If you're seeing this message, it means we're having trouble loading external resources on our website.

If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.

## Basic geometry and measurement

### Unit 13: Lesson 4

Pythagorean theorem and distance between points

# Distance formula

Walk through deriving a general formula for the distance between two points.
The start color #11accd, start text, d, i, s, t, a, n, c, e, end text, end color #11accd between the points left parenthesis, start color #1fab54, x, start subscript, 1, end subscript, end color #1fab54, comma, start color #e07d10, y, start subscript, 1, end subscript, end color #e07d10, right parenthesis and left parenthesis, start color #1fab54, x, start subscript, 2, end subscript, end color #1fab54, comma, start color #e07d10, y, start subscript, 2, end subscript, end color #e07d10, right parenthesis is given by the following formula:
square root of, left parenthesis, start color #1fab54, x, start subscript, 2, end subscript, minus, x, start subscript, 1, end subscript, end color #1fab54, right parenthesis, squared, plus, start color #e07d10, left parenthesis, y, start subscript, 2, end subscript, minus, y, start subscript, 1, end subscript, end color #e07d10, right parenthesis, squared, end square root

## Deriving the distance formula

Let's start by plotting the points left parenthesis, start color #1fab54, x, start subscript, 1, end subscript, end color #1fab54, comma, start color #e07d10, y, start subscript, 1, end subscript, end color #e07d10, right parenthesis and left parenthesis, start color #1fab54, x, start subscript, 2, end subscript, end color #1fab54, comma, start color #e07d10, y, start subscript, 2, end subscript, end color #e07d10, right parenthesis.
The first quadrant of a coordinate plane with two tick marks on the x axis labeled x one and x two. There are two tick marks on the y axis labeled y one and y two. There is a point at x one, y one and another point at x two, y two.
The length of the segment between the two points is the start color #11accd, start text, d, i, s, t, a, n, c, e, end text, end color #11accd between them:
The first quadrant of a coordinate plane with two tick marks on the x axis labeled x one and x two. There are two tick marks on the y axis labeled y one and y two. There is a point at x one, y one and another point at x two, y two. A line connects the two points.
We want to find the start color #11accd, start text, d, i, s, t, a, n, c, e, end text, end color #11accd. If we draw a right triangle, we'll be able to use the Pythagorean theorem!
The first quadrant of a coordinate plane with two tick marks on the x axis labeled x one and x two. There are two tick marks on the y axis labeled y one and y two. There is a point at x one, y one and another point at x two, y two. A line connects the two points. A third unlabeled point is at x two, y one with a line connecting from it to the point at x two, y two and another line connecting from it to the point at x one, y one forming a right triangle.
An expression for the length of the base is start color #1fab54, x, start subscript, 2, end subscript, minus, x, start subscript, 1, end subscript, end color #1fab54:
The first quadrant of a coordinate plane with two tick marks on the x axis labeled x one and x two. There are two tick marks on the y axis labeled y one and y two. There is a point at x one, y one and another point at x two, y two. A line connects the two points. A third unlabeled point is at x two, y one with a line connecting from it to the point at x two, y two and another line connecting from it to the point at x one, y one forming a right triangle. The hypotenuse of the right triangle is unknown and the side made from the point at x one, y one and x two, y one is labeled x two minus x one.
Similarly, an expression for the length of the height is start color #e07d10, y, start subscript, 2, end subscript, minus, y, start subscript, 1, end subscript, end color #e07d10:
The first quadrant of a coordinate plane with two tick marks on the x axis labeled x one and x two. There are two tick marks on the y axis labeled y one and y two. There is a point at x one, y one and another point at x two, y two. A line connects the two points. A third unlabeled point is at x two, y one with a line connecting from it to the point at x two, y two and another line connecting from it to the point at x one, y one forming a right triangle. The hypotenuse of the right triangle is unknown and the side made from the point at x one, y one and x two, y one is labeled x two minus x one. The third side is labeled y two minus y one.
Now we can use the Pythagorean theorem to write an equation:
start color #11accd, question mark, end color #11accd, squared, equals, left parenthesis, start color #1fab54, x, start subscript, 2, end subscript, minus, x, start subscript, 1, end subscript, end color #1fab54, right parenthesis, squared, plus, start color #e07d10, left parenthesis, y, start subscript, 2, end subscript, minus, y, start subscript, 1, end subscript, end color #e07d10, right parenthesis, squared
We can solve for start color #11accd, question mark, end color #11accd by taking the square root of each side:
start color #11accd, question mark, end color #11accd, equals, square root of, left parenthesis, start color #1fab54, x, start subscript, 2, end subscript, minus, x, start subscript, 1, end subscript, end color #1fab54, right parenthesis, squared, plus, start color #e07d10, left parenthesis, y, start subscript, 2, end subscript, minus, y, start subscript, 1, end subscript, end color #e07d10, right parenthesis, squared, end square root
That's it! We derived the distance formula!
Interestingly, a lot of people don't actually memorize this formula. Instead, they set up a right triangle, and use the Pythagorean theorem whenever they want to find the distance between two points.

## Want to join the conversation?

• How do you find the distance between two points if it is just a line?
• To find the distance between to points if it is just a line, you simply draw a dot where the line ends, then you make a number line and find the coordinates for both points. Finally, you follow the distance formula, plug the values in, and solve.

I hope this helps and wasn't a bore!
• I still don't understand any of this... :I
(1 vote)
• Are you alive?
• what is the formula that is used to find distance between two points
• bro are u crazy its right above you in the beginning of the lesson
• I prefer the straight up: draw a right triangle, use pythagorean theorem method. Anyone agree?
• The distance formula is an application of the Pythagorean Theorem, so what are you doing differently that makes it seem different to you? Are you just finding the x2-x1 and y2-y1 from the graph before putting it into the Pythagorean Theorem?
• who came up with this formula?
(1 vote)
• .Nevertheless, the theorem came to be credited to Pythagoras. It is also proposition number 47 from Book I of Euclid's Elements. According to the Syrian historian Iamblichus (c. 250–330 ce), Pythagoras was introduced to mathematics by Thales of Miletus and his pupil Anaximander. Brittanica.com states “ Pythagorean theorem, the well-known geometric theorem that the sum of the squares on the legs of a right triangle is equal to the square on the hypotenuse (the side opposite the right angle)—or, in familiar algebraic notation, a2 + b2 = c2. Although the theorem has long been associated with Greek mathematician-philosopher Pythagoras (c. 570–500/490 BCE), it is actually far older. Four Babylonian tablets from circa 1900–1600 BCE indicate some knowledge of the theorem, with a very accurate calculation of the square root of 2 (the length of the hypotenuse of a right triangle with the length of both legs equal to 1) and lists of special integers known as Pythagorean triples that satisfy it (e.g., 3, 4, and 5; 32 + 42 = 52, 9 + 16 = 25). The theorem is mentioned in the Baudhayana Sulba-sutra of India, which was written between 800 and 400 BCE. Nevertheless, the theorem came to be credited to Pythagoras.” I know that’s a lot of words but I had to do it...even tho this isn’t ELA class 😀😂. Hope this helps rose!
• Sooooo, if I have two points, (1, 2) and (-1, 4), it does not matter in which order I subtract as long as I do the x with the x, and so on? Because it doesn't look that way.
(1 vote)
• Yes it doesn't really matter, because you end up squaring it and ending up with a positive number.
• what is one of the points are a fraction? like how would that work
• You would still use the distance formula or the Pythagorean theorem, except that you would perform the arithmetic with the fractions. Remember that when you square a fraction, you need to square both top and bottom; when you take the square root of a fraction, you need to take the square root of both top and bottom.

Have a blessed, wonderful day!