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I am working through some questions to prepare for an upcoming exam and got stumped by a question involving a spherical mirror where we are only provided with the image height (1 cm), object height (0.2 cm) and that the image is erect.

I believe the image must be virtual to be smaller and erect and hence a convex lens must be used but from this information is there any way to calculate the radius of the lens?

Using basic trigonometry I can relate the ratio of the image and object heights to the ratio of the distances but this is not enough to solve for the radius of the lens.

I found $$\tan(\theta) = \frac{oh}{f} = \frac{ih}{s'}$$ Where $oh$ is the object height and $ih$ is the image height. $f$ is focal length and $s'$ the object distance but I have too many unknowns to be able to solve for $r$ in $$\frac{1}{s} + \frac{1}{s'} = \frac{1}{f} = \frac{2}{r}$$

Is there some other information I am missing? Thanks in advance!

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You are correct. You don't have enough information to know the focal length of the mirror (or its radius). All you do know is the magnification: $m=h_{image}/h_{object}$. This also determines the ratio of the image and object distance. However, there are many solutions which will achieve this: The radius of the mirror could be 50 mm, which implies the object and image distances are 100 mm and 20 mm, respectively.

But if we just take any system with magnification = 0.2, and scale all lengths and focal lengths by a given factor, it will make another system with magnification = 2.

It is ambiguous to use terms erect or inverted. Those terms were commonly used when telescopes, used by humans, were the optical system most people had interaction with. It is better to say the system forms a real image or a virtual image. This is less ambiguous. Certain prism systems will invert images, but that is separate from the question of forming real or virtual images.

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