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Last updated on August 26, 2025
In geometry, the 3D distance formula is used to determine the distance between two points in a three-dimensional space. It is a fundamental concept used to solve various problems involving spatial relations. In this topic, we will learn the formula for calculating the distance between two points in 3D space.
The 3D distance formula is essential for finding spatial relationships between points in three-dimensional space.
Let’s learn the formula to calculate this distance.
The 3D distance formula calculates the distance between two points \((x_1, y_1, z_1)\) and \((x_2, y_2, z_2)\) in a three-dimensional space.
It is given by: \[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2} \]
In math and real life, the 3D distance formula helps analyze and understand spatial relationships. Here are some important points about its usage:
- It is used in various fields such as physics, engineering, and computer graphics to calculate distances in 3D space.
- Understanding this formula helps solve problems related to geometry and spatial analysis.
- It is crucial for applications involving 3D modeling and design.
Students may find the 3D distance formula challenging to remember. Here are some tips and tricks to master it:
- Visualize the concept by imagining or drawing a 3D coordinate system.
- Relate it to the 2D distance formula by adding the third dimension.
- Use mnemonic devices or practice problems regularly to reinforce memory.
The 3D distance formula plays a major role in various real-life applications. Here are some examples:
- In architecture, it is used to determine the distances between structural points in 3D models.
- In navigation, it helps calculate the shortest path between locations in 3D space.
- In computer graphics, the formula is essential for rendering scenes and calculating object positions.
Students make errors when using the 3D distance formula. Here are some mistakes and ways to avoid them, to master it.
What is the distance between the points (1, 2, 3) and (4, 5, 6)?
The distance is 5.2
Substitute the coordinates into the 3D distance formula: \[ d = \sqrt{(4 - 1)^2 + (5 - 2)^2 + (6 - 3)^2} = \sqrt{9 + 9 + 9} = \sqrt{27} \approx 5.2 \]
Find the distance between the points (7, 8, 9) and (10, 11, 12).
The distance is 5.2
Using the formula: \[ d = \sqrt{(10 - 7)^2 + (11 - 8)^2 + (12 - 9)^2} = \sqrt{9 + 9 + 9} = \sqrt{27} \approx 5.2 \]
Calculate the distance between the points (-1, -2, -3) and (-4, -5, -6).
The distance is 5.2
Substituting into the formula: \[ d = \sqrt{(-4 + 1)^2 + (-5 + 2)^2 + (-6 + 3)^2} = \sqrt{9 + 9 + 9} = \sqrt{27} \approx 5.2 \]
Determine the distance between (2, 3, 4) and (5, 6, 7).
The distance is 5.2
Using the 3D distance formula: \[ d = \sqrt{(5 - 2)^2 + (6 - 3)^2 + (7 - 4)^2} = \sqrt{9 + 9 + 9} = \sqrt{27} \approx 5.2 \]
Jaskaran Singh Saluja is a math wizard with nearly three years of experience as a math teacher. His expertise is in algebra, so he can make algebra classes interesting by turning tricky equations into simple puzzles.
: He loves to play the quiz with kids through algebra to make kids love it.