The Lens Maker's Formula

A lens has no numbers stamped on it. You are handed a disc of glass with two curved faces and asked: where will it focus sunlight? The last article gave you the tool to answer — refraction at one spherical surface — and a lens is simply two such surfaces, front and back, with a sliver of glass between. Send light through the first surface, let the image it would have formed become the object for the second surface, and the intermediate image cancels out. What survives is one of the most quietly powerful equations in optics, the lens maker's formula:

\frac{1}{f} = (n - 1)\!\left(\frac{1}{R_1} - \frac{1}{R_2}\right)

It says the focal length of a thin lens is fixed by three things you can measure with calipers and a refractometer: the refractive index $n$ of the glass, and the radii of curvature $R_1$ and $R_2$ of its two faces. Nothing about the object, nothing about where you put it — this is the lens's own property, computed before any light is used. It is why Fraunhofer, the greatest lens-maker of the nineteenth century, could grind a telescope objective to a calculated focal length rather than guessing and testing. Two consequences make this more than an accounting formula. First, the signs of $R_1$ and $R_2$ decide whether the lens converges ( $f > 0$ ) or diverges ( $f < 0$ ) — a biconvex lens gathers light, a biconcave one spreads it. Second, because $f$ scales as $1/(n-1)$ , the same lens has a different focal length in water than in air — and a glass lens can even flip from converging to diverging if you surround it with something denser than itself. This page derives the formula from the single-surface result, uses it on every shape a board can ask about, and shows exactly why an air bubble in water is a diverging lens.

Key takeaways

The lens maker's formula $\dfrac{1}{f} = (n-1)\!\left(\dfrac{1}{R_1} - \dfrac{1}{R_2}\right)$ builds a thin lens's focal length from its glass index $n$ and its two radii $R_1$ , $R_2$ — a property of the lens alone.
It is derived by applying the single-surface equation twice (front then back), letting the intermediate image cancel.
Signs (New Cartesian): a biconvex lens has $R_1 > 0$ , $R_2 < 0$ , so the bracket adds and $f > 0$ (converging); a biconcave lens gives $f < 0$ (diverging).
Anchor: equiconvex crown glass, $n = 1.5$ , $R_1 = +20$ , $R_2 = -20$ cm gives $\dfrac{1}{f} = 0.5\!\left(\dfrac{1}{20}+\dfrac{1}{20}\right) = 0.05$ , so $f = +20$ cm.
Because $f \propto 1/(n-1)$ , the surrounding medium matters: the same lens is four times weaker in water, and a "convex" air bubble in water actually diverges light.
Key idea: a lens's focal length is manufactured, not measured — set by the glass and the two curvatures through $1/f = (n-1)(1/R_1 - 1/R_2)$ , and it changes when you change the medium around it.

The core insight

Applying the single-surface refraction equation to the front and back faces of a thin lens and cancelling the intermediate image yields the lens maker's formula $1/f = (n-1)(1/R_1 - 1/R_2)$ , so a lens's focal length is determined by its glass index and its two radii — and by the medium it sits in, since $f$ scales as $1/(n-1)$ .

Where this lives today

The lens maker's formula is the first line of every optical-design program on Earth. When an engineer at a camera company specifies a lens, they begin exactly here — choosing a glass index and two radii to hit a target focal length — and only then hand the residual aberrations to a computer that stacks a dozen such elements. Spectacle lenses are cut to a prescribed power by choosing $R_1$ and $R_2$ for a given plastic index; a high-index plastic ( $n = 1.67$ ) lets an optician grind a thinner, flatter lens for the same power, which is why strong prescriptions no longer mean thick "bottle-bottom" glasses. Contact lenses and the intraocular lenses implanted during cataract surgery are computed with the immersed form of the formula, since they sit against the eye's watery media, not in air — an ophthalmologist who used the in-air focal length would leave the patient badly out of focus. And the whole art of the achromatic doublet — pairing a converging crown lens with a diverging flint lens so their colour errors cancel while their powers add — is Fraunhofer's craft, still made today by choosing indices and radii through this one equation, with chromatic aberration (a coming chapter) as the thing it is built to defeat.

The physical picture

A lens is not a new optical object; it is a sandwich of two spherical surfaces you already understand. Trace the light in two stages. It meets the first surface (air-to-glass) and, by the single-surface equation, heads toward an image $I_1$ somewhere — real or virtual, in general inside or beyond the glass. But before it can get there it meets the second surface (glass-to-air), which refracts it again. The trick that makes the lens simple is this: the image the first surface was trying to form, $I_1$ , becomes the object for the second surface. Because the lens is thin — its two faces are separated by a gap small compared with every other distance — we measure both surfaces' object and image distances from essentially the same point, the optical centre, and the intermediate distance to $I_1$ appears once with a plus and once with a minus. It cancels. Two surface equations collapse into one lens equation. What is left, $1/f = (n-1)(1/R_1 - 1/R_2)$ , contains only what the lens is: the glass, through $n-1$ , and the shape, through the two curvatures. This is why the formula is called the lens maker's formula and not the lens user's formula — it is the equation a manufacturer uses to decide what to grind, long before anyone places an object in front of it. The user's equation, which locates the image once the lens exists, is the thin-lens formula of the next article.

Figure

A thin biconvex lens is two spherical surfaces back to back: the front face has radius R1 positive with its centre of curvature C1 on the far side, the back face has radius R2 negative with its centre C2 on the near side, and the light converges to the focus F at distance f. The lens maker formula reads one over f equals n minus one times one over R1 minus one over R2.

The formula, and the sign that makes a lens

Two surface equations, added and simplified (the derivation does it in full), give the whole of it:

\frac{1}{f} = (n - 1)\!\left(\frac{1}{R_1} - \frac{1}{R_2}\right) \qquad (1)

where: $n$ is the refractive index of the lens glass relative to its surroundings; $R_1$ is the radius of curvature of the surface the light meets first, $R_2$ of the surface it meets second; and $f$ is the focal length of the thin lens, the same on both sides because the lens sits in one medium. Every $R$ carries its New Cartesian sign — positive if that surface's centre of curvature is on the outgoing-light side, negative if not. The factor $(n-1)$ , not $n$ , is what does the bending: only the contrast between glass and surroundings refracts light, which is the seed of every "lens in a different medium" surprise below. The sign convention is once more the New Cartesian convention of The Mirror Formula — but note the shape of equation (1) against the mirror's. A mirror obeys $\dfrac{1}{v} + \dfrac{1}{u} = \dfrac{1}{f}$ with a plus; a lens's image equation (next article) is $\dfrac{1}{v} - \dfrac{1}{u} = \dfrac{1}{f}$ with a minus and $m = v/u$ rather than the mirror's $m = -v/u$ . The lens maker's formula here computes the $f$ that then feeds that minus-sign image equation. Keep the convention shared, the equations distinct. The signs of $R_1$ and $R_2$ are the whole personality of the lens. Take a biconvex lens, bulging out on both sides: the front surface is convex to the incoming light, so its centre of curvature is beyond it and $R_1 > 0$ ; the back surface is convex too, but its centre lies back toward the object, so $R_2 < 0$ . Then $\dfrac{1}{R_1} - \dfrac{1}{R_2}$ becomes a sum of two positive terms, $\dfrac{1}{f} > 0$ , and the lens converges. A biconcave lens reverses both signs, the bracket goes negative, $f < 0$ , and the lens diverges. This is why the humble minus sign in equation (1) matters: with $R_2 < 0$ it turns into a plus, and a symmetric convex lens gets the full bending power of both faces.

Derivation AThe lens maker's formula from two surfaces

The goal is to derive $\dfrac{1}{f} = (n-1)\!\left(\dfrac{1}{R_1} - \dfrac{1}{R_2}\right)$ by applying the single-surface equation of the previous article to the front and back faces of a thin lens in air, and to watch the intermediate image cancel. Nothing new is needed — only the surface equation used twice, plus the thin-lens promise that both faces sit at effectively the same place. This is the payoff of having done one surface carefully.

Step 1. Refract at the first surface. Light in air ( $n_1 = 1$ ) meets the front face (radius $R_1$ ) and enters the glass ( $n_2 = n$ ). If the object is at $u$ , the single-surface equation gives the intermediate image at $v_1$ : $\dfrac{n}{v_1} - \dfrac{1}{u} = \dfrac{n - 1}{R_1}$ . This $v_1$ is where the light would focus if the glass went on forever.

Step 2. Refract at the second surface. The glass does not go on forever; the light reaches the back face (radius $R_2$ ) and re-enters air. The intermediate image $I_1$ now acts as the object for this surface. Because the lens is thin, we measure from the same optical centre, so this object is at distance $v_1$ inside the glass. The surface equation, now glass-to-air ( $n_1 = n$ , $n_2 = 1$ ), reads $\dfrac{1}{v} - \dfrac{n}{v_1} = \dfrac{1 - n}{R_2}$ , where $v$ is the final image distance.

Step 3. Add the two equations. The awkward middle term $n/v_1$ appears as $+n/v_1$ in Step 1's result and as $-n/v_1$ in Step 2's — so adding the two equations cancels it exactly: $\dfrac{1}{v} - \dfrac{1}{u} = \dfrac{n - 1}{R_1} + \dfrac{1 - n}{R_2} = (n - 1)\!\left(\dfrac{1}{R_1} - \dfrac{1}{R_2}\right)$ . The intermediate image has vanished; only the object, the final image, and the lens's own properties remain.

Step 4. Read off $f$ . By definition the focal length is the image distance for an object at infinity: set $u \to -\infty$ , so $1/u \to 0$ and $v \to f$ . Then $\dfrac{1}{f} = (n-1)\!\left(\dfrac{1}{R_1} - \dfrac{1}{R_2}\right)$ , the lens maker's formula. A quick test: equiconvex $n = 1.5$ , $R_1 = +20$ , $R_2 = -20$ gives $1/f = 0.5(0.05 + 0.05) = 0.05$ , so $f = +20$ cm. The very same addition, kept general with $u$ finite, has already handed us the thin-lens formula $\dfrac{1}{v} - \dfrac{1}{u} = \dfrac{1}{f}$ of the next article — the two results are one derivation. Verified: the equiconvex $f = +20$ cm and every shape in the worked table are recomputed in verify_hs2-lens-makers-formula.py.

Go deeper · Derivation B · 1 minWhy the same lens is weaker in water

Derivation BWhy the same lens is weaker in water

The goal is to prove that a lens's focal length depends on the medium around it, and to get the factor $f_{\text{water}}/f_{\text{air}} = (n-1)/(n/n_w - 1)$ that made the equiconvex lens four times weaker underwater. This is the most exam-worthy consequence of the formula and the one students find most counter-intuitive, because a lens looks the same in air and in water.

Step 1. See where the medium enters. In the derivation above, the " $n$ " in every step was really the glass index relative to the surroundings — the ratio $n_2/n_1$ at each surface. In air that ratio is $n/1 = n$ ; in a medium of index $n_m$ it is $n/n_m$ . So equation (1) is properly $\dfrac{1}{f} = \left(\dfrac{n}{n_m} - 1\right)\!\left(\dfrac{1}{R_1} - \dfrac{1}{R_2}\right)$ , and the radii never change.

Step 2. Take the ratio. Divide the underwater version by the in-air version; the radius bracket cancels, leaving $\dfrac{f_{\text{water}}}{f_{\text{air}}} = \dfrac{n - 1}{\,n/n_w - 1\,}$ . For crown glass in water, $\dfrac{0.5}{\,1.5/(4/3) - 1\,} = \dfrac{0.5}{0.125} = 4$ .

Step 3. Read the physics. Underwater the glass-to-medium contrast shrinks from $0.5$ to $0.125$ , so the lens bends light only a quarter as hard, and to bend a given ray by less it needs four times the focal length. Nothing about the glass changed — only how different it is from what surrounds it. This is also why swimmers are long-sighted underwater: the eye's own lens, tuned to air, loses most of its power once water replaces the air in front of the cornea.

Step 4. Take it to the sign flip. If $n_m > n$ the factor $n/n_m - 1$ turns negative and $f$ changes sign — a converging glass lens becomes diverging. An air bubble in water is the clean example: $n/n_m = 1/(4/3) = 0.75$ , factor $-0.25$ , so a bubble curved like a convex lens diverges light. The lens maker's formula predicts it exactly. Verified: the ratio $f_{\text{water}}/f_{\text{air}} = 4$ and the air-bubble sign flip are recomputed in verify_hs2-lens-makers-formula.py.

Figure

The same equiconvex crown-glass lens focuses at 20 cm in air but at 80 cm in water, because the bending factor drops from n minus 1 equal to 0.5 to n over n-water minus 1 equal to 0.125, a fourfold weakening, and the radii never change.

Limiting cases

Case	Result	Meaning
biconvex, $R_1 > 0$ , $R_2 < 0$	$1/f > 0$ , $f > 0$	converging — the bracket adds two positive terms
biconcave, $R_1 < 0$ , $R_2 > 0$	$1/f < 0$ , $f < 0$	diverging
plano-convex, $R_2 \to \infty$	$1/f = (n-1)/R_1$	one surface bends; a flat face contributes nothing
$n \to 1$ (glass matches medium)	$1/f \to 0$ , $f \to \infty$	no contrast, no lens — light passes straight through
immersed in medium $n_m$	$f \propto 1/(n/n_m - 1)$	weaker in water ( $\times 4$ ); sign flips if $n_m > n$
equal radii $R_1 = R_2$	$1/f = 0$	a "lens" of uniform thickness does not focus

Common trapThe signs of the radii — and the minus that becomes a plus

Every wrong answer on the lens maker's formula is a sign mistake on $R_1$ or $R_2$ , and the trap is baited by the minus sign in $(1/R_1 - 1/R_2)$ . For a biconvex lens the second surface has $R_2 < 0$ , so the term $-1/R_2$ becomes $+1/|R_2|$ , and the two faces add their bending — feed in a positive $R_2$ by reflex and you will compute a focal length far too long, or the wrong sign entirely. Decide each radius from where its centre of curvature sits: on the outgoing side, $R > 0$ ; on the incoming side, $R < 0$ . A biconvex lens is $(R_1 > 0,\ R_2 < 0)$ ; a biconcave is $(R_1 < 0,\ R_2 > 0)$ ; a plano face is $R = \infty$ . Two more habits protect you. Use $(n - 1)$ , never $n$ — only the contrast bends light. And when a problem changes the surrounding medium, change only the factor to $(n/n_m - 1)$ and leave the radii alone; the glass has not been reground.

Worked examples

Board level — five lens shapes from one formula (CBSE and NEET pattern)

A crown-glass lens has $n = 1.5$ . Find its focal length when it is (a) equiconvex with $R_1 = +20$ cm, $R_2 = -20$ cm; (b) equiconvex with radii $\pm 10$ cm; (c) plano-convex with $R_1 = +15$ cm and a flat back; (d) biconcave with $R_1 = -20$ cm, $R_2 = +20$ cm.

Step 1. Write equation (1) once and feed each shape its signed radii. Throughout, $n - 1 = 0.5$ .

Step 2. Part (a), equiconvex $\pm 20$ : $\dfrac{1}{f} = 0.5\!\left(\dfrac{1}{20} - \dfrac{1}{-20}\right) = 0.5\!\left(\dfrac{1}{20} + \dfrac{1}{20}\right) = 0.5 \times 0.1 = 0.05\ \mathrm{cm^{-1}}$ , so $f = \boxed{+20\ \mathrm{cm}}$ — converging. Notice the minus in the formula became a plus because $R_2$ is negative.

Step 3. Part (b), radii $\pm 10$ : halving both radii doubles the bending, $\dfrac{1}{f} = 0.5\!\left(\dfrac{1}{10} + \dfrac{1}{10}\right) = 0.1\ \mathrm{cm^{-1}}$ , so $f = \boxed{+10\ \mathrm{cm}}$ . Part (c), plano-convex with a flat back: a flat surface has $R_2 = \infty$ , so $1/R_2 = 0$ and $\dfrac{1}{f} = 0.5 \times \dfrac{1}{15} = \dfrac{1}{30}$ , giving $f = \boxed{+30\ \mathrm{cm}}$ — only one surface bends, so it is weaker.

Step 4. Part (d), biconcave $R_1 = -20$ , $R_2 = +20$ : $\dfrac{1}{f} = 0.5\!\left(\dfrac{1}{-20} - \dfrac{1}{20}\right) = 0.5 \times (-0.1) = -0.05\ \mathrm{cm^{-1}}$ , so $f = \boxed{-20\ \mathrm{cm}}$ — diverging, the negative sign flagging it. Same glass, same radius magnitudes as (a), opposite curvatures, opposite lens.

Competitive level — a meniscus, and the same lens in water (JEE pattern)

A meniscus lens (convex front, concave back) of crown glass $n = 1.5$ has $R_1 = +10$ cm and $R_2 = -15$ cm. (a) Find its focal length in air. (b) Now immerse it in water ( $n_w = 4/3$ ). By what factor does its focal length change?

Step 1. In air: $\dfrac{1}{f} = (1.5 - 1)\!\left(\dfrac{1}{10} - \dfrac{1}{-15}\right) = 0.5\!\left(\dfrac{1}{10} + \dfrac{1}{15}\right) = 0.5\!\left(0.1 + 0.0\overline{6}\right) = 0.5 \times 0.1\overline{6} = 0.083\overline{3}\ \mathrm{cm^{-1}}$ , so $f = \boxed{+12\ \mathrm{cm}}$ — converging, despite one concave face, because the convex front is the tighter curve.

Step 2. In water, the index in the formula is the glass index relative to the surroundings: $n_{\text{rel}} = n/n_w = 1.5/(4/3) = 1.125$ , so the bending factor becomes $n_{\text{rel}} - 1 = 0.125$ instead of $0.5$ .

Step 3. Only the factor changes, not the radii, so focal lengths scale inversely with it: $\dfrac{f_{\text{water}}}{f_{\text{air}}} = \dfrac{n - 1}{n_{\text{rel}} - 1} = \dfrac{0.5}{0.125} = \boxed{4}$ . The lens becomes four times weaker in water — its focal length grows from $12$ cm to $48$ cm — because the glass-to-water contrast is only a quarter of the glass-to-air contrast.

Step 4. Push the idea to its limit. If the surrounding medium were denser than the glass, $n_{\text{rel}} < 1$ , the factor $n_{\text{rel}} - 1$ would go negative, and every sign of $f$ would flip: a converging lens would diverge. An air bubble ( $n = 1$ ) in water ( $n_w = 4/3$ ) has $n_{\text{rel}} = 0.75$ , factor $-0.25$ — so a bubble shaped like a convex lens actually spreads light. That is why divers see the world through swarms of tiny diverging lenses. Timing note: about 90 seconds for both parts once you see that immersion touches only the factor $(n-1)$ , never the radii. The single competitive skill is the ratio $f_{\text{water}}/f_{\text{air}} = (n-1)/(n/n_w - 1)$ ; memorise its shape and the medium-change question is one line.

Exam traps

Sign of $R_2$ . The most common lost mark. A biconvex lens has $R_2 < 0$ , which turns the formula's minus into a plus. Read each radius's sign from the position of its centre of curvature, not from the word "convex".

Using $n$ instead of $(n-1)$ . Only the contrast between glass and surroundings refracts light. The factor is $(n-1)$ in air, $(n/n_m - 1)$ in a medium — never a bare $n$ .

Forgetting the medium. Focal length is not a fixed property of the glass; it depends on what surrounds the lens. In water a crown-glass lens is four times weaker; in a denser medium it can reverse. If a question mentions immersion, the factor changes.

A plano surface is $R = \infty$ , not $R = 0$ . A flat face has infinite radius, so $1/R = 0$ and it contributes nothing to the bending. Writing $R = 0$ (a point) is a different, impossible object.

Board scope. The lens maker's formula is examined by CBSE, ICSE, JEE Main, JEE Advanced and NEET. No mainstream A-Level derives $f$ from the radii — A-Level, AP Physics 2 and IGCSE treat focal length as a measured property of a given lens and never compute it from $n$ , $R_1$ , $R_2$ . So this derivation is Indian-syllabus territory; if you sit only those other boards, you will use focal lengths but never build them, and this page is enrichment that explains where they come from.

Origins · 1 minJoseph von Fraunhofer and how this was discovered

Joseph von Fraunhofer

1787–1826 · Munich

Fraunhofer was the artisan who made the lens maker's formula pay. Orphaned at eleven and apprenticed to a Munich mirror-maker, he was buried in the 1801 collapse of the workshop and dug out by a rescue party that included the Elector of Bavaria, who then funded his education. He became the finest optical craftsman of his age, grinding achromatic telescope objectives to computed curvatures, pairing crown and flint glasses of measured index so their focal lengths combined to the value he wanted. To do it he measured refractive index more precisely than anyone before, and so charted the dark lines in the Sun's spectrum that still bear his name. The formula on this page is classical geometry; Fraunhofer is the physicist-craftsman who turned it into instruments. His spectral lines belong to the coming wave-optics chapter.

History

1611

— Johannes Kepler, in Dioptrice, works out the elementary theory of lenses and the astronomical telescope.

1637

— René Descartes, in La Dioptrique, analyses the ideal lens surface that focuses light without aberration.

1690

— Christiaan Huygens grinds his own lenses and combines them into compound eyepieces for his telescopes.

1814

— Joseph von Fraunhofer measures refractive indices to new precision and grinds achromatic objectives to calculated focal lengths.

1873

— Ernst Abbe founds the modern theory of lens design at Zeiss, extending the paraxial formula to real, aberration-corrected optics.

Going further

Texts. NCERT Physics Class 12, Ch. 9 ("Ray Optics and Optical Instruments"), §9.5 "Refraction by a Lens" and the lens maker's formula · H. C. Verma, Concepts of Physics Vol. 1, "Geometrical Optics" · D. C. Pandey, Optics and Modern Physics, lens maker's formula problems.

Provenance. Standard paraxial optics, $n = 1.5$ unless noted. Recompute: equiconvex $R = \pm 20 \to f = +20$ cm; equiconvex $R = \pm 10 \to f = +10$ cm; plano-convex $R_1 = +15$ , flat $\to f = +30$ cm; biconcave $R_1 = -20$ , $R_2 = +20 \to f = -20$ cm; meniscus $R_1 = +10$ , $R_2 = -15 \to f = +12$ cm. Medium change: $f_{\text{water}}/f_{\text{air}} = (n-1)/(n/n_w - 1) = 0.5/0.125 = 4$ , so the equiconvex $f$ grows $20 \to 80$ cm and the meniscus $12 \to 48$ cm; air bubble in water gives factor $n/n_w - 1 = 0.75 - 1 = -0.25$ , a sign flip.

Prerequisites. Refraction at a Spherical Surface — the single-surface equation this page applies twice. Leads to. The Thin Lens Formula, which uses the $f$ built here to locate images; and Lens Power & Magnification, where $1/f$ becomes the dioptre.

Go deeper (UG). Why real lens-makers pair two glasses — the achromatic doublet that cancels colour error while adding power — is developed in Chromatic Aberration and Achromats in the university optics track. It is enrichment; every result on this page follows from the single-surface equation applied twice.

Which surface is R1 and which is R2?

$R_1$ is the surface the light reaches first, $R_2$ the surface it reaches second — decided by the direction the light travels, not by which side you happen to call the front. If you flip the lens end to end, $R_1$ and $R_2$ swap, but because the formula is symmetric in a certain way the focal length comes out the same for a thin lens, which is why a thin lens works equally well either way round. What must not change is that each radius keeps the sign set by its own centre of curvature. Draw the light going left to right, label the first face $R_1$ and the second $R_2$ , and sign each from where its centre sits.

Why is it (n minus 1) and not just n?

Because only the difference between the glass and its surroundings bends light. If the glass had the same index as the air around it, a ray would pass straight through with no refraction at all — there would be no lens, however curved the surfaces. The bending at each surface is proportional to the index jump across it, which is $n - 1$ for glass in air (or $n/n_m - 1$ in a medium). Put another way, a lens works because glass is optically different from air; the size of that difference, $n - 1$ , is exactly how hard it works. A glass with $n = 1.5$ has a contrast of $0.5$ ; double the contrast and you halve the focal length.

Does the lens maker's formula tell me where the image is?

No — and this is the key distinction. The lens maker's formula computes the lens's focal length $f$ from its glass and shape; it never mentions an object or an image. To find where a given object is imaged you take that $f$ to the thin-lens formula, $1/v - 1/u = 1/f$ , in the next article. Think of it as two stages: first the maker's formula builds the lens (gives $f$ ), then the thin-lens formula uses the lens (gives $v$ ). Many students blur the two and try to put an object distance into the maker's formula; there is nowhere for it to go. One equation makes the tool, the other wields it.

What is a meniscus lens, and can it still converge?

A meniscus lens is curved the same way on both faces — convex on one side, concave on the other, like a shallow bowl. Whether it converges or diverges depends on which curve is tighter. In the worked example, $R_1 = +10$ cm (a sharp convex front) beats $R_2 = -15$ cm (a gentler concave back), so the bracket is positive and $f = +12$ cm — converging. Most spectacle and camera lenses are meniscus-shaped because the bent form reduces aberrations, yet they still converge or diverge exactly as the formula says. Do not assume "has a concave face, therefore diverges"; compute the bracket.

How can a flat surface be part of the formula? It has no radius.

A flat surface is the limit of a spherical one whose radius has grown infinite — a sphere so large that any small patch of it looks flat. So a plano face has $R = \infty$ , and $1/R = 0$ : it contributes nothing to the bending. A plano-convex lens is then just its one curved surface doing all the work, which is why the worked example gives it a longer focal length ( $30$ cm) than the equiconvex lens ( $20$ cm) of the same glass and radius. Do not write $R = 0$ for a flat face — that would mean a point, an infinitely tight curve, the exact opposite of flat.

If I grind a lens more steeply, does it always get stronger?

Yes, within the thin-lens picture: a smaller radius of curvature (a tighter, steeper face) means a larger $1/R$ , a larger $1/f$ , and hence a shorter focal length — a stronger lens. Halving both radii of the equiconvex lens took its focal length from $20$ cm to $10$ cm. But there are limits in practice. Steeper surfaces bend outer rays too much and worsen spherical aberration, and a very short focal length in a thick lens breaks the thin-lens approximation this formula assumes. Real strong lenses are therefore built from several gently curved elements rather than one steep one — but the trend, steeper means stronger, is exactly what the formula predicts.