Lens & Mirror Calculator

Free Online 7 Optical Elements Ray Diagrams Step-by-Step

Free lens equation calculator with interactive ray diagrams. Supports 7 optical elements — biconvex, biconcave, plano-convex, plano-concave lenses, concave, convex, and plane mirrors. Solve 1/f = 1/v − 1/u with step-by-step solutions, magnification, diopters, and radius of curvature.

Lens & Mirror Calculator
Positive = converging, Negative = diverging
Negative (object on left, sign convention)

Ray Diagram

Results

1/f = 1/v − 1/u

Enter values to calculate

Solve lens and mirror equations with interactive ray diagrams.

Step-by-Step Solution

What is the Thin Lens Equation?

The thin lens equation 1/f = 1/v − 1/u relates the focal length (f), object distance (u), and image distance (v) for any thin lens or spherical mirror. It is the foundation of geometric optics, used to predict where images form and whether they are real or virtual.

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Key Insight: The same equation works for lenses and mirrors — only the sign conventions differ. Converging elements have positive focal lengths; diverging elements have negative focal lengths.

Types of Optical Elements

Understanding the seven optical elements supported by this calculator.

Biconvex Lens (Converging)

Both surfaces curve outward. Positive f. Forms real or virtual images. Used in cameras, magnifying glasses, and farsightedness correction.

Biconcave Lens (Diverging)

Both surfaces curve inward. Negative f. Always forms virtual, upright, diminished images. Used in nearsightedness correction and peepholes.

Plano-Convex Lens

One flat surface, one convex. Positive f. Behaves like a converging lens. Common in laser optics, condensers, and imaging systems.

Plano-Concave Lens

One flat surface, one concave. Negative f. Behaves like a diverging lens. Used in beam expanders and to correct spherical aberration.

Concave Mirror

Curves inward (R = 2f). Positive f. Forms real or virtual images. Used in telescopes, headlights, and shaving mirrors.

Convex Mirror

Curves outward (R = 2f). Negative f. Always forms virtual, upright, diminished images. Used in vehicle side mirrors and security mirrors.

Plane Mirror

Flat reflecting surface. f = ∞, P = 0 D. Always forms virtual, upright, same-size images at equal distance behind the mirror. The simplest optical element.

Real-World Applications

Eyeglasses & Contact Lenses

Optometrists prescribe lenses in diopters (P = 1/f). Positive diopters correct farsightedness; negative correct nearsightedness.

Cameras & Telescopes

Camera lenses use the thin lens equation to focus light on the sensor. Telescopes combine converging lenses/mirrors for magnification.

Microscopes & Projectors

Compound microscopes use two converging lenses for extreme magnification. Projectors create enlarged real images on screens.

Frequently Asked Questions

The thin lens equation is 1/f = 1/v − 1/u, where f is the focal length, u is the object distance, and v is the image distance. It predicts where an image forms for any thin lens or spherical mirror. Converging elements (biconvex lens, plano-convex lens, concave mirror) have positive f. Diverging elements (biconcave lens, plano-concave lens, convex mirror) have negative f. A plane mirror has f = infinity.
Magnification m = v/u, which also equals image height divided by object height (h'/h). When m is negative the image is inverted; when positive it is upright. If |m| > 1 the image is magnified; if |m| < 1 it is diminished. For example, m = -2 means an inverted image twice the object's size. A plane mirror always gives m = 1.
Real images form where light rays actually converge and can be projected onto a screen (v is positive). Virtual images form where rays only appear to diverge from and cannot be projected (v is negative). Converging lenses and concave mirrors can form both types. Diverging lenses, convex mirrors, and plane mirrors always form virtual images.
For spherical mirrors, the radius of curvature R equals twice the focal length: R = 2f. A concave mirror with f = 15 cm has R = 30 cm. This relationship comes from the geometry of reflection at a curved surface. Lenses use a more complex relation involving both surface radii and refractive index (lensmaker's equation).
Lens power P is the reciprocal of focal length in meters: P = 1/f(m), measured in diopters (D). A converging lens with f = 50 cm has power +2D. Positive diopters mean converging, negative mean diverging. Optometrists prescribe eyeglasses in diopters. For lenses in contact, total power is P_total = P1 + P2.
When the object is placed exactly at the focal point (u = -f), the image forms at infinity. The light rays emerge parallel after passing through the lens or reflecting off the mirror. This principle is used in searchlights and collimators. For converging thin lenses (biconvex or plano-convex), the calculator draws a parallel-ray (collimated) diagram instead of hiding the diagram.
Plano-convex lenses have one flat surface and one curved outward surface. They converge light (positive f) and are common in laser optics and condensers. Plano-concave lenses have one flat and one inward-curved surface, diverging light (negative f). Both follow the same thin lens equation as biconvex and biconcave lenses.

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