Formulas · Mathematics

Three-dimensional geometry

3 formulas, each with worked examples. Revision sheet · Practise these formulas

Class 11

Distance between two points in space

d = (x_{2} - x_{1})^{2} + (y_{2} - y_{1})^{2} + (z_{2} - z_{1})^{2}

What each letter means
$d$	the distance between the points P(x₁, y₁, z₁) and Q(x₂, y₂, z₂)
$x_{1}$	the x-coordinate of P
$y_{1}$	the y-coordinate of P
$z_{1}$	the z-coordinate of P
$x_{2}$	the x-coordinate of Q
$y_{2}$	the y-coordinate of Q
$z_{2}$	the z-coordinate of Q

The distance between two points in space is the square root of the sum of the squares of the differences of their three coordinates.

Why it works

Go from P to Q in two moves: first across the floor (the x and y differences), which is √((x₂ − x₁)² + (y₂ − y₁)²) long by the flat distance formula, then straight up by z₂ − z₁. The two moves are at right angles, so by Pythagoras their squares add once more.

When to use it

Lengths of segments in space, the diagonal of a box, and testing whether points lie on one line or make a given triangle.

How to use it

Take away the coordinates of P from those of Q, one pair at a time.
Square the three differences and add them.
Take the square root.

To remember: The flat distance formula with one more square under the root.

Other forms

Special case $d = x^{2} + y^{2} + z^{2}$ When P is the origin, and Q is (x, y, z).

Worked examples

Story

A drone rises from (0, 0, 0) to (6, 2, 3), in metres. How far is it from where it started?

$6^{2} + 2^{2} + 3^{2} = 49$
$49 = 7$

Answer: 7

Picture

Find the length of the long diagonal of a box 3 cm by 4 cm by 12 cm.

$9 + 16 + 144 = 13$

Answer: 13

Direct

Find the distance between (1, −3, 4) and (−4, 1, 2).

$(- 4 - 1)^{2} + (1 + 3)^{2} + (2 - 4)^{2} = 45$
$45 = 35$

Answer: 6.708203932499369

Reverse

The point (3, 4, z), with z positive, is 13 units from the origin. Find z.

Try it first, then show the working

$3^{2} + 4^{2} + 1 2^{2} = 13$

Answer: 12

Exam

Show that (−2, 3, 5), (1, 2, 3) and (7, 0, −1) lie on one line.

Try it first, then show the working

AB = √14, BC = √56 = 2√14 and AC = √126 = 3√14.
$14 + 214 = 314$
AB + BC = AC, so the three points lie on one line.

Answer: AB + BC = AC, so they are collinear

Common mistake: Forgetting the z term, or squaring before taking away (x₂² − x₁² is not (x₂ − x₁)²).

Class 11

Section formula in space

P = (\frac{m _{1} x _{2} + m _{2} x _{1}}{m _{1} + m _{2}}, \frac{m _{1} y _{2} + m _{2} y _{1}}{m _{1} + m _{2}}, \frac{m _{1} z _{2} + m _{2} z _{1}}{m _{1} + m _{2}})

What each letter means
$P$	the point that divides AB in the ratio m1 : m2 (AP : PB = m1 : m2)
$m_{1}$	the first part of the ratio (the share next to A)
$m_{2}$	the second part of the ratio (the share next to B)
$x_{1}$	the x-coordinate of A
$y_{1}$	the y-coordinate of A
$z_{1}$	the z-coordinate of A
$x_{2}$	the x-coordinate of B
$y_{2}$	the y-coordinate of B
$z_{2}$	the z-coordinate of B

The point that splits AB in the ratio m1 : m2 is found one coordinate at a time, with the same weighted average as on a flat page, now for z as well.

Why it works

Drop perpendiculars from A, P and B onto a coordinate plane: their feet divide the shadow of AB in the same ratio, so the flat section formula gives the x and y. Doing the same on another plane gives z.

When to use it

Finding the point a given fraction of the way along a segment in space, the midpoint, and the ratio in which a plane or a point divides a segment.

How to use it

Weight each end by the part of the ratio on the far side: m1 goes with B, m2 with A.
Do this for x, for y and for z, dividing each by m1 + m2.

To remember: Cross the ratio over: m1 with the far end.

Other forms

Special case $P = (\frac{x _{1} + x _{2}}{2}, \frac{y _{1} + y _{2}}{2}, \frac{z _{1} + z _{2}}{2})$ When m1 = m2: the midpoint.

Worked examples

Story

A cable runs straight from the top of a pole at (0, 0, 12) to a peg at (9, 12, 0), in metres. A bead sits one third of the way along from the top. Where is it?

One third of the way along means the ratio 1 : 2.
$x = \frac{1 \times 9 + 2 \times 0}{3} = 3$
$y = \frac{1 \times 12 + 2 \times 0}{3} = 4$
$z = \frac{1 \times 0 + 2 \times 12}{3} = 8$

Answer: (3, 4, 8)

Picture

Find the midpoint of the join of (2, −1, 4) and (4, 3, −2).

$x = \frac{2 + 4}{2} = 3$
$y = \frac{- 1 + 3}{2} = 1$
$z = \frac{4 - 2}{2} = 1$

Answer: (3, 1, 1)

Direct

Find the point that divides the join of (1, −2, 3) and (3, 4, −5) in the ratio 2 : 3.

$x = \frac{2 \times 3 + 3 \times 1}{2 + 3} = \frac{9}{5}$
$y = \frac{2 \times 4 + 3 \times ( - 2 )}{5} = \frac{2}{5}$
$z = \frac{2 \times ( - 5 ) + 3 \times 3}{5} = - \frac{1}{5}$

Answer: (9/5, 2/5, −1/5)

Reverse

In what ratio does the yz-plane divide the join of (−2, 4, 7) and (3, −5, 8)?

Try it first, then show the working

On the yz-plane x = 0. With the ratio k : 1, (3k − 2) ÷ (k + 1) = 0, and so k = 2/3.
The ratio is 2 : 3. Check: x = (2 × 3 + 3 × (−2)) ÷ 5 = 0.
$\frac{2 \times 3 + 3 \times ( - 2 )}{5} = 0$

Answer: 2 : 3

Exam

Using the section formula, show that A(2, −3, 4), B(−1, 2, 1) and C(0, 1/3, 2) lie on one line.

Try it first, then show the working

Let C divide AB in the ratio k : 1. The x-coordinate gives (−k + 2) ÷ (k + 1) = 0, so k = 2.
$y = \frac{2 \times 2 - 3}{3} = \frac{1}{3}$
$z = \frac{2 \times 1 + 4}{3} = 2$
The y and z of C agree as well, so C is on AB: the points are collinear.

Answer: C divides AB in the ratio 2 : 1, so they are collinear

Common mistake: Pairing m1 with A's coordinates; m1 multiplies B's, since the point is closer to the end with the smaller share.

Class 11

Centroid of a triangle in space

G = (\frac{x _{1} + x _{2} + x _{3}}{3}, \frac{y _{1} + y _{2} + y _{3}}{3}, \frac{z _{1} + z _{2} + z _{3}}{3})

What each letter means
$G$	the centroid: where the three medians of the triangle meet
$x_{1}$	the x-coordinate of the first corner
$y_{1}$	the y-coordinate of the first corner
$z_{1}$	the z-coordinate of the first corner
$x_{2}$	the x-coordinate of the second corner
$y_{2}$	the y-coordinate of the second corner
$z_{2}$	the z-coordinate of the second corner
$x_{3}$	the x-coordinate of the third corner
$y_{3}$	the y-coordinate of the third corner
$z_{3}$	the z-coordinate of the third corner

The centroid of a triangle is the average of its three corners, taken one coordinate at a time.

Why it works

The centroid lies on the median from the first corner A, two thirds of the way from A to the midpoint M of the opposite side. The section formula with the ratio 2 : 1 from A to M gives ((x₁ + x₂ + x₃) ÷ 3, …), and the same for y and z.

When to use it

Finding the balance point of a triangle in space, and finding a missing corner when the centroid is known.

How to use it

Add the three x-coordinates and divide by 3.
Do the same for y and for z.

To remember: The centroid is the average corner.

Other forms

To remember itThe centroid is two thirds of the way from each corner to the midpoint of the opposite side.

Worked examples

Story

Three drones hover at (2, 4, 6), (4, 0, 9) and (6, 2, 3), in metres. A camera is set at the centroid of their triangle. Where is it?

$x = \frac{2 + 4 + 6}{3} = 4$
$y = \frac{4 + 0 + 2}{3} = 2$
$z = \frac{6 + 9 + 3}{3} = 6$

Answer: (4, 2, 6)

Picture

For A(0, 0, 0), B(6, 0, 0) and C(0, 6, 6), show that the centroid G is two thirds of the way from A to M, the midpoint of BC.

G = (2, 2, 2), and M, the midpoint of BC, is (3, 3, 3).
$\frac{2}{3} \times 3 = 2$
Each coordinate of G is two thirds of the way from A to M.

Answer: G = (2, 2, 2), two thirds of the way from A to M = (3, 3, 3)

Direct

Find the centroid of the triangle with corners (3, −5, 7), (−1, 7, −6) and (1, 1, 2).

$x = \frac{3 - 1 + 1}{3} = 1$
$y = \frac{- 5 + 7 + 1}{3} = 1$
$z = \frac{7 - 6 + 2}{3} = 1$

Answer: (1, 1, 1)

Reverse

A triangle has centroid (1, 1, 1) and two corners (3, −5, 7) and (−1, 7, −6). Find the third corner.

Try it first, then show the working

For each coordinate, take three times the centroid and subtract the two known corners.
$x_{3} = 3 - 3 + 1 = 1$
$y_{3} = 3 + 5 - 7 = 1$
$z_{3} = 3 - 7 + 6 = 2$

Answer: (1, 1, 2)

Exam

The origin is the centroid of the triangle with corners (2a, 2, 6), (−4, 3b, −10) and (8, 14, 2c). Find a, b and c.

Try it first, then show the working

Each coordinate of the centroid is 0: 2a − 4 + 8 = 0, 2 + 3b + 14 = 0 and 6 − 10 + 2c = 0.
Solving each: a = −2, b = −16/3 and c = 2.

Answer: a = −2, b = −16/3, c = 2

Common mistake: Dividing by 2 as for a midpoint; three corners means dividing by 3.