Mirror Equation Calculator

Apply the mirror equation 1/f = 1/u + 1/v for spherical mirrors. Compute image distance, magnification, image type (real/virtual), orientation, and size. Visualize object & image positions with sign convention compliance.

Concave mirrors converge light; convex mirrors diverge light. Focal length sign follows Cartesian sign convention.
Always positive for real objects (standard convention).
Concave mirror → f > 0  |  Convex → f < 0
? Concave: u=30, f=15
? Concave: u=10 (inside focus)
? Convex: u=20, f=-12
⚡ At focus (u=f) → infinite distance
? Concave: u=45, f=20
Privacy-first: All calculations and ray diagrams run locally in your browser. No data transmission.

Mirror Equation: Core Optics Principle

The mirror equation relates object distance u, image distance v, and focal length f for spherical mirrors: 1/f = 1/u + 1/v. Derived from geometry under paraxial approximation, it forms the basis of geometrical optics. The linear magnification m = -v/u determines image size and orientation.

? Sign convention (Cartesian):
• Focal length: Concave → f > 0   |  Convex → f < 0
• Object distance u > 0 (real object in front of mirror)
• Image distance v > 0 → real image (in front, inverted)
• v < 0 → virtual image (behind mirror, upright)

Common Pitfalls in Mirror Calculations
  • Forgetting the sign of f – Using a positive f for a convex mirror incorrectly predicts real images. Always set f negative for convex mirrors.
  • Using u negative – For standard real objects, u is always positive. Negative u is only for virtual objects (rare in introductory problems).
  • Mixing units – Ensure u and f are in the same unit (both cm or both m). The calculator does not auto‑convert.
  • Misinterpreting magnification sign – Negative m means inverted image; positive m means upright. Do not confuse sign with size.
  • Ignoring the u = f special case – For concave mirrors, when u equals f, no image is formed (rays exit parallel). Our calculator correctly warns you.

Derivation & Validity

Using similar triangles for paraxial rays, the mirror equation emerges independent of mirror curvature to first order. For concave mirrors, the focal point is real (converging); for convex mirrors, the focus is virtual (diverging). This formula accurately predicts image formation for telescopes, shaving mirrors, rearview mirrors, and astronomical instruments.

Real-World Applications

  • Automotive mirrors: Convex side mirrors provide wider field, objects appear smaller and upright (virtual).
  • Concave mirrors: Used in shaving/makeup mirrors (enlarged virtual image when u < f), solar concentrators, and reflecting telescopes.
  • Dental mirrors: Small concave mirrors give magnified images for oral examination.

Case Study: Designing a Solar Cooker

A concave mirror with focal length 30 cm is used to concentrate sunlight. To produce a highly focused real image (intense spot) at the focal plane, the object (sun) is at infinity → image at focus. For extended objects, the cooker positions the pot near the focus. Our calculator helps determine how image distance changes with object placement, optimizing energy collection.

Condition Image Distance (v) Magnification Image Properties
Concave, u > 2f f < v < 2f |m| < 1, m negative Real, inverted, diminished
Concave, u = 2f v = 2f |m| = 1, negative Real, inverted, same size
Concave, f < u < 2f v > 2f |m| > 1, negative Real, inverted, enlarged
Concave, u < f v negative (virtual) |m| > 1, positive Virtual, upright, magnified
Convex (any u) v negative, |v| < |f| 0 < m < 1 Virtual, upright, diminished

Optics Pedagogy & Historical Context

The mirror equation was formalized by Ibn al-Haytham (Alhazen) in the 11th century and later developed by Kepler and Descartes. Its modern form is taught in every introductory physics course. This interactive tool promotes active learning, aligning with Next Generation Science Standards (NGSS).

If object distance equals focal length (u = f) for a concave mirror, the mirror equation gives 1/v = 0, meaning v → ∞. No image is formed (rays emerge parallel).

Consistent signs ensure correct predictions of image location (real/virtual) and orientation. The Cartesian system used here is standard in most optics textbooks.

Yes, as long as all distances (u, f) share the same unit. The calculator works unit‑independently.
References: Hecht, E. "Optics" (5th ed.); University Physics (OpenStax); Wolfram Demonstrations. Interactive validation: compare results with PhET Geometric Optics simulation.
Data integrity: Last code audit: April 2026. All calculations follow the standard Cartesian sign convention as defined in ISO 80000-7.

Expert reviewer: Developed by the GetZenQuery Physics Team — providing research‑grade interactive simulations with educational depth, peer‑reviewed equations, and seamless user experience.