Calculators / Calculus 3 / 3D Section Formula Calculator

3D Section Formula Calculator

Published on: January 25, 2026

This 3D Section Formula Calculator helps you find the coordinates of the point that divides the line segment between two points in three-dimensional space in a given ratio. It can be used for both internal division and external division. Substitute the coordinates and ratio into the correct 3D section formula, then simplify each coordinate to get the final point. It is a simple way to check answers, understand the formula clearly, and practise coordinate geometry step by step.

Step-by-step method

  1. Identify x₁, y₁, z₁, x₂, y₂, z₂, the ratio, and whether it is internal or external division.
  2. Write the correct 3D section formula.
  3. Substitute the values into the formula.
  4. Simplify to get the final dividing point.

Formula (Internal division):

P = (
mx2 + nx1
m + n
, 
my2 + ny1
m + n
, 
mz2 + nz1
m + n
)

Formula (External division):

P = (
mx2 − nx1
m − n
, 
my2 − ny1
m − n
, 
mz2 − nz1
m − n
)

Example 1: (1,2,3),(4,6,3),2:1

Step 1 - Identify the inputs.

In this problem: Ratio = 2 : 1, Type = Internal division

x1 = 1, y1 = 2, z1 = 3
x2 = 4, y2 = 6, z2 = 3

Step 2 - Write the formula.

In this problem: Use the 3D section formula for internal division.

P = (
mx2 + nx1
m + n
, 
my2 + ny1
m + n
, 
mz2 + nz1
m + n
)

Step 3a - Substitute the values into the formula.

In this problem: Replace x₁, y₁, z₁, x₂, y₂, z₂, m, and n with the given values.

P = (
2·4 + 1·1
2 + 1
, 
2·6 + 1·2
2 + 1
, 
2·3 + 1·3
2 + 1
)

Step 3b - Evaluate the numerator and denominator.

In this problem: After evaluating, no further simplification is needed.

P = (
9
3
, 
14
3
, 
9
3
)

Step 4 - Simplify and state the final answer.

In this problem: This is the required dividing point.

P = ( 3, 
14
3
, 3 )

Final answer: ( 3, 14/3, 3 )

Example 2: (1,0,2),(5,4,6),3:1,external

Step 1 - Identify the inputs.

In this problem: Ratio = 3 : 1, Type = External division

x1 = 1, y1 = 0, z1 = 2
x2 = 5, y2 = 4, z2 = 6

Step 2 - Write the formula.

In this problem: Use the 3D section formula for external division.

P = (
mx2 − nx1
m − n
, 
my2 − ny1
m − n
, 
mz2 − nz1
m − n
)

Step 3a - Substitute the values into the formula.

In this problem: Replace x₁, y₁, z₁, x₂, y₂, z₂, m, and n with the given values.

P = (
3·5 − 1·1
3 − 1
, 
3·4 − 1·0
3 − 1
, 
3·6 − 1·2
3 − 1
)

Step 3b - Evaluate the numerator and denominator.

In this problem: After evaluating, no further simplification is needed.

P = (
14
2
, 
12
2
, 
16
2
)

Step 4 - Simplify and state the final answer.

In this problem: This is the required dividing point.

P = ( 7, 6, 8 )

Final answer: ( 7, 6, 8 )