103 LearnersLast updated on July 4th, 2025
Distance Formula
A distance formula helps us find the length of the straight line connecting two points. The distance between two points is just the length of the line segment joining them. It is based on the Pythagorean theorem and works in both two-dimensional and three-dimensional spaces. In co-ordinate geometry, it helps calculate distance between two points on the XY plane.
List of Math Formulas for Distance
To determine the distance between two points, there are a few formulas that children should learn. Let’s look at the formulas related to distance:
Formula 1: Distance Formula in a 2D plane
Distance = √((x2 -x1)2 + (y2 - y1)2)
This formula is used to find the straight line distance between two points (x1 , y1) and (x2 ,y2) on a two-dimensional plane.
For example, Points A (3 , 4) and B (7 , 1) = Distance = (7 -3)2 + (1 - 4)2 = 5
Formula 2: Distance Formula in a 3D plane
Distance = √((x2 -x1)2 + (y2 - y1)2 + (z2-z1)2)
This formula is used to find the straight line distance between two points (x1 , y1 , z1) and (x2 , y2 , z2) in a three-dimensional plane.
For example, Points A(1 , 2 , 3) and B(4 , 6 , 8) = Distance = '√((4 -1)2 + (6 - 2)2+(8-3)2) = 7.07
Formula 3: Distance between a point and a line (2D)
Distance = Ax1 + By1+C/'√(A2 + B2)
This formula is used to find the shortest distance between a point (x1 , y1) and a line given by the equation Ax + By + C =0.
For Example, Point P(1 , 2) and line 3x-4y+5=0 ; Distance 3(1)-4(2)+5/ '√(32+(-4)2) = 1.
Formula 4: Distance between two parallel lines
Distance = C2 - C3/ '√(A2+B2)
This formula is used to calculate the perpendicular distance between two parallel lines Ax + By + C1 = 0 and Ax + By + C2 = 0.
Importance of Distance Math Formulas
Distance formulas are very useful in math and everyday life. They make solving problems easier and help us think logically.
- Distance formulas help us calculate the shortest distance between two points in 2D and 3D spaces.
- Understanding distance formulas is essential for learning advanced topics in mathematics, such as vectors, trigonometry, calculus, and coordinate geometry.
- Distance formulas are widely used in real-world applications, like navigation (GPS systems), architecture, urban planning, and robotics.
- Distance formulas are not limited to mathematics; they’re also used in physics (motion, force) , computer graphics (game development), and data science.
Tips and Tricks to Memorize Distance Math Formulas
Memorizing math formulas can sometimes be tricky, but with the right methods, it becomes easier and more manageable. When kids use these tips and tricks , they will not only memorize formulas faster but also be able to apply them confidently to solve problems. Let’s look at some ways to make students understand better:
- Instead of memorizing the formula, understand the formula’s concept. For 2D distance formula √((x2 -x1)2 + (y2 - y1)2) , think of it as finding the length of the hypotenuse of a right triangle formed by the differences in the x and y coordinates.
- Break the formula into smaller parts. Focus first on understanding the differences in the x and y coordinates, then square them, and finally add them.
Common Mistakes and How to Avoid Them While Using Distance Math Formulas
While learning distance formulas, students often make a few common mistakes. Understanding these mistakes and how to avoid them can help students get a better understanding of distance formulas. Let’s look at a few common mistakes and how to avoid them.
Mistake 1
Forgetting to square the differences
Children sometimes forget to square the difference between the x and y coordinates. Children should understand the formula and square the difference of x and y coordinates before adding them. For example, For points A (1 , 2) and B (4 , 6), the correct formula is: instead of writing √((4 -1)2 + (6 - 2)2) = √(9 + 16) = √25 = 5 , they write it as(4 -1) + (6 - 2) = √(3 + 4) = √7.
Mistake 2
Incorrect use of the Pythagorean Theorem
Confusing the Pythagorean theorem’s application when using the distance formula, especially when calculating the distance between points in different dimensions (2D or 3D). For example, For points A(2, 1, 3) and B(5, 6, 8), apply the 3D distance formula:
Distance = (5-2)2 + (6-1)2+(8-3)2 = 9+25+25 = 59.
Mistake 3
Using wrong formula for distance from a point to a line
Children use the general formula instead of the specific formula for the distance from a point to a line. They should be using the correct formula for a specific coordinate. For example, the correct formula is Ax1 + By1+CA2 + B2.
Mistake 4
Forgetting to simplify square roots
Students sometimes forget to simplify square roots. They should always simplify square roots when possible. For example, 9= 3, not just leaving it as 9.
Mistake 5
Confusing the sign of the result.
Students ignore the value when applying formulas, which leads to negative distances. They should use absolute value when calculating distances. For example, instead of writing, 3(1) -4(2)+5/ √ (32 + (-4)2)=1 they write 3(1) -4(2)+5 / √(32 + (-4)2) =-1.
Examples of Problems Using Distance Math Formulas
Problem 1
Find the distance between the points P (3,4) and Q (7,1) in a 2D plane.
The distance between the points P(3 , 4) and Q(7 , 1) is 5 units.
Explanation
Use the 2D distance formula:
D= √((x2 -x1)2 + (y2 - y1)2)
Substitute the coordinates of points P(3 , 4) and Q (7 , 1):
√((7 -3)2 + (1 - 4)2) = √(42 + (-3)2)
√(16 + 9) = √25 = 5.
Problem 2
Find the distance between the points A(1, 2, 3) and B(4, 6, 8) in 3D space.
The distance between A (1, 2, 3) and B (4, 6, 8) is approximately 7.07 units.
Explanation
Use the 3D distance formula:
D= √((x2 -x1)2 + (y2 - y1)2 + (z2-z1)2)
Substitute the coordinates of points A (1, 2 , 3) and B (4 , 6, 8):
√ ((4-1)2 + (6-2)2+(8-3)2) = √ (9+ 16+ 25)
= √ 50 = 7.07.
Problem 3
Find the distance between P (0 , 0) and Q (6 , 8) on a straight line.
The distance between P (0 , 0) and Q ( 6, 8) is approximately 10 units.
Explanation
D = √((6-0)2 +(8-0)2) = √(36+64) = √100 = 10
Problem 4
Find the distance from A (3 , 4) to the origin (0 , 0).
The distance between A(3 , 4) to the origin (0, 0) 5 units.
Explanation
D = √((3)2 +(4)2) = √(9+16) = √25 = 5
FAQs on Distance Math Formulas
1.What is the distance of the point 5,9 from Y axis?
2.What happens if the two points are the same?
3.Can the distance formula be used for points on a straight line?
4.What is the distance between the points (-1,-2) and (3,4)?
5.How can children in Australia use numbers in everyday life to understand Distance Formula ?
6.What are some fun ways kids in Australia can practice Distance Formula with numbers?
7.What role do numbers and Distance Formula play in helping children in Australia develop problem-solving skills?
8.How can families in Australia create number-rich environments to improve Distance Formula skills?
Glossary for Distance Math Formulas
- Coordinates: A pair in 2D and a triplet in 3D of values that define the position of a point on a coordinate plane or space. In 2D, coordinates are written as (x,y) and in 3D as (x, y, z).
- Origin: The point where the x-axis and y-axis intersect. In 2D, the origin is at (0, 0) and in 3D is at (0, 0, 0).
- Straight Line: The shortest distance between two points
Explore More numbers
![Important Math Links Icon]()
Previous to Distance Formula
Hiralee Lalitkumar Makwana
About the Author
Hiralee Lalitkumar Makwana has almost two years of teaching experience. She is a number ninja as she loves numbers. Her interest in numbers can be seen in the way she cracks math puzzles and hidden patterns.
Fun Fact
: She loves to read number jokes and games.
