thick lenses or compound systems without checking validity.
Treating the lens equation result as a precise prediction. Real lenses have aberrations the simple equation can't capture.

Frequently Asked Questions

Sources & further reading

Geometrical optics — Hecht (textbook reference) — Pearson — Optics by Eugene Hecht

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