= Lensmaker's equation
The <lensmaker's equation> connects <focal length> to the <refractive index> and shape of a <thin lens>. For a <thin lens> in air with surfaces shaped as parts of <spheres>, <refractive index> $n$, and incident <light> travelling to the right,
$$
\frac1f=(n-1)\left(\frac1{R_a}-\frac1{R_b}\right).
$$
Each signed <radius of curvature> is positive when its centre of <curvature> lies to the right of that surface, and negative when it lies to the left. A symmetric biconvex <thin lens> with $n=1.5$, $R_a=1\,\mathrm m$, and $R_b=-1\,\mathrm m$ therefore has $1/f=0.5(1+1)\,\mathrm{m^{-1}}=1\,\mathrm{m^{-1}}$, giving $f=1\,\mathrm m$ and <optical power> $+1\,\mathrm D$. This is an ideal shape example consistent with the <thin lens> diagrams, not a specification of their finite thickness.
In a uniform surrounding medium of <refractive index> $n_s$ on both sides, replace $n-1$ by $n/n_s-1$. Thus the same shape reverses its focusing sign if its <refractive index> is lower than its surroundings: for $n=1.5$ and $n_s=2$, the same radii give $1/f=-0.5\,\mathrm{m^{-1}}$, or $f=-2\,\mathrm m$. Positive <focal length> therefore cannot be inferred from convex shape in every environment. A proof outline applies <Snell's law> with the <small-angle approximation> at each surface, adds their direction changes, and neglects the distance between them. https://www.feynmanlectures.caltech.edu/I_27.html[Feynman, Leighton and Sands, The Feynman Lectures on Physics I, §27-3, equations 27.10–27.12]{ref}.
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