# You are standing 8 feet from a flat mirror. In the mirror, you hold a ruler that is exactly 12-inches long. How long will the image of the ruler be in the mirror?

Oct 13, 2014

The ruler will appear to be 16 ft away. It will look just the same as a 12-inch ruler would look if you moved 16 ft away from it.

You've been asked a question that is a bit of a riddle. And it isn't clear what your teacher had in mind. It is simple to calculate the angular size of the ruler by imagining a circle at a 16 foot radius from your eye. The circumference of the circle will be 2$\pi$r. And you can determine the fraction of that circle that the ruler would fill and calculate an angle.

circumference = $2 \pi \cdot 16 = 100.5$ft

So the ruler will appear to your line of site to have an angular size of about $\frac{1}{100}$th of a circle, or about 3.6 degrees. That's roughly the apparent size of the width of three fingers held at arm's length. The human body is a useful reference here because A) everybody has one and B) the length of your arm and the size of your hand are in roughly the same proportion.

For comparison, the apparent angular size of the moon is about 0.5 degrees. You can easily block it from your field of view with a single finger held at arm's length.

So, what do you want to compare the ruler to? It will look like every other ruler held at a 16 ft distance. It will be a similar size to your hand at arm's length - about 2 or 3 inches. But it will be much smaller than that same hand held just in front of your eye. Try it -- peek through your fingers if you need to. Your hand can cover your entire field of view (more than 180 degrees) when held just in front of your eye. If we compare to a ruler held near your eye, the size of the distant ruler may be measured in fractions of a millimeter.

The question is flawed, you can not an answer in terms of a length measurement. The best you can do is answer with the apparent angular size.