Lensmaker's Equation Calculator
Free calculate focal length from radii of curvature, refractive index, and lens thickness. Get instant, accurate results with our easy-to-use calculator.
Input Parameters
Typical: Glass ~1.5, Water ~1.33, Air ~1.0
Positive for convex (toward light), negative for concave
Positive for concave (away from light), negative for convex
Results
Enter parameters to calculate
What is the Lensmaker's Equation?
The Lensmaker's Equation relates the focal length of a lens to its physical properties: the radii of curvature of its two surfaces, the refractive index of the lens material, and (for thick lenses) the lens thickness.
This equation is fundamental in optical design, allowing engineers and physicists to calculate how a lens will focus light based on its geometry and material properties. It's used to design camera lenses, eyeglasses, microscopes, telescopes, and other optical instruments.
The sign convention for radii is important: R₁ is positive for surfaces convex toward the light source, and R₂ is positive for surfaces concave toward the light source. A plano surface (flat) has infinite radius.
Lensmaker's Equation Formulas
Thin Lens
Thick Lens
Where: f = focal length, n = refractive index, R₁, R₂ = radii of curvature, d = lens thickness
How to Calculate
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1
Convert all units to meters
Convert radii and thickness to meters for consistent calculations.
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2
Apply sign convention
R₁ positive for convex toward light, R₂ positive for concave toward light. Flat surfaces: R = ∞ (1/R = 0).
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3
Calculate 1/f
Use thin lens formula for thin lenses, or thick lens formula if thickness is significant.
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4
Find focal length
f = 1 / (1/f). Positive f = converging lens, negative f = diverging lens.
Practical Examples
Example 1: Double Convex Thin Lens
Glass lens (n=1.5), R₁ = 20 cm (convex), R₂ = -20 cm (convex). Calculate focal length.
Solution:
1/f = (1.5-1)(1/0.2 - 1/(-0.2)) = 0.5(5 + 5) = 5
f = 1/5 = 0.2 m = 20 cm (converging lens)
Example 2: Plano-Convex Lens
Glass lens (n=1.5), R₁ = 30 cm (convex), R₂ = ∞ (flat).
Solution:
1/f = (1.5-1)(1/0.3 - 0) = 0.5 × 3.33 = 1.67
f = 0.6 m = 60 cm
Applications
Optical Design
Designing camera lenses, telescopes, microscopes, and other optical instruments with specific focal lengths.
Eyeglasses
Calculating lens prescriptions, determining curvature needed for vision correction, and lens manufacturing.
Scientific Instruments
Designing precision optical systems for research, measurement, and analysis in laboratories.
Education
Teaching optics, understanding lens behavior, and demonstrating geometric optics principles.
Frequently Asked Questions
What is the sign convention for radii?
R₁ is positive when the surface is convex toward the incoming light (center of curvature on the side of incoming light). R₂ is positive when concave toward incoming light. This convention ensures consistent calculations.
When should I use the thick lens formula?
Use the thick lens formula when the lens thickness is significant compared to the focal length (typically d > f/10). For most thin lenses, the thickness term is negligible and the thin lens formula is sufficient.
What does a negative focal length mean?
A negative focal length indicates a diverging lens (concave). It spreads out parallel light rays. Positive focal length means a converging lens (convex) that focuses parallel rays to a point.
How does refractive index affect focal length?
Higher refractive index (n) increases the bending power, resulting in shorter focal length for the same radii. The factor (n-1) directly affects the lens power - higher n means stronger lens.
What if one surface is flat (plano)?
For a flat surface, R = ∞, so 1/R = 0. Simply omit that term in the equation. For a plano-convex lens: 1/f = (n-1)/R₁, where R₁ is the curved surface radius.