The Lens Maker's Equation, and the Sign Convention That Decides the Answer

Q: Why is R2 negative for a convex lens when both faces are convex?

Because both radii are measured from the same origin, the optical centre, by the same rule: positive if the centre of curvature lies on the side the light is going towards. The first face bulges towards the incoming light, so its centre is downstream and R₁ is positive. The second face bulges away from the incoming light, so its centre is upstream and R₂ is negative. The faces look the same from outside; what differs is which way each one is measured. HyperPhysics puts it as R₁ “positive since it is measured from the front surface” and R₂ negative because it extends leftward.

Q: Why does a glass lens get so much weaker in water?

Because the bracket is n/nm − 1, not n − 1. In air that is 1.5 − 1 = 0.5; in water it is 1.5/1.333 − 1 = 0.125, four times smaller, so the focal length is four times longer. The physics is that refraction at a surface depends on the step in index, not on the index itself, and water makes the step much smaller. Push it further: a lens made of something with a lower index than its surroundings reverses entirely, so an air bubble in water is a diverging lens even though it is shaped like a converging one. Try n = 1.0003 with nm = 1.333 above and watch the sign flip.

1/f = (n − 1)(1/R₁ − 1/R₂) is two marks if you know the signs and nothing if you do not. This page implements the Cartesian convention that NCERT and most boards use, says so everywhere it matters, and draws the lens you have actually described — converging or diverging, with both centres of curvature marked on the side the sign puts them.

Last updated 11 Nov 2026

Focal length from the two radii, the index and the medium

Cartesian signs, surface powers, and the lens in water

Refractive index of the lens material1.5 is the usual school value for glass and is the figure the Cornell lecture cited on the sibling Brewster page uses for its own worked example. Real glasses run from about 1.5 to 1.6 and every one of them disperses, so use the index the question gives you. Note that n only ever appears as n − 1 (or n/n_m − 1), so a lens with n = 1 has no power at all — which is why a glass rod in an exactly index-matched liquid vanishes.

Refractive index of the surrounding medium1.0 for air, which is what almost every question means. Set it to 1.333 to put the lens under water and watch the focal length roughly quadruple: the factor is (n − 1)/(n/n_m − 1), which for n = 1.5 is 0.5/0.1253 ≈ 4. HyperPhysics says the same thing qualitatively — in water “its lens strength will be diminished”. If n_m equals n the lens disappears optically and the page refuses to answer, which is correct rather than broken.

Radius of curvature of the first surfaceThe surface the light hits first. Signed, Cartesian: positive if the centre of curvature lies on the far side (the outgoing side), negative if it lies on the near side. For a double convex lens with light arriving from the left, R₁ is positive — HyperPhysics says exactly that, “positive since it is measured from the front surface”. For a double concave lens it is negative. A flat surface has R = infinity, which this engine cannot hold, so type a very large number such as 1000000 instead; the 1/R term then contributes a millionth of a dioptre, which is nothing. R = 0 is not a surface and the page says so. Be honest about what that stand-in costs: at R = 1000000 mm the surface still carries 0.0005 D, so a lens that ought to be exactly 100 mm comes out as 100.005 mm — five parts in a hundred thousand, invisible at any precision a question will ask for, but not zero.

Radius of curvature of the second surfaceThe surface the light leaves by, measured by the same rule from the same origin — not from the second surface and not with the sign flipped. For a double convex lens R₂ is negative, because its centre of curvature is back on the incoming side. That is the NCERT note BC₁ = +R₁, DC₂ = −R₂ and it is the single most common place to lose the mark. The default pair +100 and −100 is an equiconvex lens; +100 and +100 would be a concentric meniscus with no power at all.

The lens you have described. It is drawn thick in the middle when f is positive and thin in the middle when f is negative, so the shape follows the sign of your answer rather than the shape of your radii. One ray comes in parallel to the axis: for a converging lens it bends to the focus F, and for a diverging lens it bends away and the dashed line shows where it only appears to come from. The two green arrows run from the lens to each centre of curvature, so an arrow pointing right is a positive radius in the Cartesian convention and an arrow pointing left is a negative one. The distances along the axis are compressed, so a 10 mm focal length and a 2 m one both fit on a phone: read the numbers, not the lengths.

100.00mmExample

An equiconvex glass lens, n = 1.5 in air, both faces ground to a radius of 100 mm

The lens maker’s equation, and the convention it only works in

1/f = (n − 1)(1/R₁ − 1/R₂) · in a medium: 1/f = (n/n_m − 1)(1/R₁ − 1/R₂) · n_m/f = F₁ + F₂ = (n − n_m)/R₁ + (n_m − n)/R₂

f: the focal length, signed. Positive is converging, negative is diverging. Measured from the optical centre, in the medium the lens sits in. This is also the f in the thin lens formula 1/v − 1/u = 1/f, NCERT’s Equation 9.23
n: the refractive index of the lens material. It only ever appears as n − 1, so a lens with the index of its surroundings has no power whatever its shape
n_m: the index of the surrounding medium, which school questions silently take as 1. Where it is not 1, the bracket becomes n/n_m − 1 and the lens weakens — a glass lens in water is about four times longer in focal length than the same lens in air
R₁, R₂: the radii of curvature of the two surfaces, both measured from the optical centre by the same rule and both signed. Positive when the centre of curvature lies on the outgoing side. NCERT’s note is BC₁ = +R₁ and DC₂ = −R₂ for a convex lens; HyperPhysics says R₁ is positive for a double convex lens and R₂ negative. A flat surface is R = ∞ and its 1/R term is 0
F₁, F₂: the powers of the two refracting surfaces separately, each (n_after − n_before)/R. HyperPhysics states that for a thin lens “the power is approximately the sum of the surface powers”, and the rows above show that F₁ + F₂ comes out exactly equal to n_m/f — which is a check on your arithmetic every single time
D: the dioptre. NCERT gives P = 1/f with 1 D = 1 m⁻¹, so a 100 mm converging lens is +10 D. Spectacle prescriptions are in these units, and they add when lenses are in contact, which is the whole reason the unit exists

Worked example

An equiconvex glass lens, n = 1.5 in air, both faces ground to a radius of 100 mm
SIGNS FIRST, BEFORE ANY ARITHMETIC. Light arrives from the left. The first face bulges towards it, so its centre of curvature is on the far side: R₁ = +100 mm. The second face bulges away, so its centre of curvature is back on the near side: R₂ = −100 mm. This is HyperPhysics's “R₁ is positive since it is measured from the front surface” and NCERT's BC₁ = +R₁, DC₂ = −R₂. Both faces are convex and the two radii have opposite signs. If that feels wrong, it is the thing this page exists for.
WHERE THE FORMULA COMES FROM. Refraction at one spherical surface is NCERT's Equation 9.16, n₂/v − n₁/u = (n₂ − n₁)/R. Apply it at the first surface with n₁ = 1 and n₂ = n to get an intermediate image, then apply it again at the second surface treating that image as the object, with n₁ = n and n₂ = 1. The intermediate distance cancels, and what is left is the lens maker's equation. The lens maker's equation is two refractions in a row with the middle crossed out.
THE SHAPE FACTOR. 1/R₁ − 1/R₂ = 1/100 − 1/(−100) = 0.01 + 0.01 = 0.02 mm⁻¹. The minus sign in the formula and the minus sign on R₂ cancel, so the two curvatures add. That is the whole reason a double convex lens is twice as strong as a plano-convex one of the same radius.
THE MATERIAL FACTOR. n/n_m − 1 = 1.5/1 − 1 = 0.5. In air this is just n − 1.
MULTIPLY. 1/f = 0.5 × 0.02 = 0.01 mm⁻¹, so f = +100 mm, or 0.1 m, or +10 D. Positive, so converging, which matches the picture.
CHECK BY SURFACE POWERS. F₁ = (1.5 − 1)/0.1 m = +5 D and F₂ = (1 − 1.5)/(−0.1 m) = +5 D. They sum to +10 D, which is 1/f. Two surfaces of five dioptres each: that is what a ten-dioptre lens actually is, and the check costs ten seconds.
NOW PUT IT IN WATER. n/n_m − 1 becomes 1.5/1.333 − 1 = 0.1253, so 1/f = 0.1253 × 0.02 and f = 399 mm — four times weaker. This is why you cannot see clearly underwater: your cornea does most of your focusing because of the index step from air to tissue, and in water that step almost vanishes. Goggles work by putting the air back.

The six lens shapes, their signs, and what each one does

**The last two rows are why the sign convention exists.** Both are meniscus lenses, both are convex on one face and concave on the other, both have two positive radii — and one converges while the other diverges. No rule of the form “convex means plus” can tell them apart; only the arithmetic can. Try +100 and +200 against +200 and +100 in the fields above and watch the diagram flip. The concentric case, R₁ = R₂, sits exactly between them and has no power at all, which is why the page refuses to answer there.
Shape	R₁	R₂	Sign of 1/R₁ − 1/R₂	f
Double convex (biconvex)	+	−	positive, the two add	positive, converging
Plano-convex, flat face second	+	∞	positive, one term only	positive, converging
Plano-convex, flat face first	∞	−	positive, one term only	positive, converging
Double concave (biconcave)	−	+	negative, the two add	negative, diverging
Meniscus, first face tighter	+	+ but larger	positive	positive, converging
Meniscus, second face tighter	+	+ but smaller	negative	negative, diverging

Why this formula is really about signs

The lens maker’s equation does something no other school optics formula does: it takes you from the shape of a piece of glass to what it will do to light. The thin lens formula tells you where the image goes once you know f. This tells you what f is in the first place, from two radii and an index — which is how a lens actually gets made.

Where it comes from. It is not a new law. It is refraction at a spherical surface, NCERT’s Equation 9.16, applied twice: once as the light enters the glass and once as it leaves. The image formed by the first surface is the object for the second, and when you write both equations down and eliminate that intermediate distance, the result is 1/f = (n − 1)(1/R₁ − 1/R₂). HyperPhysics states the same thing a different way: for a thin lens the power is the sum of the two surface powers, F₁ + F₂. The rows on this page show both, and they agree exactly, which is the fastest check you have.

The convention, stated plainly, because this page has to pick one. This calculator uses the Cartesian sign convention: all distances measured from the optical centre, positive in the direction the light travels, negative against it. That is NCERT’s section 9.2.1 and it is what nearly every board now examines. The older real-is-positive convention measures a real object distance as positive and flips several signs, so the same lens produces a differently signed answer and the thin lens formula is written 1/v + 1/u = 1/f rather than 1/v − 1/u = 1/f. Both are self-consistent; mixing them is what loses marks. If your textbook writes a plus in the thin lens formula, it is not using the convention on this page, and you should not read numbers off here into that textbook’s equations.

The thing almost everybody gets wrong. For a double convex lens, R₁ is positive and R₂ is negative — even though both faces are convex and look identical. The radii are measured from the same origin by the same rule, so a face that bulges towards the light has its centre of curvature downstream (positive) and a face that bulges away has its centre upstream (negative). The minus sign already in the formula then turns that into an addition, which is why the two curvatures reinforce each other. Write both signs down before you touch the arithmetic; the diagram above draws the centres of curvature exactly where your signs put them, so if the green arrows point the wrong way, so do your signs.

Why the medium matters more than you would think. In a medium of index n_m the bracket becomes n/n_m − 1. For glass in water that falls from 0.5 to 0.125, so the same lens is four times weaker. This is not a curiosity: your eye focuses mostly at the cornea, where light crosses from air into tissue, and underwater that step almost disappears, which is why everything is blurred until you put goggles on and restore the air. For the syllabus itself, the CBSE 2025-26 Class XII curriculum names the lens maker’s formula in Unit VI by name, so this is examinable content rather than enrichment.

Where this goes wrong in an exam. Four places. Using the same sign for both radii of a biconvex lens. Leaving the radii in centimetres and the answer in metres. Forgetting that a plano surface contributes nothing, and then dividing by zero. And quoting a converging answer for a meniscus without checking which of its two positive radii is the tighter one. Next steps: the thin lens and mirror formula takes this f and finds the image, the sign convention interpreter works through the conventions on their own, and Snell’s law is the single refraction this whole equation is built out of.

Frequently asked questions

Why is R2 negative for a convex lens when both faces are convex?

Because both radii are measured from the same origin, the optical centre, by the same rule: positive if the centre of curvature lies on the side the light is going towards. The first face bulges towards the incoming light, so its centre is downstream and R₁ is positive. The second face bulges away from the incoming light, so its centre is upstream and R₂ is negative. The faces look the same from outside; what differs is which way each one is measured. HyperPhysics puts it as R₁ “positive since it is measured from the front surface” and R₂ negative because it extends leftward.

What do I type for a flat surface? The formula wants infinity.

Type a very large number — 1000000 mm works, and the field accepts it. That surface then still has a power of 0.0005 D rather than exactly zero, so a lens that should be 100.000 mm comes out as 100.005 mm — five parts in a hundred thousand, which no question will ever ask you to resolve, but it is not nothing and this page will not pretend it is. The engine behind this page is numeric and has no infinity, so this is a genuine limitation stated rather than hidden. Do not type 0: a radius of zero is not a flat surface, it is a point, and the page will correctly refuse to answer. There is a row above that gives you the plano-convex answer directly if that is what you wanted.

Does it matter which way round I put the lens?

For the focal length, no — and that is worth knowing. Turn the lens round and R₁ becomes −R₂_old and R₂ becomes −R₁_old, so the bracket 1/R₁ − 1/R₂ is unchanged and f is the same. A plano-convex lens has the same focal length whichever face points at the light. The image quality is not the same: aberrations depend on how the bending is shared between the two surfaces, which is why a plano-convex lens is used with the curved face towards the parallel beam. The thin lens equation on this page cannot see that; it only knows the focal length.

Why does a glass lens get so much weaker in water?

Because the bracket is n/n_m − 1, not n − 1. In air that is 1.5 − 1 = 0.5; in water it is 1.5/1.333 − 1 = 0.125, four times smaller, so the focal length is four times longer. The physics is that refraction at a surface depends on the step in index, not on the index itself, and water makes the step much smaller. Push it further: a lens made of something with a lower index than its surroundings reverses entirely, so an air bubble in water is a diverging lens even though it is shaped like a converging one. Try n = 1.0003 with n_m = 1.333 above and watch the sign flip.

Is the lens maker’s formula actually on the syllabus?

On the CBSE Class XII curriculum for 2025-26 it is, by name: Unit VI Optics, Chapter 9 Ray Optics lists the lens maker’s formula alongside the thin lens formula, magnification, power of a lens and combination of thin lenses in contact. NCERT’s Chapter 9 derives it as Equation 9.21 in section 9.5.2. Citing those documents is not a claim that any board has approved this page — it is a statement about what the documents contain, which anyone can check. If you are following a different board, check its own syllabus; the derivation is the same everywhere and only the sign convention varies.

Related calculators

References

NCERT, Physics, Class XII, Part II, Chapter 9, Ray Optics and Optical Instruments (rationalised 2023–24 printing; the copy read was a third-party hosted PDF of the NCERT chapter, not a file served by ncert.nic.in, and that is stated because it matters). Section 9.2.1 gives the Cartesian sign convention: all distances are measured from the pole of the mirror or the optical centre of the lens, distances “measured in the same direction as the incident light are taken as positive”, those “measured in the direction opposite to the direction of incident light are taken as negative”, heights above the axis positive and below it negative. Section 9.5.1 is refraction at a single spherical surface (Equation 9.16) and 9.5.2 is refraction by a lens, whose result is Equation (9.21), the lens maker’s formula. The printed body of Equation 9.21 did not survive text extraction from this copy — the page shows the equation number with nothing beside it — so it is not quoted here as a verbatim equation; the formula on this page is derived from scratch in the worked example instead. What did read, and is quoted, is the sign note “By the sign convention, BC₁ = + R₁, DC₂ = –R₂” and the statement that for a concave lens “R₁ is negative, R₂ positive and therefore, f is negative”, which is the check the second golden test on this page performs. Section 9.5.3 gives P = 1/f with 1 D = 1 m⁻¹ and says power is “positive for a converging lens and negative for a diverging lens”. Citing the chapter is not a claim that any board has approved this page.
HyperPhysics, Department of Physics and Astronomy, Georgia State University, Lens-Maker’s Formula. States the thing this page’s surface-power rows are built on: “For a thin lens, the power is approximately the sum of the surface powers”, which is the lens maker’s equation rearranged and is why F₁ + F₂ on this page comes out exactly equal to n_m/f. It uses the Cartesian convention and spells out the case that confuses everyone: “For a double convex lens the radius R₁ is positive since it is measured from the front surface”, while R₂ is negative because it extends leftward from the second surface. On the medium it says only that if the lens is in something other than air, such as water, “its lens strength will be diminished”. The formula itself is published on that page as an image, so it is not quoted here as text, and the version implemented on this page is derived in the worked example rather than copied.
Central Board of Secondary Education, Physics (Code No. 042), Classes XI–XII, Senior Secondary Curriculum 2025-26, read from cbseacademic.nic.in. Unit VI Optics, Chapter 9 Ray Optics lists the lens maker’s formula by name, alongside the thin lens formula, magnification, power of a lens, thin lenses in contact and refraction through a prism. Chapter 10 Wave Optics lists Huygens’ principle, the proof of the laws of reflection and refraction, coherent sources, interference, “Young’s double slit experiment and expression for fringe width (No derivation final expression only)” and “diffraction due to a single slit, width of central maxima (qualitative treatment only)” — and then stops. “Polarisation”, “Malus” and “Brewster” appear nowhere in the Optics unit. The whole document was searched: the only occurrence of the word at all is “Dielectrics and electric polarization” in Unit I, which is electrostatics and a different thing. This is reported because it is load-bearing: on that document the lens maker’s formula is named examinable content and the two polarisation laws are not.
The refractive indices on these pages, and how far each can be trusted. n = 1.5 for glass is the figure the Cornell lecture above uses for its own worked Brewster angle, so it is sourced. n = 1.333 for water at visible wavelengths is the figure in common tabulation and this batch did not reach a primary standards publication for it; it is consistent with the Cornell example to the precision that example is quoted at. Real glass is not one number: crown glass, flint glass and polycarbonate run from roughly 1.5 to 1.6 and every one of them disperses, so the index you should use is the one the question gives you. Water’s index falls slightly as it warms and rises towards the blue end of the spectrum, which is why a Brewster angle quoted to four figures is false precision. The indices used in the comparison tables are marked in the table itself with where each came from, and the ones that could not be sourced say so rather than being dressed up.