How to find distance between the parallel planes

How to find distance between the parallel planes

At the solution of geometrical and practical tasks sometimes it is required to find distance between the parallel planes. So, for example, height of the room is, actually, a distance between a ceiling and a floor which represent the parallel planes. An example of the parallel planes are also opposite walls, covers of the book, a wall of boxes and many other things.

It is required to you

  • - ruler;
  • - drawing triangle with a right angle;
  • - calculator;
  • - compasses.


1. To find distance between two parallel planes: • draw the straight line perpendicular to one of the plane; • define points of intersection of this straight line from each of the planes; • measure distances between these points.

2. To draw the straight line perpendicular to the plane, use the following method borrowed from descriptive geometry: • choose any point on the plane; • draw through this point two the crossed straight lines; • construct a straight line perpendicular to at the same time both crossed straight lines.

3. If the parallel planes are located horizontally, for example, a floor and a ceiling of the house, then for measurement of distance use a plumb. For this purpose: • take a thread, length obviously bigger the measured distance; • tie a small small weight to one of its ends; • throw thread through the tack or a wire located near a ceiling, or hold thread with a finger; • lower a small weight until it does not concern a floor; • record a thread point when the small weight falls to a floor (for example, tie a small knot); • measure distance between a mark and the end of thread with cargo.

4. If the planes are set by the analytical equations, then find distance between them as follows: • let A1*kh + V1*u + C1*z + D1 = 0 and A2*kh + V2*u + C2*z + D2 = 0 – the equations of the planes in space; • as for the parallel planes multipliers at coordinates are equal, rewrite these equations in the following look: A*kh + V*u + C*z + D1 = 0 and A*kh + V*u + C*z + D2 = 0; • use the following formula for finding of distance between these parallel planes: s = |D2-D1| / √ (A²+B²+C²), where: | | - standard designation of the module (absolute value) of expression.

5. Example: Define distance between the parallel planes set by the equations: the 6th +6u-3z+10=0 and the 6th +6u-3z+28=0. Decision: Substitute parameters from the equations for the planes in the above-stated formula. It will turn out: s = |28-10| / √ (6²+6²+ (-3)²) = 18 / √ 81 = 18/9 = 2. Answer: Distance between the parallel planes – 2 (units of measure).

Author: «MirrorInfo» Dream Team