What is the thin lens equation?

The thin lens equation is 1/f = 1/do + 1/di, where f is the focal length, do is the distance from the object to the lens, and di is the distance from the lens to the image. It holds for both converging (convex) and diverging (concave) lenses and for curved mirrors, provided the thin-lens (paraxial) approximation is satisfied — meaning all rays travel close to and nearly parallel with the optical axis.

What sign convention does this calculator use?

The real-is-positive convention (also called the Cartesian sign convention in many physics curricula). Object distances do are positive for real objects on the incoming side. Image distances di are positive for real images on the outgoing side and negative for virtual images. Focal lengths f are positive for converging lenses and negative for diverging lenses. Magnification m = −di/do is positive for upright images and negative for inverted images.

What is a real image versus a virtual image?

A real image forms when refracted rays actually converge on the far side of the lens; di is positive and the image can be projected onto a screen. A virtual image forms when the rays diverge after the lens but appear to come from a point behind it; di is negative and you can only see it by looking back through the lens (as with a magnifying glass). A concave (diverging) lens always produces a virtual, upright, reduced image for a real object.

How is optical power measured in diopters?

Optical power P = 1/f, where f is in metres, giving units of dioptres (D). A +2 D lens has a focal length of 0.5 m and converges light. A −4 D lens has a focal length of −0.25 m and diverges light. Ophthalmologists and opticians prescribe corrective lenses in diopters because powers of lens combinations simply add (P_total = P1 + P2) when lenses are in contact.

What does the lens maker's equation calculate?

The Lens Maker's Equation 1/f = (n−1)[1/R₁ − 1/R₂] links a lens's focal length to its physical construction: n is the refractive index of the glass (typically 1.45–1.9), R₁ is the radius of curvature of the first surface (positive if its centre is to the right), and R₂ is the radius of the second surface. This lets optical engineers design lenses from scratch — choosing glass type and grinding radii to hit a target focal length or diopter value.

How does magnification work for lenses?

Lateral magnification m = −di/do. When |m| > 1 the image is larger than the object (magnified); when |m| < 1 it is smaller (reduced). A positive m means the image is upright relative to the object; a negative m means it is inverted. For a simple magnifying glass (object inside the focal length), di is negative and m is positive and greater than 1 — a virtual, upright, magnified image.

Lens Equation Calculator

The lens equation calculator solves the fundamental thin-lens relationship for any unknown — focal length, object distance, or image distance — in a single step, then extends the result to magnification, optical power in diopters, image classification, and a schematic ray diagram. A second Lens Maker’s Equation tab lets you derive a focal length directly from glass refractive index and surface radii.

The thin lens equation

All thin-lens (paraxial) optics flows from one elegant formula:

1/f = 1/do + 1/di

where f is the focal length of the lens, do is the distance from the object to the lens, and di is the distance from the lens to the image. The equation applies equally to converging (convex) and diverging (concave) lenses, and to spherical mirrors, as long as the paraxial approximation holds — rays stay close to the optical axis.

Rearranging for each unknown:

Solve for image distance: di = (f × do) / (do − f)
Solve for object distance: do = (f × di) / (di − f)
Solve for focal length: f = (do × di) / (do + di)

Magnification describes how the image relates to the object:

m = −di / do

A positive m means upright; negative means inverted. |m| > 1 is magnified; |m| < 1 is reduced.

Optical power is simply the reciprocal of focal length in metres:

P = 1/f (unit: dioptre, D)

Optometrists write prescriptions in dioptres because lens powers add linearly when lenses are in contact.

The Lens Maker’s Equation

The Lens Maker’s Equation connects focal length to the physical glass properties:

1/f = (n − 1) × [1/R₁ − 1/R₂]

where n is the refractive index of the glass (crown glass ≈ 1.52, flint glass ≈ 1.62, borosilicate ≈ 1.47), R₁ is the radius of curvature of the first surface (positive if its centre of curvature lies to the right), and R₂ is the radius of the second surface. This equation lets optical engineers design lenses from scratch to hit a target focal length or dioptre specification.

Worked example: camera lens

A camera prime lens has a focal length of f = 50 mm = 0.050 m. A subject is do = 1.00 m away from the lens. Where does the sharp image form?

Using di = (f × do) / (do − f):

di = (0.050 × 1.000) / (1.000 − 0.050)
   = 0.050 / 0.950
   ≈ 0.05263 m  (≈ 52.6 mm behind the lens)

Magnification: m = −di/do = −0.05263/1.000 ≈ −0.053. The image is real, inverted, and reduced to about 5% of the subject’s size — exactly what a sensor captures.

Focal length	Object distance	Image distance	Magnification
50 mm	1.00 m	52.6 mm	−0.053×
50 mm	0.20 m	100 mm	−0.50×
50 mm	0.10 m	∞ (at focus)	—
−50 mm (diverging)	1.00 m	−47.6 mm	+0.048×

Sign convention note

This calculator uses the real-is-positive (Cartesian) convention:

Real objects: do > 0 (object on the incoming side)
Real images: di > 0 (image on the outgoing side, can be projected)
Virtual images: di < 0 (rays appear to diverge; only visible through the lens)
Converging lens: f > 0
Diverging lens: f < 0

Every result includes step-by-step algebra so you can check each substitution, and a colour-coded SVG ray diagram that updates live as you type. No data leaves your browser — the entire calculation runs in JavaScript.