Chapter 7 · The coordinate plane

Draw Polygons and Find Side Lengths from Coordinates

Plot the vertices of a polygon on the coordinate plane and join them in order. Then use the coordinates to find the lengths of its horizontal and vertical sides.

12 min read

A polygon is a closed flat shape made of straight sides that meet only at their endpoints. Each corner, where two sides meet, is a vertex (plural: vertices).

The coordinate plane is the grid built on a horizontal xx-axis and a vertical yy-axis. Every vertex of a polygon drawn there sits at a point, so an ordered pair (x,y)(x, y) names it. The first number, xx, tells how far the point is to the right or left of the yy-axis. The second number, yy, tells how far it is above or below the xx-axis.

So a short list of ordered pairs can describe a whole polygon. In this lesson you will turn such a list into a drawing. Then you will use the same numbers to find the lengths of the polygon's horizontal and vertical sides.

Each ordered pair in the list marks one corner. The list also tells you where the sides go. A side joins two neighbors, vertices that are next to each other in the list. The last vertex and the first vertex are neighbors too, because a polygon is closed.

So to draw a polygon from its vertices:

  1. Plot each vertex as a point.
  2. Join each vertex to the next one in the list with a straight line segment.
  3. Join the last vertex back to the first.

This works when the vertices are listed in order as you travel around the polygon, as they always are in this lesson. Joining the same points in another order can pair up different neighbors and change the shape.

Worked example

Draw a polygon from its vertices

Draw the polygon with vertices A(1,1)A(1, 1), B(6,1)B(6, 1), C(6,4)C(6, 4) and D(1,4)D(1, 4), listed in order around the polygon.

  1. Plot each vertex. For A(1, 1), start at the origin (0, 0), where the axes cross. Move 1 unit right and 1 unit up, and mark the point. Plot the others the same way: B(6, 1) is 6 right and 1 up, C(6, 4) is 6 right and 4 up, and D(1, 4) is 1 right and 4 up. The first coordinate gives the left-right move and the second gives the up-down move.

  2. Join each vertex to the next one in the list: draw straight line segments from A to B, from B to C and from C to D. Vertices next to each other in the list are neighbors, so a side joins them.

    A→B→C→DA \to B \to C \to D

  3. Close the shape with a segment from D back to A. A polygon is closed, so the last vertex and the first vertex are neighbors too.

    D→AD \to A

  4. Describe the sides from the coordinates, not from how the drawing looks. Points A and B both have y = 1, and points C and D both have y = 4, so sides AB and CD are horizontal. Points B and C both have x = 6, and points D and A both have x = 1, so sides BC and DA are vertical. A horizontal side and a vertical side meet at a right angle, so every corner is a right angle.

Answer

The polygon is ABCDABCD, a quadrilateral (a polygon with four sides). Its four corners are right angles, so it is a rectangle. The figure below shows it.

Coordinate grid from x 0 to 8 and y 0 to 6, with a grid step of 1, showing 1 plotted element. Details: polygon with 4 vertices.
Polygon ABCDABCD with vertices (1,1)(1, 1), (6,1)(6, 1), (6,4)(6, 4) and (1,4)(1, 4). Bottom side ABAB, right side BCBC, top side CDCD, left side DADA.

Now measure a side. A side length is the distance between the two vertices at the ends of a side. You already know how to find the distance between two points that share a coordinate. A side whose two vertices share a coordinate is exactly that situation.

In the figure above, the bottom side runs from A(1,1)A(1, 1) to B(6,1)B(6, 1). Both points have yy-coordinate 11, so the side lies on one horizontal gridline. Follow it from x=1x = 1 to x=6x = 6 and you pass 55 grid spaces. The gridlines in this figure are 11 unit apart, so ABAB is 55 units long.

Subtracting gives the same number without counting: 6−1=56 - 1 = 5. This works because only the xx-coordinate changes along a horizontal side, so the change in xx is the whole distance.

Key concept

From vertices to sides and side lengths

The vertices of a polygon, joined in order around it, give its sides. For one side, let (x1,y1)(x_1, y_1) and (x2,y2)(x_2, y_2) be the vertices at its two ends. One unit is the distance from 00 to 11 on an axis.

  • Same yy-coordinate: the side is horizontal. Its length is the distance between the xx-coordinates, ∣x2−x1∣\lvert x_2 - x_1 \rvert units.
  • Same xx-coordinate: the side is vertical. Its length is the distance between the yy-coordinates, ∣y2−y1∣\lvert y_2 - y_1 \rvert units.
  • Neither coordinate the same: the side is slanted. Subtracting coordinates does not give its length. In this lesson, a slanted side's length is always given.

The bars mean absolute value, a number's distance from 00, so a length is never negative.

horizontal side: ∣x2−x1∣vertical side: ∣y2−y1∣\text{horizontal side: } |x_2 - x_1| \qquad \text{vertical side: } |y_2 - y_1|

Worked example

Find the other side lengths of rectangle ABCD

Rectangle ABCDABCD has vertices A(1,1)A(1, 1), B(6,1)B(6, 1), C(6,4)C(6, 4) and D(1,4)D(1, 4). You found that side ABAB is 55 units long. Find the lengths of sides BCBC, CDCD and DADA.

  1. Side BC joins B(6, 1) and C(6, 4). Both vertices have x = 6, so the side is vertical and only y changes along it. Subtract the y-coordinates.

    ∣4−1∣=∣3∣=3|4 - 1| = |3| = 3

  2. Side CD joins C(6, 4) and D(1, 4). Both vertices have y = 4, so the side is horizontal. Subtract the x-coordinates. The result -5 is negative because x drops by 5 as you move left from C to D. A distance has no direction, so take the absolute value. Subtracting in the other order, 6 - 1, gives 5 right away.

    ∣1−6∣=∣−5∣=5|1 - 6| = |-5| = 5

  3. Side DA joins D(1, 4) and A(1, 1). Both vertices have x = 1, so the side is vertical. Subtract the y-coordinates and take the absolute value.

    ∣1−4∣=∣−3∣=3|1 - 4| = |-3| = 3

  4. Check with what you know about the shape. The first example showed that ABCD is a rectangle, and opposite sides of a rectangle have equal lengths. CD is opposite AB, and DA is opposite BC, so the lengths should match in pairs. They do.

    CD=AB=5DA=BC=3CD = AB = 5 \qquad DA = BC = 3

Answer

Side BCBC is 33 units long, side CDCD is 55 units long and side DADA is 33 units long.

Worked example

Draw and measure a polygon that crosses the axes

Draw the polygon with vertices P(−4,−2)P(-4, -2), Q(3,−2)Q(3, -2), R(3,2)R(3, 2) and S(−1,2)S(-1, 2), listed in order around the polygon. You are told that side SPSP is 55 units long. Find the lengths of sides PQPQ, QRQR and RSRS.

  1. Plot the vertices. A negative x-coordinate puts a point left of the y-axis. A negative y-coordinate puts it below the x-axis. From the origin, P(-4, -2) is 4 left and 2 down, Q(3, -2) is 3 right and 2 down, R(3, 2) is 3 right and 2 up, and S(-1, 2) is 1 left and 2 up.

  2. Join each vertex to the next one in the list, then close the polygon by joining S back to P.

    P→Q→R→S→PP \to Q \to R \to S \to P

  3. Sort the sides by comparing coordinates. P and Q both have y = -2, so PQ is horizontal. Q and R both have x = 3, so QR is vertical. R and S both have y = 2, so RS is horizontal. S(-1, 2) and P(-4, -2) share neither coordinate, so SP is slanted. Use its given length, 5 units, and do not subtract.

  4. PQ is horizontal, so find the distance between the x-coordinates -4 and 3. Subtracting -4 is the same as adding 4. The x-axis shows why: -4 and 3 are on opposite sides of 0, so the side covers 4 units from -4 to 0 and then 3 more units from 0 to 3.

    ∣3−(−4)∣=∣3+4∣=∣7∣=7|3 - (-4)| = |3 + 4| = |7| = 7

  5. QR is vertical, so find the distance between the y-coordinates -2 and 2. Subtracting -2 is the same as adding 2. These numbers are on opposite sides of 0 too: the side covers 2 units from -2 up to 0 and then 2 more units up to 2.

    ∣2−(−2)∣=∣2+2∣=∣4∣=4|2 - (-2)| = |2 + 2| = |4| = 4

  6. RS is horizontal, so subtract the x-coordinates of R(3, 2) and S(-1, 2). From R to S, x drops from 3 to -1. That is 3 units left to 0 and then 1 more unit left to -1, so the change is -4. The length is its absolute value.

    ∣−1−3∣=∣−4∣=4|-1 - 3| = |-4| = 4

Answer

Side PQPQ is 77 units long, side QRQR is 44 units long and side RSRS is 44 units long. Side SPSP is 55 units long, as given. The figure below shows PQRSPQRS.

Coordinate grid from x -6 to 6 and y -4 to 4, with a grid step of 1, showing 1 plotted element. Details: polygon with 4 vertices.
Polygon PQRSPQRS with vertices (−4,−2)(-4, -2), (3,−2)(3, -2), (3,2)(3, 2) and (−1,2)(-1, 2). Bottom side PQPQ, right side QRQR, top side RSRS, slanted left side SPSP.

Takeaway. A list of vertices, in order around a polygon, is enough to draw it and to measure its horizontal and vertical sides. Plot the points, join each one to the next and close the shape. A side whose two vertices share a coordinate is horizontal or vertical, and its length is the distance between the coordinates that differ. A slanted side shares neither coordinate, so subtracting does not work and you need its length to be given.

Next, you will add side lengths to find a polygon's perimeter, the total distance around it. You will also use coordinates to solve word problems and to find a missing vertex.

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Perimeters, Word Problems, and Missing Vertices

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