Objects in Mirror Are Closer Than They Appear

The passenger-side outside rear-view mirror of a car is inscribed with “Caution: objects in mirror are closer than they appear”. What is the reason that objects appear farther in the mirror than they are? The distance that an object appears to be from an observer is determined by the angle that the object (or its image) subtends. The outside mirror is convex as you will soon find out by looking closely at it, or by the analyzing the following calculation.

Let’s compare the angle subtended by the image of an object formed by a convex mirror as opposed to a flat mirror. The upper diagram below shows the image formed of an object through a concave mirror. The image is the stumpy little black arrow to the left of the mirror is the image of the larger black arrow to the right. The magenta arrow on the left is the image formed by a plane mirror. The lower figure shows the eye of an observer and the perceived angular height of the image formed by the curved mirror compared with that of a plane mirror. The curved mirror image reveals a smaller angle, and therefore it appears to be further away. Notice that this is in spite of the fact that the image is actually closer to the observer than it is for a plane mirror. The fact is, the apparent distances and sizes of objects (or images) are gauged by the angular size and by their sizes relative to each other, not by their actual distances and sizes.

We can easily calculate the angular size of the image formed by the curved mirror, assuming the observer is a distance d from the mirror:

(The first minus sign is simply to make the angle be positive). A plane mirror corresponds to . The second term in the denominator vanishes giving the angular size of the object seen through a plane mirror. A convex mirror has a negative focal length, and therefore the angular size is less than that for a plane mirror, as qualitatively concluded above.