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Draw Polygons on the
Coordinate Plane

Learn to plot vertices, connect them to form polygons, and calculate side lengths using coordinates.

6th Grade Math

What Is the Coordinate Plane?

The coordinate plane is a flat surface formed by two number lines that cross at a right angle. The horizontal line is the \(x\)-axis and the vertical line is the \(y\)-axis. They meet at the origin \((0, 0)\).

0 −2 2 −4 4 2 −2 x y (0, 0) Quadrant I Quadrant II Quadrant III Quadrant IV

What Is a Polygon?

A polygon is a closed figure made of straight line segments called sides. The points where the sides meet are called vertices. Triangles, rectangles, and hexagons are all polygons.

Triangle
3 sides, 3 vertices
Rectangle
4 sides, 4 right angles
Pentagon
5 sides, 5 vertices

Why Coordinates Matter

Using coordinates makes geometric thinking precise. Instead of estimating lengths with a ruler, you can calculate exact side lengths from the numbers alone. This is the foundation of analytic geometry.

Precision
Exact positions: no guessing, no ruler needed
Calculation
Find lengths, perimeters, and areas from numbers

Finding Side Lengths from Coordinates

When two points share the same \(x\)-coordinate (vertical side) or the same \(y\)-coordinate (horizontal side), the length is found by subtracting and taking the absolute value.

(1, 2) (4, 2)
\(|4 - 1| = 3\)
 (4, 4)
\(|4 - 2| = 2\)
 0 1 2 3 4 0 1 2 3 4

The Two Formulas You Need

Horizontal Side
Same \(y\)-coordinate:
$$\text{length} = |x_2 - x_1|$$
Vertical Side
Same \(x\)-coordinate:
$$\text{length} = |y_2 - y_1|$$

Horizontal vs. Vertical Sides

Horizontal Side Vertical Side
What's shared Same \(y\)-coordinate Same \(x\)-coordinate
Direction Left ↔ Right Up ↔ Down
Formula \(|x_2 - x_1|\) \(|y_2 - y_1|\)
Example \((1,3)\) to \((5,3)\) → 4 \((2,1)\) to \((2,6)\) → 5

How to Draw and Measure

1
Plot each vertex. Locate every ordered pair \((x, y)\) on the grid.
2
Connect in order. Draw straight segments from one vertex to the next, closing the shape at the end.
3
Identify shared coordinates. For each side, check whether the endpoints share an \(x\)- or \(y\)-value.
4
Subtract and take absolute value. Use \(|x_2 - x_1|\) or \(|y_2 - y_1|\) to find the length.
5
Find perimeter (if asked). Add all side lengths together.

Example 1: Draw a Rectangle

Vertices: \(A(1, 1)\), \(B(1, 5)\), \(C(6, 5)\), \(D(6, 1)\). Plot, connect, and find the perimeter.

0 1 2 3 4 5 6 1 2 3 4 5 4 5 A(1,1) B(1,5) C(6,5) D(6,1)

Example 2: A Triangle with Measurable Sides

Vertices: \(P(−3, 1)\), \(Q(4, 1)\), \(R(4, 5)\). Find the lengths of sides \(PQ\) and \(QR\).

0 −3 −2 −1 1 2 3 4 1 2 3 4 5
\(PQ = |4 - (-3)| = 7\)
\(QR = |5 - 1| = 4\)
 P(−3,1) Q(4,1) R(4,5)

Example 3: Perimeter of an L-Shape

An L-shaped hexagon has vertices: \(A(0,0)\), \(B(0,4)\), \(C(2,4)\), \(D(2,2)\), \(E(5,2)\), \(F(5,0)\). Find the perimeter.

0 1 2 3 4 5 1 2 3 4 4 2 2 3 2 5 A B C D E F

Crossing the Axes: Negative Coordinates

When a side crosses from negative to positive coordinates, absolute value takes care of the sign. Just subtract and ignore the negative.

Example
\((−3, 2)\) to \((4, 2)\)
Length: \(|4 - (-3)| = |7| = 7\)
Example
\((1, -2)\) to \((1, 5)\)
Length: \(|5 - (-2)| = |7| = 7\)

Real-World: Garden Plot

A rectangular garden has corners at \((2, 1)\), \((2, 7)\), \((8, 7)\), and \((8, 1)\). Each grid unit represents 1 meter. How much fencing is needed?

1
Width: \(|8 - 2| = 6\) meters
2
Height: \(|7 - 1| = 6\) meters
3
Perimeter: \(2(6) + 2(6) = 24\) meters of fencing

Common Mistakes to Avoid

✕ Swapping \(x\) and \(y\)
Moving up first then right instead of right first then up. Always: \(x\) is horizontal, \(y\) is vertical.
✕ Forgetting Absolute Value
Getting a negative length like \(1 - 5 = -4\). Lengths are always positive: use \(|1 - 5| = 4\).
✕ Wrong Coordinates Subtracted
Subtracting \(x\)-values when the side is vertical. Match the shared coordinate to the direction: shared \(y\) → subtract \(x\)'s.

Quick Reference Summary

Concept Key Idea Formula / Rule
Ordered pair \((x, y)\) Right/left, then up/down
Horizontal side Same \(y\) \(|x_2 - x_1|\)
Vertical side Same \(x\) \(|y_2 - y_1|\)
Perimeter Add all sides Sum of all lengths
Negatives Absolute value \(a - (-b) = a + b\)

Question 1 of 5

What is the length of the segment from \((2, 3)\) to \((2, 9)\)?

Question 2 of 5

Points \((−4, 5)\) and \((3, 5)\) form a horizontal segment. What is its length?

Question 3 of 5

A rectangle has vertices at \((1, 2)\), \((1, 8)\), \((5, 8)\), and \((5, 2)\). What is the perimeter?

Question 4 of 5

Which formula finds the length of a vertical segment?

Question 5 of 5

A triangle has vertices \(A(0, 0)\), \(B(0, 6)\), and \(C(8, 0)\). What is the length of side \(AC\)?