How to Use Distance Formula to Find the Length of a Line

Reviewed by Grace Imson, MA

Last Updated: September 29, 2024 Fact Checked

You can measure the length of a vertical or horizontal line on a coordinate plane by simply counting coordinates; however, measuring the length of a diagonal line is trickier. You can use the Distance Formula to find the length of such a line. This formula is basically the Pythagorean Theorem, which you can see if you imagine the given line segment as the hypotenuse of a right triangle. ^{[1]
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Research source} By using a basic geometric formula, measuring lines on a coordinate path becomes a relatively easy task.

Part 1

Part 1 of 2:

Setting up the Formula

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The formula states that $d={\sqrt {(x_{{2}}-x_{{1}})^{{2}}+(y_{{2}}-y_{{1}})^{{2}}}}$ , where $d$ equals the distance of the line, $(x_{{1}},y_{{1}})$ equal the coordinates of the first endpoint of the line segment, and $(x_{{2}},y_{{2}})$ equal the coordinates of the second endpoint of the line segment. ^{[2]
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2
Find the coordinates of the line segment’s endpoints. These might already be given. If not, count along the x-axis and y-axis to find the coordinates. ^{[3]
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- The x-axis is the horizontal axis; the y-axis is the vertical axis.
- The coordinates of a point are written as $(x,y)$ .
- For example, a line segment might have an endpoint at $(2,1)$ and another at $(6,4)$ .
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3
Plug the coordinates into the Distance Formula. Be careful to substitute the values for the correct variables. The two $x$ coordinates should be inside the first set of parentheses, and the two $y$ coordinates should be inside the second set of parentheses. ^{[4]
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- For example, for points $(2,1)$ and $(6,4)$ , your formula would look like this: $d={\sqrt {(6-2)^{{2}}+(4-1)^{{2}}}}$

Part 2

Part 2 of 2:

Calculating the Distance

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1
Calculate the subtraction in parentheses. By using the order of operations, any calculations in parentheses must be completed first. ^{[5]
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- For example:
  $d={\sqrt {(6-2)^{{2}}+(4-1)^{{2}}}}$
  $d={\sqrt {(4)^{{2}}+(3)^{{2}}}}$
2
Square the value in parentheses. The order of operations states that exponents should be addressed next. ^{[6]
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- For example:
  $d={\sqrt {(4)^{{2}}+(3)^{{2}}}}$
  $d={\sqrt {16+9}}$
3
Add the numbers under the radical sign. You do this calculation as if you were working with whole numbers. ^{[7]
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- For example:
  $d={\sqrt {16+9}}$
  $d={\sqrt {25}}$
4
Solve for $d$ . To reach your final answer, find the square root of the sum under the radical sign. ^{[8]
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- Since you are finding a square root, you may have to round your answer.
- Since you are working on a coordinate plane, your answer will be in generic “units,” not in centimeters, meters, or another metric unit.
- For example:
  $d={\sqrt {25}}$
  $d=5$ units

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What do we call points that are on same line?

Donagan

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Points on the same straight line are said to be "collinear" or "colinear."

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How do I simplify the square root?

Donagan

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Look for a perfect square inside the radical sign, find its square root, and put that square root out in front of the radical sign, indicating that it's to be multiplied by the radical. For example: √50 = √(2 x 25) = 5√2.

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The rise is 4 inches. The angle is 90 degrees from the height to the base. What is the slope?

Donagan

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To find the slope, you have to know the horizontal distance over which the rise occurs. The 90° angle is not relevant.

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Remember the order of operations when calculating your answer. Subtract first, then square the differences, then add, and then find the square root.

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Do not confuse this formula with others, like the Midpoint Formula, Slope Formula, Equation of a Line or Line Formula.

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How to Use Distance Formula to Find the Length of a Line

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Setting up the Formula

Calculating the Distance

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