How can focal length be equal to radius of curvature?

Hello!

It is not clear whether you speak about a (spherical) mirror or about a lens. Let's consider both options.

1. For a spherical mirror with the radius of curvature R the focal length is equal (up to a sign) to the half of `R,` i.e. `R/2.` Therefore the radius of curvature and the focal length cannot be equal.

2. There are more degrees of freedom for a lens with spherical surfaces: the radius of curvature of the second lens' side and the index of refraction (which depends on the lens' material).

Lensmaker's equation says that for a thin double convex lens in air

`1/(f) approx (n-1)*(1/R_1+1/R_2),`

where `ngt1` is the refractive index of a material.

(here `R_1` and `R_2` are considered positive to not think about the sign convention).

If `f=R_1` as you ask then  `1/(R_1)*(2-n)=1/R_2,`

so for `1ltnlt2` it is possible for `R_2=R_1/(2-n).`

One can also consider non-thin lens.