Lens & Mirror Equation Calculator

Q: How do I tell if the image is real or virtual?

Positive dᵢ → real, inverted image (can be projected). Negative dᵢ → virtual, upright image (seen through the lens/mirror). The tool states the image type.

Q: How is this different from a camera focal-length tool?

This solves the imaging equation (object/image distance, magnification). Camera FOV tools relate focal length to sensor size and angle of view - a different question.

Q: Where does a concave mirror form the image, and what happens at 2f?

A concave mirror forms its real image on the same side as the object (in front of the mirror), whereas a converging lens forms its real image on the opposite side. When the object sits at 2f - the center of curvature, since R = 2f - the image also lands at 2f, the same size and inverted. Move the object inside the focal point and a concave mirror gives a virtual, upright, magnified image behind the mirror, which is how a shaving or makeup mirror works.

Frequently Asked Questions

What is the thin-lens (and mirror) equation?

1/f = 1/dₒ + 1/dᵢ, relating focal length, object distance, and image distance. Magnification m = −dᵢ/dₒ. The same equation (with sign conventions) covers mirrors.

What do the signs mean?

Conventions: converging lens / concave mirror f > 0; diverging lens / convex mirror f < 0. A negative image distance means a virtual image on the same side as the object.

How do I tell if the image is real or virtual?

Positive dᵢ → real, inverted image (can be projected). Negative dᵢ → virtual, upright image (seen through the lens/mirror). The tool states the image type.

How is this different from a camera focal-length tool?

This solves the imaging equation (object/image distance, magnification). Camera FOV tools relate focal length to sensor size and angle of view - a different question.

Where does a concave mirror form the image, and what happens at 2f?

A concave mirror forms its real image on the same side as the object (in front of the mirror), whereas a converging lens forms its real image on the opposite side. When the object sits at 2f - the center of curvature, since R = 2f - the image also lands at 2f, the same size and inverted. Move the object inside the focal point and a concave mirror gives a virtual, upright, magnified image behind the mirror, which is how a shaving or makeup mirror works.

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How This Calculator Works

The thin-lens (and mirror) equation relates the focal length f, the object distance dₒ, and the image distance dᵢ through 1÷f = 1÷dₒ + 1÷dᵢ. Give any two and the calculator solves the third by simple algebra on the reciprocals. The lateral magnification is m = −dᵢ÷dₒ: its sign tells orientation (negative → inverted, positive → upright) and its magnitude tells size (greater than 1 enlarged, less than 1 reduced). The tool then classifies the image as real (dᵢ positive - light actually converges there, can be projected on a screen) or virtual (dᵢ negative - rays only appear to diverge from it). A sign-convention selector switches between a converging÷diverging lens and a concave÷convex mirror, since the equation is identical but the geometry of where positive distances lie differs. This is the thin-lens÷mirror imaging equation - not a camera field-of-view or focal-length-from-sensor tool.

Worked Example: Object Outside the Focal Point

A converging lens has f = 10 cm; an object sits 30 cm in front. Solve dᵢ: 1÷dᵢ = 1÷f − 1÷dₒ = 1÷10 − 1÷30 = 3÷30 − 1÷30 = 2÷30, so dᵢ = 15 cm. Magnification m = −dᵢ÷dₒ = −15÷30 = −0.5. The image is therefore real (positive dᵢ), inverted (negative m), and reduced to half height - exactly how a camera lens projects a smaller, flipped image of a distant scene onto film. Move the object inside f (say 6 cm): dᵢ = −15 cm, m = +2.5 → a virtual, upright, magnified image, which is how that same lens works as a magnifying glass.

Where This Is Used

Eyeglasses & contacts: lens power in dioptres is 1÷f (metres); this equation sizes the corrective lens for myopia or hyperopia.

Microscopes & telescopes: objective and eyepiece are chained, each step solved with the same relation.

Mirrors: car side mirrors (convex, reduced wide view), shaving÷makeup mirrors (concave, magnified), and reflecting telescopes.

Projectors & cameras: placing the object just outside f to throw a large real image, or focusing onto a sensor.

Common Mistakes & Edge Cases

Sign errors: a diverging lens or convex mirror has negative f; a virtual image has negative dᵢ. Mixing conventions flips the answer.

Object exactly at the focal point → rays leave parallel, image at infinity (1÷dᵢ = 0); the calculator flags this rather than dividing by zero.

Assuming every lens magnifies - an object beyond 2f gives a reduced image; a diverging lens always reduces.

Confusing magnification sign with size: negative m means inverted, not smaller; |m| sets the size.

Applying the thin-lens equation to thick or compound lenses without using principal planes - it assumes negligible lens thickness.

Related Concepts & Limits

This model is paraxial (small-angle) and thin-lens÷mirror: it ignores lens thickness, spherical and chromatic aberration, and assumes rays close to the optical axis. Real optics use the lensmaker's equation 1÷f = (n−1)(1÷R₁ − 1÷R₂) to get f from surface curvatures and refractive index, and multi-element designs to cancel aberrations. Related quantities: optical power P = 1÷f (dioptres, shown when f is in cm), the Newtonian form xₒxᵢ = f² measured from the focal points, and ray-tracing for compound systems. The mirror equation is the same as the lens equation with a sign-convention shift, which is why one selector covers both. Distances may use any consistent length unit; results carry six significant figures.

Lens & Mirror Equation Calculator

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