How to Find the Distance From a Point to a Plane

The distance from a plane to a single point is actually the combination of three distances. The distance between the point's and the plane's x-, y- and z-coordinates work together to calculate the total distance. The point's coordinates are always given as a set, with the coordinates placed alphabetically. The plane's coordinates are given in its equation, with each coordinate placed in front of its respective variable. Finding the distance from a point to a plane is accomplished by a specialized distance equation.

Instructions

    • 1

      Plug the coordinates of the point into the equation for the plane and solve. For example, the point is (2, 2, 3) and the equation is 4x - 4y + 7z + 6 = 0. Plugging the coordinates into the equation results in 4(2) - 4(2) - 7(3) - 6. Solving the equation results in 8 - 8 - 21 - 6, which is -27.

    • 2

      Calculate the absolute value for that number. The absolute value is the positive amount of that number, which means for negative numbers that the negative sign is omitted; for positive numbers, no action is necessary. For the example, -27 becomes 27.

    • 3

      Square each of the coefficients of the plane's equation, and add those squares together. For the example, the coefficients are 4, 4 and 7. Squaring these numbers results in 16, 16 and 49. Adding them together equals 81.

    • 4

      Calculate the square root of that number, and then divide it into the number from Step 2. For the example, the square root of 81 is 9, and 27 divided by 9 equals 3. For the example, the distance from the point to the plane is 3.

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