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Horizontal and vertical distance

Two places on a grid map sit on the same grid line, one on each side of an axis. How far apart are they, without a ruler?

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When you reach for it

On a grid map, many lengths run along grid lines: a trail drawn straight down the page, a runway laid straight across it, the side of a fenced rectangle. In the first quadrant such a length is the larger of two numbers minus the smaller. Once points sit in all four quadrants, one end may lie left of an axis and the other right of it, and that subtraction would subtract a negative.

This lesson finds such lengths by counting squares and, with no grid to count, from absolute values. Part 1 measures one segment. Part 2 uses four of them as the sides of a rectangle: its perimeter, its area and a missing corner.

The procedure

Part 1: the length of a segment straight across or straight up

Spot the shared number. Two points with the same second number are at the same height, so the segment joining them is horizontal: it runs straight across, along one grid line. Two points with the same first number sit one above the other, so the segment is vertical. The shared number only says which grid line the segment lies on. The length is how far apart the other two numbers are. For (-4, 3) and (5, 3), the 3 puts the segment 3 units above the horizontal axis, and the length is the gap between -4 and 5.

Count, when there is a grid. Each square is one unit, so the length is the number of squares between the ends. For points far apart, or with no grid drawn, counting is slow.

Use absolute values, when there is not. The absolute value of a first number is how far its point is from the vertical axis, and the absolute value of a second number is how far its point is from the horizontal axis. That is what absolute value means: distance from 0 on a number line, whichever side. A horizontal segment either crosses the vertical axis or stays on one side of it, and a vertical segment does the same with the horizontal axis. The signs say which.

  • Opposite sides of the axis: add. In Figure 1, P(-4, 3) is \lvert -4 \rvert = 4 units left of the vertical axis and Q(5, 3) is \lvert 5 \rvert = 5 units right of it. The axis cuts PQ into two pieces, one of 4 and one of 5, so PQ = 4 + 5 = 9 units.
  • Same side of the axis: subtract. R(3, -1) and S(3, -6) are both below the horizontal axis, \lvert -1 \rvert = 1 and \lvert -6 \rvert = 6 units below it. The 6 units from the axis down to S pass through R, and the first 1 of them, the dashed piece, is not part of RS. So RS = 6 - 1 = 5 units: the larger absolute value minus the smaller.
  • One end on the axis. A point on the axis is 0 units from it, so the length is the absolute value of the other end’s number. From (0, 2) to (7, 2) is 7 units.
−6 −5 −4 −3 −2 −1 0 1 2 3 4 5 6 −6 −5 −4 −3 −2 −1 0 1 2 3 4 5 6 P Q R S 4 5 1 5
Figure 1PQ crosses the vertical axis, so its two pieces add, 4 + 5 = 9. R and S are both below the horizontal axis, so the dashed piece is left out, 6 - 1 = 5.

Both rules work with absolute values only, so the result is a distance and is never negative. It does not matter which end is named first: PQ and QP are the same length.

Part 2: rectangles on the grid

A rectangle drawn with its sides along grid lines has two horizontal sides and two vertical sides, so Part 1 gives the length of every side. The two corners (vertices) at the ends of a horizontal side share their second number; the two at the ends of a vertical side share their first number. Opposite sides of a rectangle are equal, so one horizontal and one vertical side are enough.

The perimeter is the distance all the way round, the sum of the four sides. The area is length times width, in square units. When each unit of the grid stands for a real length, turn the side lengths into that length first and multiply after, because a square unit is then a square of that size on both sides: at 20 meters per unit, one square unit is 20 meters by 20 meters, which is 400 square meters.

A missing vertex is found by moving from a known vertex by a known length along a grid line. A move that crosses an axis is split there: from -2, a move of 7 to the right uses 2 units to reach 0 and ends 5 past it, at 5. A move can go either way along the line, so a word like “below” or a second fact has to choose.

Worked examples

Examples 1 to 4 go with Part 1. Examples 5 to 7 go with Part 2.

Example 1

Two segments, two cases

Find the length of each segment. (a) From (-6, -2) to (3, -2). (b) From (4, 7) to (4, 1).

Solution

(a) Both points have second number -2, so the segment is horizontal, along the grid line 2 units below the horizontal axis. Compare the first numbers. -6 is negative, so its point is left of the vertical axis, \lvert -6 \rvert = 6 units away; 3 is positive, so its point is right of it, 3 units away. They are on opposite sides, so the axis splits the segment into pieces of 6 and 3:

6 + 3 = 9 \text{ units}.

Counting the squares from -6 across to 3 gives the same 9.

(b) Both points have first number 4, so the segment is vertical. The second numbers, 7 and 1, are both positive, so both points are above the horizontal axis, 7 and 1 units up. They are on the same side, so the nearer point lies on the way to the further one:

7 - 1 = 6 \text{ units}.
Example 2

The route down a ski trail map

Figure 2 is a ski resort’s trail map, drawn on a grid where each unit is 50 meters. Sten starts at the gondola top G(-5, 6) and skis straight down the map to the hut H(-5, 1), straight across to the café C(4, 1), and straight down to the lift base B(4, -3). How long is his route?

−6 −5 −4 −3 −2 −1 0 1 2 3 4 5 6 −6 −5 −4 −3 −2 −1 0 1 2 3 4 5 6 G H C B
Figure 2Sten’s route in Example 2, from the gondola top G to the lift base B, at 50 meters per unit.
Solution

Measure each leg in units, add the legs, and change to meters once at the end.

G to H: both first numbers are -5, so the leg is vertical. The second numbers 6 and 1 are both above the horizontal axis, same side: 6 - 1 = 5 units.

H to C: both second numbers are 1, so the leg is horizontal. The first numbers -5 and 4 are on opposite sides of the vertical axis, 5 and 4 units from it: 5 + 4 = 9 units.

C to B: both first numbers are 4, so the leg is vertical. C is 1 unit above the horizontal axis and B is \lvert -3 \rvert = 3 units below it, opposite sides: 1 + 3 = 4 units.

The route is 5 + 9 + 4 = 18 units, and each unit is 50 meters:

18 \times 50 = 900 \text{ meters}.

Counting along the route in the figure, 5 squares down, 9 across and 4 down, agrees.

Example 3

Landing lights along a runway

On the plan of an airfield, the control tower is at the origin and each unit is 100 meters. A row of landing lights runs straight along the runway from (-7.5, -3) to (4.5, -3), with a light at each end and one every 1.5 units in between. How long is the row, how many lights are there, and is there a light directly below the tower on the plan?

Solution

Both ends have second number -3, so the row is horizontal. The first numbers -7.5 and 4.5 are on opposite sides of the vertical axis, 7.5 and 4.5 units from it:

7.5 + 4.5 = 12 \text{ units}.

At 100 meters per unit, that is 12 \times 100 = 1200 meters. The lights are 1.5 units apart, so the row is cut into 12 \div 1.5 = 8 gaps. Each gap has a light at both ends and neighboring gaps share a light, so there is one more light than gaps: 9 lights.

A light directly below the tower would sit on the vertical axis, at (0, -3). From the west end (-7.5, -3) to (0, -3), one end is on the axis, so the distance is \lvert -7.5 \rvert = 7.5 units. Since 7.5 \div 1.5 = 5 is a whole number of gaps, the sixth light lands exactly on the axis: yes, there is a light at (0, -3), 3 units or 300 meters below the tower on the plan.

Example 4

Working backwards: points a given distance away

Which points are exactly 5 units from A(-3, 2), straight across or straight up or down?

Solution

There is one point in each of the four directions.

Right: a horizontal move keeps the second number 2. From -3, the first 3 units reach the vertical axis and the other 5 - 3 = 2 carry past it, ending 2 units right: (2, 2). Check: opposite sides, 3 + 2 = 5.

Left: the move goes away from the vertical axis, so the distance from it grows from 3 to 3 + 5 = 8, still on the left: (-8, 2). Check: same side, 8 - 3 = 5.

Up: a vertical move keeps the first number -3. A is 2 above the horizontal axis and moves further away, to 2 + 5 = 7: (-3, 7).

Down: 2 units reach the horizontal axis and 5 - 2 = 3 more go below it: (-3, -3). Check: 2 + 3 = 5.

The four points are (2, 2), (-8, 2), (-3, 7) and (-3, -3).

Example 5

Perimeter and area across all four quadrants

Rectangle ABCD has vertices A(-4, 3), B(2, 3), C(2, -5) and D(-4, -5), as in Figure 3. Find its perimeter and area.

−6 −5 −4 −3 −2 −1 0 1 2 3 4 5 6 −6 −5 −4 −3 −2 −1 0 1 2 3 4 5 6 A B C D 4 2 3 5
Figure 3Rectangle ABCD of Example 5. Each axis cuts two of its sides into pieces that add.
Solution

AB is horizontal, since both second numbers are 3. Its first numbers -4 and 2 are on opposite sides of the vertical axis: AB = 4 + 2 = 6 units. BC is vertical, since both first numbers are 2. Its second numbers 3 and -5 are on opposite sides of the horizontal axis: BC = 3 + 5 = 8 units. Opposite sides of a rectangle are equal, so CD = 6 and DA = 8.

\text{perimeter} = 6 + 8 + 6 + 8 = 28 \text{ units}.
\text{area} = 6 \times 8 = 48 \text{ square units}.

The axes give a check. They cut the rectangle into four smaller rectangles, one in each quadrant, of 4 by 3, 2 by 3, 4 by 5 and 2 by 5. Their areas 12 + 6 + 20 + 10 make 48, as the distributive property says they must.

Example 6

Fencing a beginners’ area

On another trail map, each unit is 20 meters. Noa fences off a beginners’ area, a rectangle with corners (-7, -1), (-2, -1), (-2, -6) and (-7, -6). How much fence does she need, and what area of snow does it enclose?

Solution

All four corners are in Quadrant III, so every side stays on one side of each axis and every length is a subtraction. The top side, from (-7, -1) to (-2, -1), is horizontal: 7 - 2 = 5 units. The right side, from (-2, -1) to (-2, -6), is vertical: 6 - 1 = 5 units. Both are 5, so the rectangle is a square.

In meters each side is 5 \times 20 = 100 meters, so the fence is 4 \times 100 = 400 meters.

The area is 100 \times 100 = 10\,000 square meters. Counted in units instead, the square holds 5 \times 5 = 25 square units, and each square unit is 20 meters by 20 meters, 400 square meters, so the area is 25 \times 400 = 10\,000 square meters again.

Example 7

A missing vertex from the area

Rectangle ABCD has A(-5, 4), B(3, 4) and an area of 56 square units. C is directly below B and D directly below A. Find C and D.

Solution

AB is horizontal, and -5 and 3 are on opposite sides of the vertical axis, so AB = 5 + 3 = 8 units. The area is length times width, so the other side is 56 \div 8 = 7 units.

C is 7 units straight below B(3, 4), so its first number is 3. B is 4 units above the horizontal axis: 4 units down reach the axis and 7 - 4 = 3 more go below it. So C is (3, -3). In the same way D is 7 units below A(-5, 4), at (-5, -3).

−6 −5 −4 −3 −2 −1 0 1 2 3 4 5 6 −6 −5 −4 −3 −2 −1 0 1 2 3 4 5 6 A B C D 4 3
Figure 4Example 7. Going 7 units down from B uses 4 units to reach the axis and 3 more below it.

Check: BC = 4 + 3 = 7 units and 8 \times 7 = 56. Had the rectangle been above AB instead, the corners would be 7 units up, at (3, 11) and (-5, 11), with the same area. The word “below” is the second fact that chooses.

Boundaries

  • The method measures only segments that run straight across or straight up and down, whose ends share a coordinate. For two points that share neither number, such as (1, 2) and (4, 6), the segment is slanted, and its length is neither the across gap nor the up gap nor their sum. That length comes from the Pythagorean theorem, in the distance formula.
  • Both methods assume each axis is a number line with equal steps. If a map uses one scale across and another up, each length is turned into real units with its own axis’s scale before lengths are compared or multiplied.
  • The rectangle rules need the sides along grid lines. A tilted rectangle has slanted sides.
  • A length is never negative, and it is 0 only when the two points are the same point.

Where this goes next

Reflecting a point across an axis makes pairs of points on opposite sides of an axis, and this lesson measures how far apart such pairs are. Distance on a number line later finds any gap between two numbers by subtracting them, once subtracting negative numbers is available. The area of a polygon from coordinates boxes a shape that is not a rectangle inside one whose sides are measured as here, and the distance formula reaches slanted segments by building a right triangle from one horizontal and one vertical segment.

Exercises

Part 1: lengths from coordinates

Find the length of the segment joining each pair of points.

  1. (2, -6) and (2, 3)
  2. (-7, 5) and (-1, 5)
  3. (-3, -8) and (-3, -2)
  4. (0, -4) and (9, -4)
  5. (-2.5, 6) and (3, 6)
  6. (8, -1) and (8, -9.5)

Part 1: on the trail map and the airfield

  1. On a trail map where each unit is 25 meters, Oren walks straight from the ski school at (-6, -4) to the rental shop at (-6, 3). How far does he walk?
  2. On an airfield plan where each unit is 50 meters, Rio checks a row of landing lights running straight from (-4, -1) to (11, -1), with a light at each end and one every 2.5 units in between. How long is the row in meters, how many lights are there, and is one of them on the vertical axis?
  3. Say whether each segment is horizontal, vertical or neither, and give the length of each one that is horizontal or vertical. (a) (3, -2) to (3, 5) (b) (-4, 1) to (1, -4) (c) (-6, -6) to (2, -6)

Part 1: working backwards

  1. Find the four points that are exactly 6 units from (4, -1), straight across or straight up or down.
  2. Point B has the same first number as A(-5, 2), lies below the horizontal axis and is 9 units from A. Find B.

Part 2: perimeter and area

Find the perimeter and area of the rectangle with these vertices.

  1. (-6, 2), (1, 2), (1, -3), (-6, -3)
  2. (2, -1), (7, -1), (7, -6), (2, -6)
  3. (-8, 7), (-2, 7), (-2, 3), (-8, 3)
  4. On a trail map where each unit is 5 meters, Mads ropes off a rectangular finish area with corners (-3, -2), (4, -2), (4, -8) and (-3, -8). How much rope goes once round it, and what is its area in square meters?
  5. On an airfield plan where each unit is 30 meters, the paved runway is a rectangle with corners (-12, 1), (15, 1), (15, -1) and (-12, -1). Find its perimeter in meters and its area in square meters.

Part 2: missing vertices

  1. Three vertices of a rectangle are (-4, 6), (3, 6) and (3, -1). Find the fourth vertex and the area.
  2. A square has one side from (-2, -3) to (4, -3) and lies above that side. Find its other two vertices and its area.
  3. A rectangle has one side from (5, 2) to (5, -6) and an area of 24 square units. Find the other two vertices. There are two answers; give both.
  4. On Sten’s trail map, a rectangular rest area has one side from (-6, 1) to (-6, -4), lies to the right of that side, and has a perimeter of 26 units. Find its other two corners.
  5. A rectangle with sides along grid lines has opposite corners at (-4, 2) and (5, -3). Find its other two corners, its perimeter and its area.

Reasoning

  1. Use a number line to explain why the length between two points on opposite sides of an axis is the sum of their absolute values, but between two points on the same side it is the difference.
  2. Reflect (5, -7) across the horizontal axis and find the distance between the point and its reflection. Explain why a point and its reflection across the horizontal axis are always twice the absolute value of the second number apart.
  3. Explain why this lesson’s methods do not give the length of the segment from (-2, 1) to (3, 5). What do they give instead, and is the straight segment longer or shorter than that?
  4. Can a rectangle with sides along grid lines have one vertex in each of the four quadrants? If it can, give an example and say how each of its side lengths is found; if not, explain why.
Answers
  1. 9 units. Vertical, since both first numbers are 2; -6 and 3 are on opposite sides of the horizontal axis, so 6 + 3 = 9.
  2. 6 units. Horizontal; -7 and -1 are both left of the vertical axis, so 7 - 1 = 6.
  3. 6 units. Vertical; -8 and -2 are both below the horizontal axis, so 8 - 2 = 6.
  4. 9 units. Horizontal; (0, -4) is on the vertical axis, so the length is \lvert 9 \rvert = 9.
  5. 5.5 units. Horizontal; opposite sides of the vertical axis, 2.5 + 3 = 5.5.
  6. 8.5 units. Vertical; both below the horizontal axis, 9.5 - 1 = 8.5.
  7. 175 meters. Vertical; 4 units below the axis and 3 above, so 4 + 3 = 7 units, and 7 \times 25 = 175.
  8. 750 meters, 7 lights, and no light on the vertical axis. The row is 4 + 11 = 15 units, and 15 \times 50 = 750. There are 15 \div 2.5 = 6 gaps, so 7 lights. The west end is 4 units from the vertical axis and 4 \div 2.5 is not a whole number, so no light lands there: the lights are at -4, -1.5, 1, 3.5, 6, 8.5 and 11.
  9. (a) Vertical, 7 units: 2 + 5 = 7. (b) Neither: the points share no number. (c) Horizontal, 8 units: 6 + 2 = 8.
  10. (10, -1), (-2, -1), (4, 5) and (4, -7). Right: 4 + 6 = 10. Left: 4 units reach the axis and 2 more go past it. Up: 1 unit reaches the axis and 5 more go above it. Down: 1 + 6 = 7 below.
  11. B(-5, -7). 2 units down reach the horizontal axis and 9 - 2 = 7 more go below it.
  12. Perimeter 24 units, area 35 square units. Width 6 + 1 = 7, height 2 + 3 = 5; 7 + 5 + 7 + 5 = 24 and 7 \times 5 = 35.
  13. Perimeter 20 units, area 25 square units. Width 7 - 2 = 5, height 6 - 1 = 5, a square; 4 \times 5 = 20 and 5 \times 5 = 25.
  14. Perimeter 20 units, area 24 square units. Width 8 - 2 = 6, height 7 - 3 = 4; 6 + 4 + 6 + 4 = 20 and 6 \times 4 = 24.
  15. 130 meters of rope, area 1050 square meters. Width 3 + 4 = 7 units, which is 35 meters; height 8 - 2 = 6 units, which is 30 meters. 35 + 30 + 35 + 30 = 130 and 35 \times 30 = 1050.
  16. Perimeter 1740 meters, area 48\,600 square meters. Length 12 + 15 = 27 units, which is 810 meters; width 1 + 1 = 2 units, which is 60 meters. 810 + 60 + 810 + 60 = 1740 and 810 \times 60 = 48\,600.
  17. (-4, -1), area 49 square units. The fourth vertex shares its first number with (-4, 6) and its second with (3, -1). Sides 4 + 3 = 7 and 6 + 1 = 7, so it is a square and 7 \times 7 = 49.
  18. (4, 3) and (-2, 3), area 36 square units. The side is 2 + 4 = 6. Going 6 up from -3, 3 units reach the axis and 3 more go above it.
  19. Either (8, 2) and (8, -6), or (2, 2) and (2, -6). The side is 2 + 6 = 8 units, so the other side is 24 \div 8 = 3. Moving 3 right from 5 gives 8; moving 3 left gives 5 - 3 = 2, still right of the axis.
  20. (2, 1) and (2, -4). The given side is 1 + 4 = 5, so the other two sides make 26 - 5 - 5 = 16 and each is 8. From -6, 6 units right reach the axis and 2 more go past it.
  21. (-4, -3) and (5, 2); perimeter 28 units, area 45 square units. Each missing corner takes its first number from one given corner and its second from the other. Width 4 + 5 = 9, height 2 + 3 = 5; 9 + 5 + 9 + 5 = 28 and 9 \times 5 = 45.
  22. Absolute value is distance from 0. Two numbers on opposite sides of 0 are joined by a path through 0, made of their two distances to 0 end to end, so the length is the sum. On the same side, the path from 0 to the further number passes the nearer one first, so the length is the longer distance with the shorter taken off. On the grid, the axis plays the part of 0.
  23. The reflection is (5, 7) and the distance is 14 units: vertical, and 7 + 7 = 14. A reflection across the horizontal axis keeps the first number and puts the point the same distance from the axis on the other side, so the length is that distance added to itself.
  24. The points share neither number, so the segment is slanted. The methods give the across gap, 2 + 3 = 5 units, and the up gap, 5 - 1 = 4 units: a path by way of (3, 1) of 5 + 4 = 9 units. The straight segment is shorter, since a straight segment is the shortest way between two points; its exact length needs the distance formula.
  25. Yes, for example (-2, 3), (4, 3), (4, -1) and (-2, -1). Every side crosses an axis, so every side length is a sum: width 2 + 4 = 6 and height 3 + 1 = 4.
kg1128 · updated 2026-10-12 · CC-BY-SA-4.0