How to Calculate Focal Length
A complete step-by-step guide to calculating the focal length of any lens using the lens maker formula. Includes examples, common mistakes to avoid, and a free calculator.
Use Calculator Now ↓The Focal Length Formula
1/f = (n-1)(1/R₁ - 1/R₂)
Identify Your Lens Type
First, determine what type of lens you have. This affects the sign convention for the radii.
Convex (Converging)
Thicker in the middle - magnifying glasses, camera lenses
Positive focal length
Concave (Diverging)
Thinner in the middle - myopia correction glasses
Negative focal length
Plano-convex/concave
One flat surface, one curved
One radius = infinity
Find the Refractive Index (n)
The refractive index describes how much light slows down in the lens material. Look up the value for your lens material.
| Material | n | Common Use |
|---|---|---|
| Crown Glass | 1.52 | Standard optical glass |
| Flint Glass | 1.62 | Higher dispersion optical glass |
| Acrylic (PMMA) | 1.49 | Lightweight plastic lenses |
| Polycarbonate | 1.59 | Impact-resistant lenses |
| Water | 1.33 | Reference medium |
Measure or Note the Radii of Curvature
Find R₁ (first surface) and R₂ (second surface). These are usually provided in lens specifications or can be measured with a spherometer.
- •R₁ is the radius of the surface that light hits first
- •R₂ is the radius of the surface that light exits from
- •Use the same units for both radii (typically meters or centimeters)
- •Flat surfaces have R = ∞ (infinity), meaning 1/R = 0
Apply the Sign Convention
The sign convention is crucial for getting the correct result. Using the Cartesian convention:
Convex surface facing light source (center of curvature on far side)
Example: R₁ > 0 for biconvex
Concave surface facing light source (center of curvature on near side)
Example: R₁ < 0 for biconcave
Flat surface
Example: 1/R = 0 in calculation
Calculate Using the Lens Maker Formula
Now apply the formula with your values:
1/f = (n - 1) × (1/R₁ - 1/R₂)
- 1Calculate (n - 1) — this is the "lens power factor"
- 2Calculate 1/R₁ — if R₁ = ∞, this term is 0
- 3Calculate 1/R₂ — if R₂ = ∞, this term is 0
- 4Calculate (1/R₁ - 1/R₂)
- 5Multiply the results to get 1/f
- 6Take the reciprocal: f = 1/(1/f)
Interpret Your Result
The sign and magnitude of f tell you about your lens:
f > 0 (positive)
Converging lens - focuses parallel light to a real point
f < 0 (negative)
Diverging lens - spreads parallel light (virtual focal point)
Large |f|
Weak lens - long focal length, gentle focusing
Small |f|
Strong lens - short focal length, powerful focusing
Complete Worked Example
Problem:
A biconvex lens made of crown glass has radii of curvature R₁ = +15 cm and R₂ = -20 cm. Calculate the focal length.
Step 1: Identify lens type
Biconvex lens (converging) — expect positive f
Step 2: Refractive index
Crown glass: n = 1.52
Step 3: Convert to meters
R₁ = +0.15 m, R₂ = -0.20 m
Step 4: Sign convention ✓
R₁ > 0 (convex facing light), R₂ < 0 (convex facing away)
Step 5: Calculate
1/f = (1.52 - 1) × (1/0.15 - 1/(-0.20))
1/f = 0.52 × (6.67 + 5.0)
1/f = 0.52 × 11.67 = 6.07
f = 1/6.07 = 0.165 m = 16.5 cm
Step 6: Interpret
✓ Positive focal length confirms converging lens
Parallel light will focus 16.5 cm behind the lens
Try It Yourself — Free Calculator
Lens Maker Formula
Calculate focal length from lens parameters
Formula
1/f = (n-1)(1/R₁ - 1/R₂)
Typical: 1.5 (glass), 1.33 (water), 1.52 (crown glass)
Positive for convex, negative for concave
Positive for convex, negative for concave
Common Mistakes to Avoid
Wrong sign convention
Fix: Remember: convex surfaces facing light are positive, concave are negative. Draw a diagram if needed.
Forgetting to convert units
Fix: Use consistent units throughout. If R is in cm, f will be in cm. Convert to meters for SI units.
Confusing R₁ and R₂
Fix: R₁ is always the surface that light hits first. For lenses, this is typically the left surface when light comes from the left.
Using n of the surrounding medium
Fix: The formula uses the refractive index of the lens material relative to air (n ≈ 1). If the lens is in water, use n_lens/n_water.
Ignoring thickness for thick lenses
Fix: For thick lenses, use the modified formula: 1/f = (n-1)[1/R₁ - 1/R₂ + (n-1)d/(nR₁R₂)]
Frequently Asked Questions
How do I calculate focal length of a lens?
Use the lens maker formula: 1/f = (n-1)(1/R₁ - 1/R₂). Find the refractive index (n) and radii of curvature (R₁, R₂), apply the correct sign convention, calculate 1/f, then take the reciprocal to get f.
What is the formula for focal length of a lens?
The focal length formula is 1/f = (n-1)(1/R₁ - 1/R₂), where f is focal length, n is refractive index of the lens material, and R₁ and R₂ are the radii of curvature of the two lens surfaces.
How do I find focal length from radius of curvature?
Use the lens maker formula with both radii. For a symmetric lens where |R₁| = |R₂| = R, the formula simplifies to f = R/(2(n-1)) for biconvex/biconcave, or f = R/(n-1) for plano-convex/plano-concave lenses.
Why is my focal length negative?
A negative focal length means you have a diverging (concave) lens. This is physically correct — diverging lenses have virtual focal points, indicated by negative f. Check your sign convention if this was unexpected.
Can I use this formula for thick lenses?
The basic lens maker formula assumes thin lenses. For thick lenses where thickness d is significant, use: 1/f = (n-1)[1/R₁ - 1/R₂ + (n-1)d/(nR₁R₂)]. Our thick lens calculator handles this automatically.