# Determining a spherical mirror setup just from the image and object heights

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!

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.