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Math Coach Amy
Amy Ferguson Moncure

IB AA · 4B Three-Dimensional Coordinate Geometry

Distance in Three Dimensions

Add the squared change in z, then take the square root.

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Key Ideas Easy Medium Hard

d = square root of (x₂ − x₁)² + (y₂ − y₁)² + (z₂ − z₁)²

What to Remember

Formula

d = square root of (x₂ − x₁)² + (y₂ − y₁)² + (z₂ − z₁)²

Same steps as the plane

Subtract, square, add, then simplify the square root.

A zero change

If one coordinate is the same for both points, that term is 0.

Exact form

Leave a simplified radical when the sum of squares is not a perfect square.

Easy

Easy

The distance is a whole number.

Example
Find the exact distance between (0, 0, 0) and (3, 6, 6).

d = square root of (x₂ − x₁)² + (y₂ − y₁)² + (z₂ − z₁)²

d = square root of (3 − 0)² + (6 − 0)² + (6 − 0)²

d = square root of 9 + 36 + 36

d = square root of 81

d = 9

Answer: 9

1
Find the exact distance between (0, 0, 0) and (2, 3, 6).
Show Answer

d = square root of (x₂ − x₁)² + (y₂ − y₁)² + (z₂ − z₁)²

d = square root of (2 − 0)² + (3 − 0)² + (6 − 0)²

d = square root of 4 + 9 + 36

d = square root of 49

d = 7

Answer: 7

2
Find the exact distance between (1, 2, 2) and (4, 6, 14).
Show Answer

d = square root of (x₂ − x₁)² + (y₂ − y₁)² + (z₂ − z₁)²

d = square root of (4 − 1)² + (6 − 2)² + (14 − 2)²

d = square root of 9 + 16 + 144

d = square root of 169

d = 13

Answer: 13

3
Find the exact distance between (0, 0, 0) and (1, 4, 8).
Show Answer

d = square root of (x₂ − x₁)² + (y₂ − y₁)² + (z₂ − z₁)²

d = square root of (1 − 0)² + (4 − 0)² + (8 − 0)²

d = square root of 1 + 16 + 64

d = square root of 81

d = 9

Answer: 9

4
Find the exact distance between (−2, 0, 1) and (4, 3, 7).
Show Answer

d = square root of (x₂ − x₁)² + (y₂ − y₁)² + (z₂ − z₁)²

d = square root of (4 − (−2))² + (3 − 0)² + (7 − 1)²

d = square root of 36 + 9 + 36

d = square root of 81

d = 9

Answer: 9

5
Find the exact distance between (2, −1, 0) and (5, 3, 4).
Show Answer

d = square root of (x₂ − x₁)² + (y₂ − y₁)² + (z₂ − z₁)²

d = square root of (5 − 2)² + (3 − (−1))² + (4 − 0)²

d = square root of 9 + 16 + 16

d = square root of 41

Answer: square root of 41

Medium

Medium

One coordinate does not change.

Example
Find the exact distance between (1, 1, 1) and (3, 5, 2).

d = square root of (x₂ − x₁)² + (y₂ − y₁)² + (z₂ − z₁)²

d = square root of (3 − 1)² + (5 − 1)² + (2 − 1)²

d = square root of 4 + 16 + 1

d = square root of 21

Answer: square root of 21

1
Find the exact distance between (0, 4, −1) and (6, 4, 7).
Show Answer

d = square root of (x₂ − x₁)² + (y₂ − y₁)² + (z₂ − z₁)²

d = square root of (6 − 0)² + (4 − 4)² + (7 − (−1))²

d = square root of 36 + 0 + 64

d = square root of 100

d = 10

Answer: 10

2
Find the exact distance between (3, 0, 2) and (3, 5, 14).
Show Answer

d = square root of (x₂ − x₁)² + (y₂ − y₁)² + (z₂ − z₁)²

d = square root of (3 − 3)² + (5 − 0)² + (14 − 2)²

d = square root of 0 + 25 + 144

d = square root of 169

d = 13

Answer: 13

3
Find the exact distance between (−1, 2, 2) and (2, 6, 5).
Show Answer

d = square root of (x₂ − x₁)² + (y₂ − y₁)² + (z₂ − z₁)²

d = square root of (2 − (−1))² + (6 − 2)² + (5 − 2)²

d = square root of 9 + 16 + 9

d = square root of 34

Answer: square root of 34

4
Find the exact distance between (4, 1, −2) and (6, 4, 4).
Show Answer

d = square root of (x₂ − x₁)² + (y₂ − y₁)² + (z₂ − z₁)²

d = square root of (6 − 4)² + (4 − 1)² + (4 − (−2))²

d = square root of 4 + 9 + 36

d = square root of 49

d = 7

Answer: 7

5
Find the exact distance between (1, −1, 2) and (5, 1, 5).
Show Answer

d = square root of (x₂ − x₁)² + (y₂ − y₁)² + (z₂ − z₁)²

d = square root of (5 − 1)² + (1 − (−1))² + (5 − 2)²

d = square root of 16 + 4 + 9

d = square root of 29

Answer: square root of 29

Hard

Hard

Simplify the square root.

Example
Find the exact distance between (0, 0, 1) and (4, 4, 8).

d = square root of (x₂ − x₁)² + (y₂ − y₁)² + (z₂ − z₁)²

d = square root of (4 − 0)² + (4 − 0)² + (8 − 1)²

d = square root of 16 + 16 + 49

d = square root of 81

d = 9

Answer: 9

1
Find the exact distance between (2, 3, −1) and (6, 6, 11).
Show Answer

d = square root of (x₂ − x₁)² + (y₂ − y₁)² + (z₂ − z₁)²

d = square root of (6 − 2)² + (6 − 3)² + (11 − (−1))²

d = square root of 16 + 9 + 144

d = square root of 169

d = 13

Answer: 13

2
Find the exact distance between (−3, 1, 4) and (1, 4, 6).
Show Answer

d = square root of (x₂ − x₁)² + (y₂ − y₁)² + (z₂ − z₁)²

d = square root of (1 − (−3))² + (4 − 1)² + (6 − 4)²

d = square root of 16 + 9 + 4

d = square root of 29

Answer: square root of 29

3
Find the exact distance between (5, 0, 2) and (5, 8, 8).
Show Answer

d = square root of (x₂ − x₁)² + (y₂ − y₁)² + (z₂ − z₁)²

d = square root of (5 − 5)² + (8 − 0)² + (8 − 2)²

d = square root of 0 + 64 + 36

d = square root of 100

d = 10

Answer: 10

4
Find the exact distance between (1, 2, 2) and (7, 5, 6).
Show Answer

d = square root of (x₂ − x₁)² + (y₂ − y₁)² + (z₂ − z₁)²

d = square root of (7 − 1)² + (5 − 2)² + (6 − 2)²

d = square root of 36 + 9 + 16

d = square root of 61

Answer: square root of 61

5
Find the exact distance between (−2, −2, 1) and (2, 1, 13).
Show Answer

d = square root of (x₂ − x₁)² + (y₂ − y₁)² + (z₂ − z₁)²

d = square root of (2 − (−2))² + (1 − (−2))² + (13 − 1)²

d = square root of 16 + 9 + 144

d = square root of 169

d = 13

Answer: 13

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