Optics: Lens / Mirror Equation Calculator

Enter focal length f, object distance d₀, and object height h₀ to compute image distance dᵢ, magnification m, and the image nature (real/virtual, upright/inverted). Non-threatening & private—everything runs in your browser.

Inputs

Equation: 1/f = 1/d₀ + 1/dᵢ, magnification m = −dᵢ/d₀ = hᵢ/h₀. Default sign rules: f > 0 for converging (convex lens, concave mirror); f < 0 for diverging (concave lens, convex mirror).

Results

Image distance dᵢ:
–
Signs tell side: + usually opposite lens’ object side (real), − same side (virtual).
Magnification m & image height hᵢ:
–
m = −dᵢ/d₀; hᵢ = m·h₀.
Image nature:
–
“Real” if rays actually converge; “Virtual” if they seem to from extensions.
Axis Element Focal points Object Image Rays

How to Use This Lens / Mirror Calculator

Pick an optical element and enter f (focal length), d₀ (object distance), and h₀ (object height) in your preferred units. The calculator applies the thin lens / spherical mirror relation 1/f = 1/d₀ + 1/dᵢ to solve for the image distance dᵢ, then computes the magnification m = −dᵢ/d₀ and image height hᵢ = m·h₀. The signs tell you if the image is real or virtual and upright or inverted.

Sign Conventions at a Glance

  • Converging elements: convex lens or concave mirror → f > 0.
  • Diverging elements: concave lens or convex mirror → f < 0.
  • Object distance d₀: positive for a real object in front of the element.
  • Image distance dᵢ: lenses—dᵢ > 0 is real on the far side; mirrors—dᵢ > 0 is real in front of the mirror.
  • Magnification: m < 0 inverted image; m > 0 upright image.

Common Scenarios

  • Object at infinity (very large d₀): light is nearly parallel and the image forms at dᵢ ≈ f.
  • Object at the focal point (d₀ = f): image is at infinity (collimated output) — useful for projectors and searchlights.
  • Object inside the focal length of a converging lens (d₀ < f): image is virtual, upright, and magnified — a simple magnifier.
  • Diverging lens: image is always virtual, upright, and reduced (useful for eyeglasses and peepholes).
  • Concave mirror: behaves like a converging element; f > 0. For a spherical mirror, f = R/2, where R is the radius of curvature.
  • Convex mirror: always forms a virtual, upright, reduced image behind the mirror — wide field of view.

Practical Tips

  • Measure distances along the optical axis from the lens plane or mirror vertex. Keep units consistent (m, cm, or mm).
  • If your goal is a particular image size, adjust d₀ to target m = hᵢ/h₀; the calculator shows m live.
  • To “focus to a screen,” look for dᵢ > 0 (real image) and position the screen at that distance on the image side.
  • For mirrors, remember that the real image forms in front of the mirror (same side as the object), while a virtual image appears behind the mirror.

Assumptions & Limitations

This tool uses the thin lens and spherical mirror models in the paraxial (small-angle) regime. It ignores thickness, aberrations, and aperture effects. Real optics may deviate, especially with large apertures or wide fields.

Educational use only — not for safety-critical design. For precise optical systems, consult detailed lens data and tolerances.

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