The distance between two points

What the distance formula measures

The distance between two points is the straight-line length of the segment joining them — how far apart they are on the grid.
Each point is written as a pair of co-ordinates $(x, y)$ : the first number is across, the second is up.
The two points are usually named $(x_1, y_1)$ and $(x_2, y_2)$ . The small 1 and 2 are just labels for "the first point" and "the second point" — they are not powers.

Where the formula comes from

Join the two points and complete a right-angled triangle, with one horizontal side and one vertical side.
The horizontal side has length $x_2 - x_1$ (the change in $x$ ).
The vertical side has length $y_2 - y_1$ (the change in $y$ ).
The distance is the hypotenuse, so it comes straight from Pythagoras' theorem $a^2 + b^2 = c^2$ .

A segment AB drawn as the hypotenuse of a right-angled triangle, with a horizontal leg of length x-two minus x-one and a vertical leg of length y-two minus y-one.

The distance formula

This gives the distance formula:

$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$

$d$ — the distance between the two points.
$x_2 - x_1$ — the horizontal gap (run). Square it, so its sign never matters.
$y_2 - y_1$ — the vertical gap (rise). Square it too.
Subtract the co-ordinates, square each difference, add, then take the square root.

Applying the formula

Label the points: pick one as $(x_1, y_1)$ and the other as $(x_2, y_2)$ . It does not matter which is which.
Substitute into the formula, keeping the brackets so negatives are handled correctly.
Leave the answer as a surd (e.g. $\sqrt{29}$ ) for an exact value, or round to a sensible number of decimal places if the context is a real measurement.

Worked ExampleThe length of a walking track

A straight section of a walking track runs from a shelter at $A(2, 3)$ to a lookout at $B(7, 15)$ , where each unit on the map is 100 m. Find the length of this section, in metres.

Step 1 — Label the points and find the gaps

Take $A(2, 3)$ as $(x_1, y_1)$ and $B(7, 15)$ as $(x_2, y_2)$ :

$x_2 - x_1 = 7 - 2 = 5, \qquad y_2 - y_1 = 15 - 3 = 12$

Step 2 — Substitute into the distance formula

$d = \sqrt{5^2 + 12^2} = \sqrt{25 + 144} = \sqrt{169}$

Step 3 — Evaluate and interpret in context

$d = 13 \text{ units}$

Each unit is 100 m, so the section is $13 \times 100 = \textbf{1300 m}$ long. (Notice $5, 12, 13$ is a Pythagorean triple — a neat whole-number answer.)

When the answer is a surd

Most real point pairs do not give a whole number. Leave the exact value as a surd unless asked to round.
Example — the distance from $(1, 2)$ to $(4, 7)$ :

$d = \sqrt{(4-1)^2 + (7-2)^2} = \sqrt{9 + 25} = \sqrt{34} \approx 5.83$

Test yourself

Practice by grade

One question each at Achieved, Merit and Excellence. Have a go, then compare with the model answer.

Achieved

Two survey pegs are at $P(1, 2)$ and $Q(6, 14)$ on a plan. Calculate the distance $PQ$ .

Merit

A drone flies in a straight line from $A(-3, 1)$ to $B(2, 13)$ . Each unit is 10 m. Calculate the distance flown, in metres.

Excellence

A triangular garden bed has corners $A(0, 0)$ , $B(6, 0)$ and $C(3, 4)$ . Show that the triangle is isosceles.

The distance between two points

Covers AS91256

What the distance formula measures

The distance between two points is the straight-line length of the segment joining them — how far apart they are on the grid.
Each point is written as a pair of co-ordinates $(x, y)$ : the first number is across, the second is up.
The two points are usually named $(x_1, y_1)$ and $(x_2, y_2)$ . The small 1 and 2 are just labels for "the first point" and "the second point" — they are not powers.

Where the formula comes from

Join the two points and complete a right-angled triangle, with one horizontal side and one vertical side.
The horizontal side has length $x_2 - x_1$ (the change in $x$ ).
The vertical side has length $y_2 - y_1$ (the change in $y$ ).
The distance is the hypotenuse, so it comes straight from Pythagoras' theorem $a^2 + b^2 = c^2$ .

A segment AB drawn as the hypotenuse of a right-angled triangle, with a horizontal leg of length x-two minus x-one and a vertical leg of length y-two minus y-one.

The distance formula

This gives the distance formula:

$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$

$d$ — the distance between the two points.
$x_2 - x_1$ — the horizontal gap (run). Square it, so its sign never matters.
$y_2 - y_1$ — the vertical gap (rise). Square it too.
Subtract the co-ordinates, square each difference, add, then take the square root.

Applying the formula

Label the points: pick one as $(x_1, y_1)$ and the other as $(x_2, y_2)$ . It does not matter which is which.
Substitute into the formula, keeping the brackets so negatives are handled correctly.
Leave the answer as a surd (e.g. $\sqrt{29}$ ) for an exact value, or round to a sensible number of decimal places if the context is a real measurement.

Worked ExampleThe length of a walking track

A straight section of a walking track runs from a shelter at $A(2, 3)$ to a lookout at $B(7, 15)$ , where each unit on the map is 100 m. Find the length of this section, in metres.

Step 1 — Label the points and find the gaps

Take $A(2, 3)$ as $(x_1, y_1)$ and $B(7, 15)$ as $(x_2, y_2)$ :

$x_2 - x_1 = 7 - 2 = 5, \qquad y_2 - y_1 = 15 - 3 = 12$

Step 2 — Substitute into the distance formula

$d = \sqrt{5^2 + 12^2} = \sqrt{25 + 144} = \sqrt{169}$

Step 3 — Evaluate and interpret in context

$d = 13 \text{ units}$

Each unit is 100 m, so the section is $13 \times 100 = \textbf{1300 m}$ long. (Notice $5, 12, 13$ is a Pythagorean triple — a neat whole-number answer.)

When the answer is a surd

Most real point pairs do not give a whole number. Leave the exact value as a surd unless asked to round.
Example — the distance from $(1, 2)$ to $(4, 7)$ :

$d = \sqrt{(4-1)^2 + (7-2)^2} = \sqrt{9 + 25} = \sqrt{34} \approx 5.83$

Test yourself

Practice by grade

One question each at Achieved, Merit and Excellence. Have a go, then compare with the model answer.

Achieved

Two survey pegs are at $P(1, 2)$ and $Q(6, 14)$ on a plan. Calculate the distance $PQ$ .

Merit

A drone flies in a straight line from $A(-3, 1)$ to $B(2, 13)$ . Each unit is 10 m. Calculate the distance flown, in metres.

Excellence

A triangular garden bed has corners $A(0, 0)$ , $B(6, 0)$ and $C(3, 4)$ . Show that the triangle is isosceles.