# Practical Math - Hanging a 48" high mirror at the proper height

How high must a 48-inch high mirror be hung if one who is six feet tall needs to see his entire image from approximately 6 feet away from the mirror?

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I think this should be moved to physics.stackexchange.com because it involves the physics of mirror images. –  000 Dec 7 '12 at 11:25
@Limitless: I think in mathematics we can get away with assuming that the angle of incidence equals the angle of reflection, without worrying particularly about the physical reasons why it is so. –  Henning Makholm Dec 7 '12 at 12:02
@HenningMakholm My physics knowledge is not that great, and I did not want to make assumptions. That is why I was concerned; I was erring on the side of caution. –  000 Dec 7 '12 at 12:07
@Limitless: not migrating is erring on the side of caution :-) –  robjohn Dec 7 '12 at 12:16
@robjohn Thanks for clarifying! :-) –  000 Dec 7 '12 at 12:17

No matter what the horizontal distance between the user and the mirror, the bottom edge of the mirror must be no higher than half the height of the user's eyes. If so, the ray of vision starting from the user's eyes and bouncing off the mirror right at the bottom edge will hit the ground just where his feet are. The horizontal distance is not important because the mirror is always exactly halfway between you and your image behind the looking-glass.

Since 48" is 121 cm, well above half of most people's height, there's a range of positions that will suffice, and you can choose among those by aesthetic preferences. I think it would would probably do well to hang the mirror such that its top edge is about a hand's width above the top of the tallest user's head. Unless that will make the bottom so high that the shortest users will have difficulty seeing themselves completely -- in which case lower it, but not more than such that the top edge is a few centimeters below the top of the tallest user's head.

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nitpick: the user of the mirror is 6-foot tall, so we don't need to consider "most people's height". Also, technically one is making the assumption that when the mirror is hung it is hung vertically, and that the mirror is planar. Both are often leveraged (especially in shoe stores) to get different heights and smaller mirrors. –  Willie Wong Dec 7 '12 at 12:21
So, $2$ feet is the required height