This mirror equation calculator applies 1/f = 1/dₒ + 1/dᵢ to concave and convex spherical mirrors — reporting image distance, magnification and character (real/virtual, upright/inverted, enlarged/reduced). The mathematics mirrors the thin lens exactly; the geometry of reflection, and where the images live, are what change.

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Reflection's Version of the Same Music

The mirror equation is the lens equation with the physics folded: instead of light passing through glass and converging on the far side, it bounces and converges on the SAME side it came from — so “real” images form in front of the mirror (positive distance, our convention), and f = R/2 ties the focal length to the mirror's radius of curvature: a concave mirror ground from a sphere of radius 20 cm focuses at 10 cm. The defaults replay the lens example in mirror form — f = 10, dₒ = 30 → dᵢ = 15 cm, m = −0.5: real, inverted, half-size, floating in front of the mirror where a screen can prove it. The convex case inverts the personality: f = −10, same object → dᵢ = −7.5 cm, m = +0.25, virtual and upright and shrunken — the passenger-mirror image whose deliberate miniaturization packs a wide scene into a small glass at the cost of lying about distance.

The shared equation raises the natural question — why two calculators? Because the sign geography and the use-cases genuinely differ: lens images form on the opposite side, mirror images in front; lenses refract twice, mirrors reflect once (no chromatic aberration — which is why great telescopes are mirrors); and the instruments divide cleanly — cameras and eyes on the lens side, telescopes, headlights, shaving mirrors and security domes on the mirror side. Our Thin Lens Calculator states its own convention beside its inputs; keep each page's convention with its page.

Worked Examples (Verified)

  • Concave, camera-like (defaults): dᵢ = 15.0 cm in front, m = −0.50, image height −2.0 cm.
  • Concave at C: f = 10, dₒ = 20 (= 2f) → dᵢ = 20 cm, m = −1 — same-size inverted, the symmetry point.
  • Convex security mirror: f = −10, dₒ = 30 → dᵢ = −7.5 cm (behind the mirror), m = +0.25 — everything fits, everything looks farther than it is.

Mirror Discipline

  • f = R/2 with the sign of the mirror: concave positive (center of curvature in front), convex negative. A problem stated with R needs the halving before the equation — the classic first slip.
  • Real images of mirrors form IN FRONT (same side as the object) — the geometry opposite to lenses; ray diagrams should be sketched when the sign output surprises you.
  • Plane mirror check: f = ∞ makes dᵢ = −dₒ, m = +1 — the equation reproduces the bathroom mirror exactly; use it as the zero-curvature sanity bound.
  • Spherical aberration lives here too: wide spherical mirrors focus marginal rays short — paraboloids (headlights, dishes, telescopes) exist to fix precisely that, at f determined by the parabola rather than R/2.
  • m's sign remains the orientation oracle: negative inverted (real), positive upright (virtual) for single mirrors, exactly as with lenses.

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Frequently Asked Questions

How do you use the mirror equation?

1/dᵢ = 1/f − 1/dₒ with f positive for concave and negative for convex mirrors (distances measured from the mirror's surface, in front positive), then m = −dᵢ/dₒ. Defaults: f = 10 cm, dₒ = 30 cm → dᵢ = 15 cm, m = −0.5. This calculator solves it and decodes the image's character from the signs.

What is the relation between focal length and radius of curvature?

f = R/2 for a spherical mirror (paraxial approximation): the mirror focuses parallel rays halfway between its surface and its center of curvature. A concave mirror from an R = 20 cm sphere has f = +10 cm — the defaults' mirror. Deeper mirrors (small R) focus shorter and collect more light, which is why telescope specifications quote f-numbers derived from exactly this halving.

Why do convex mirrors say “objects are closer than they appear”?

Because a convex mirror always reduces (|m| < 1): the image is smaller, and your visual system reads smallness as distance. Our example shrinks the object to 0.25× — the brain concludes “far away” for a car that may be close. The trade is deliberate: reduction buys field of view (more of the scene fits the glass), and the warning label is the price of the bargain.

Concave mirror vs convex mirror — which forms real images?

Only concave (converging) mirrors can form real images — when the object sits beyond the focal point. Inside f, even concave mirrors go virtual (the makeup-mirror case). Convex (diverging) mirrors form only virtual, upright, reduced images of real objects, at any distance — a one-regime instrument, like the diverging lens on the refraction side.

How is the mirror equation different from the lens equation?

Formally identical — 1/f = 1/dₒ + 1/dᵢ and m = −dᵢ/dₒ — with different physical geography and sign conventions: mirror real images form in front (object side), lens real images behind (opposite side); mirror f signs follow concavity, lens f signs follow convergence (which coincides); mirrors suffer no chromatic aberration. Learn one and you own both — but respect each page's stated convention when entering numbers.

What happens with the object at the center of curvature?

dₒ = 2f images at dᵢ = 2f (the center itself) with m = −1: life-size and inverted — the configuration of some optical test benches and a neat self-check (object and image coincide in position). Move the object slightly in and the image runs out enlarged; slightly out and it shrinks toward f — the reciprocal dance the equation encodes.

Can a mirror magnify like a lens?

Yes — the concave mirror inside its focal length is a magnifier (m = f/(f − dₒ) > 1, upright, virtual): shaving and makeup mirrors are typically f ≈ 20–30 cm used at 15–25 cm, giving gentle 2–4× enlargement. Telescopes use concave primaries differently (real images at the focal plane, then an eyepiece lens magnifies THAT) — a two-stage system across both of this cluster's calculators.

Is this mirror equation calculator free?

Yes — free, instant, no sign-up. Enter signed focal length, object distance and optional height for image distance, magnification and character.