Do I have to plot a polygon in a specific order?

Yes, you should connect the vertices in the order they are given. This ensures you draw the correct shape. If you connect them out of order, the sides may cross each other or the shape may look wrong. Always go from the first vertex to the second, second to third, and so on, ending by connecting the last vertex back to the first.

What if my distance formula gives me a square root that doesn't come out even?

That's completely normal. Not all side lengths work out to whole numbers. For example, the distance from (0, 0) to (1, 1) is sqrt(2) units, which is about 1.41 units. You can leave the answer as a square root, simplify it if possible, or give a decimal approximation. Your teacher will let you know which form they prefer.

How does the distance formula connect to the Pythagorean theorem?

When you find the distance between two points, you're creating an invisible right triangle where the horizontal distance is one leg, the vertical distance is the other leg, and the actual distance between the points is the hypotenuse. The distance formula d = sqrt((x_2 - x_1)^2 + (y_2 - y_1)^2) is just the Pythagorean theorem c = sqrt(a^2 + b^2) applied to the coordinate plane.

What's the difference between distance and length?

In this context, they mean the same thing. The distance between two points equals the length of the line segment connecting them. When we talk about a side of a polygon, the length of that side is the distance between its two endpoints on the coordinate plane.

M6MATH-9.2

Polygons in the Coordinate Plane

Learn to plot polygons on a coordinate plane using vertices and calculate side lengths using the distance formula for Grade 6.

What you'll do in this lesson

A voice-first session with the Crimsora tutor on Polygons in the Coordinate Plane, then targeted practice and FRQs — with the tutor adapting to where you get stuck.

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What this lesson covers

Polygons aren't just shapes you draw on paper—they can be precisely located using coordinates. In this lesson, you'll learn to plot a polygon on the coordinate plane by marking its vertices, connect them to form the shape, and use coordinates to find exact side lengths. These skills connect geometry to the number system and form the foundation for more advanced work with shapes in later grades.

Understanding the Coordinate Plane

The coordinate plane is a grid formed by a horizontal number line (the

x

-axis) and a vertical number line (the

y

-axis) that cross at the origin, labeled as

(0, 0)

. Every point on the plane has an address written as an ordered pair

(x, y)

, where

x

tells you how far left or right to move from the origin, and

y

tells you how far up or down. Positive

x

values move right, negative move left. Positive

y

values move up, negative move down. When you're given coordinates for a polygon's vertices, you're essentially getting the exact locations where the corners of that shape sit on the grid. For example, the triangle with vertices at

(1, 2)

(5, 2)

, and

(3, 6)

has one corner 1 unit right and 2 units up, another 5 units right and 2 units up, and a third 3 units right and 6 units up. Once you plot all the vertices carefully, you connect them in order (usually listed in sequence) to draw the complete polygon.

Plotting Vertices and Drawing the Polygon

To draw a polygon from coordinates, follow these steps: First, set up your coordinate plane with a scale that makes sense for your points—if your largest coordinate is 10, mark the axes at least to 10. Second, plot each vertex by starting at the origin, moving the correct distance along the

x

-axis, then moving parallel to the

y

-axis. Mark each point clearly. Third, connect the vertices in the order they were listed. For a triangle with vertices

A = (2, 1)

B = (6, 1)

, and

C = (4, 5)

, you would plot each point, then draw a line from

A

B

B

C

, and

C

back to

A

. A common mistake is plotting points in the wrong order or forgetting to close the polygon by connecting the last vertex back to the first. Another error is mixing up which number is

x

and which is

y

—remember:

x

first (left/right), then

y

(up/down). Your finished drawing should show a recognizable shape with all sides visible.

Finding Side Lengths Using the Distance Formula

Once your polygon is plotted, you can find the length of any side using the distance formula. For two points

(x_1, y_1)

and

(x_2, y_2)

, the distance is:

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

. This formula comes from the Pythagorean theorem—when you connect two points, you create a right triangle where the two legs are the horizontal and vertical distances, and the hypotenuse is the actual side length. For example, to find the distance from

(1, 2)

(4, 6)

: calculate the horizontal change as

4 - 1 = 3

and the vertical change as

6 - 2 = 4

, then

d = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5

units. This method works for any pair of points, whether they lie on a horizontal line, a vertical line, or a diagonal. Students often forget to square both differences, or they forget to take the square root at the end. Always check that your answer makes sense by looking at the plotted points—if they are 3 units apart horizontally and 4 units apart vertically, a distance of 5 is reasonable, but a distance of 3 would be too small.

Special Cases: Horizontal and Vertical Sides

When a side of your polygon is perfectly horizontal or vertical, finding its length is much simpler. For a horizontal side, both points have the same

y

-coordinate, so you only need to find the difference in

x

-coordinates. For example, from

(2, 3)

(8, 3)

, the length is

|8 - 2| = 6

units. For a vertical side, both points have the same

x

-coordinate, so you find the difference in

y

-coordinates: from

(5, 1)

(5, 7)

gives a length of

|7 - 1| = 6

units. You can still use the full distance formula for these cases—it will give you the same answer—but recognizing horizontal and vertical sides saves time. Many rectangles and right triangles on the coordinate plane have sides aligned this way, which makes them easier to work with. When a polygon has sides that are not horizontal or vertical (diagonal sides), you must use the full distance formula.

Putting It All Together: From Coordinates to Measurements

Working with polygons in the coordinate plane means combining plotting skills with distance calculations. You start with a list of vertices, plot them to visualize the shape, then use coordinates to find exact measurements without needing a ruler. This is powerful because coordinates are precise: you can communicate a polygon's exact shape and size using just numbers, and someone else can recreate it perfectly. Squares, rectangles, triangles, and more complex shapes can all be defined and analyzed this way. The coordinate plane is a bridge between pure number work and geometry—it lets you use arithmetic (subtraction, squaring, square roots) to solve spatial problems. In later lessons, you'll use these same coordinates to find area and perimeter, and even to determine whether sides are parallel or perpendicular to each other. The skills you're building now form the foundation for all of that.

Key terms

Coordinate plane.: A two-dimensional grid formed by a horizontal $x$ -axis and a vertical $y$ -axis that intersect at the origin $(0, 0)$ , used to locate points using ordered pairs.
Ordered pair.: A pair of numbers $(x, y)$ that tells the location of a point on the coordinate plane, where $x$ is the horizontal coordinate and $y$ is the vertical coordinate.
Vertex.: A corner or endpoint of a polygon, located at a specific coordinate on the plane.
Distance formula.: The formula $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$ used to find the straight-line distance between two points on the coordinate plane.
Origin.: The point where the $x$ -axis and $y$ -axis intersect, located at $(0, 0)$ .
Polygon.: A closed two-dimensional shape made up of straight line segments (sides) that connect vertices.

Worked example

Plot the quadrilateral with vertices at

A = (1, 1)

B = (5, 1)

C = (5, 4)

, and

D = (1, 4)

. Then find the lengths of sides

AB

and

AD

Start by setting up a coordinate plane with

x

and

y

axes marked from 0 to at least 6. Plot point

A

(1, 1)

by moving 1 unit right and 1 unit up from the origin. Plot

B

(5, 1)

by moving 5 units right and 1 unit up—notice it's on the same horizontal line as

A

. Plot

C

(5, 4)

by moving 5 units right and 4 units up—same vertical line as

B

. Plot

D

(1, 4)

by moving 1 unit right and 4 units up—same vertical line as

A

. Now connect them in order: draw a line from

A

B

, then

B

C

, then

C

D

, then

D

back to

A

. You should see a rectangle. To find the length of side

AB

: both points have

y = 1

, so this is a horizontal side. The length is

|5 - 1| = 4

units. You can verify with the distance formula:

d = \sqrt{(5 - 1)^2 + (1 - 1)^2} = \sqrt{16 + 0} = \sqrt{16} = 4

units. To find the length of side

AD

: both points have

x = 1

, so this is a vertical side. The length is

|4 - 1| = 3

units. Using the distance formula:

d = \sqrt{(1 - 1)^2 + (4 - 1)^2} = \sqrt{0 + 9} = \sqrt{9} = 3

units. Your quadrilateral is a rectangle with dimensions 4 units by 3 units.

Practice questions

Triangle

PQR

has vertices at

P = (2, 3)

Q = (8, 3)

, and

R = (5, 7)

. Using the distance formula, find the length of side

PQ

Answer: 6 units

Since both

P

and

Q

have

y

-coordinate 3, side

PQ

is horizontal. The length is the difference in

x

-coordinates:

|8 - 2| = 6

units. Using the full distance formula:

d = \sqrt{(8 - 2)^2 + (3 - 3)^2} = \sqrt{36 + 0} = \sqrt{36} = 6

units.

What are the coordinates of a point that lies on a vertical line passing through

(4, 2)

$(4, 5)$
$(6, 2)$
$(2, 4)$
$(4, 2)$

Answer: $(4, 5)$

A vertical line has the same

x

-coordinate for all points on it. Since the given point is

(4, 2)

, any point on the same vertical line must also have

x = 4

. The point

(4, 5)

has

x = 4

, so it lies on the vertical line. The point

(6, 2)

has a different

x

-coordinate,

(2, 4)

has

x = 2

, and while

(4, 2)

is the original point, only

(4, 5)

is a different point on the same vertical line.

A rectangle is plotted on the coordinate plane with vertices at

(0, 0)

(6, 0)

(6, 4)

, and

(0, 4)

. Find the length of the diagonal from

(0, 0)

(6, 4)

using the distance formula.

Answer: The diagonal is $\sqrt{52}$ or approximately 7.21 units

Use the distance formula with

(x_1, y_1) = (0, 0)

and

(x_2, y_2) = (6, 4)

d = \sqrt{(6 - 0)^2 + (4 - 0)^2} = \sqrt{36 + 16} = \sqrt{52}

. This can be simplified to

\sqrt{4 \cdot 13} = 2\sqrt{13}

units, or approximately 7.21 units. Note that the diagonal is longer than either side (which are 6 and 4 units), which makes sense geometrically.

FAQ

Do I have to plot a polygon in a specific order?: Yes, you should connect the vertices in the order they are given. This ensures you draw the correct shape. If you connect them out of order, the sides may cross each other or the shape may look wrong. Always go from the first vertex to the second, second to third, and so on, ending by connecting the last vertex back to the first.
What if my distance formula gives me a square root that doesn't come out even?: That's completely normal. Not all side lengths work out to whole numbers. For example, the distance from $(0, 0)$ to $(1, 1)$ is $\sqrt{2}$ units, which is about 1.41 units. You can leave the answer as a square root, simplify it if possible, or give a decimal approximation. Your teacher will let you know which form they prefer.
How does the distance formula connect to the Pythagorean theorem?: When you find the distance between two points, you're creating an invisible right triangle where the horizontal distance is one leg, the vertical distance is the other leg, and the actual distance between the points is the hypotenuse. The distance formula $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$ is just the Pythagorean theorem $c = \sqrt{a^2 + b^2}$ applied to the coordinate plane.
What's the difference between distance and length?: In this context, they mean the same thing. The distance between two points equals the length of the line segment connecting them. When we talk about a side of a polygon, the length of that side is the distance between its two endpoints on the coordinate plane.

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