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How to Derive a Distance Formula in Three-Space

To derive a distance formula in three-dimensional space, you must use a simple formula--twice. Think of going to the 70th floor of a building in New York. From your current location, you go five blocks in one direction, take a left, then go nine blocks in the new direction to reach the building. Fortunately, the elevator is on the outside of the building, and it's only a foot from the street, which simplifies the illustration. Now think triangles.

Things You'll Need

  • Calculator
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Instructions

  1. First Triangle

    • 1

      Calculate the distance back to the starting point from the door. Multiply 7 by itself. Multiply 5 by itself. Add the products like this: 49 + 25 =74. Take what is called the "square root."

    • 2

      Get a calculator either online or a physical device. Type in "74." Press the square root sign. Type in "square root sign" on a search engine to find the math symbol. Look for that symbol on the calculator.

    • 3

      Write the result, which is 8.60. Record that you have calculated the "hypotenuse" of a "right triangle." Remember that the "right angle" is the left turn you made, which was a 90-degree turn.

    Second Triangle

    • 4

      Calculate a second right triangle, and simply use the distance 8.60 from the first right triangle as one of the sides. Estimate the distance to the 70th floor to be about four blocks.

    • 5

      Calculate the distance from the 70th floor to your starting point by using the right triangle formula. Multiply 8.6 by itself. 8.6 X 8.6 = 73.96. Multiply 4 by itself. 4 X 4 = 16. Add the two results. 73.96+16=89.96. Calculate the square root of 89.96, which is 9.48. Record this number as the distance in three-dimensional space between your starting point and a second point on the 70th floor.

    • 6

      Memorize the right triangle formula.

    • 7

      Understand that it applies for the second triangle because of the right angle of 90 degrees you turn in 3-D space when you go up the elevator.

    • 8

      Record that you can always calculate distance in space between 3-D space by calculating two right triangles. Record that the distance you seek is the hypotenuse of the second right triangle.

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