Learn on PengiPengi Math (Grade 6)Chapter 5: Coordinate Plane & Graphs

Lesson 3: Polygons on the Coordinate Plane

In this Grade 6 Pengi Math lesson from Chapter 5, students learn to draw polygons on a coordinate plane by plotting and connecting ordered vertex coordinates. They calculate horizontal and vertical side lengths using coordinate differences and absolute value, and determine missing vertices to complete shapes like rectangles and right triangles with sides parallel to the axes. Students also use coordinate-based side lengths to verify geometric properties such as equal opposite sides in axis-aligned rectangles.

Section 1

Finding a Missing Vertex of a Polygon

Property

Use the properties of a geometric figure, such as parallel or perpendicular sides, to determine the coordinates of a missing vertex. For polygons with sides parallel to the axes, a missing vertex often shares an $x$ -coordinate with one adjacent vertex and a $y$ -coordinate with another.

Examples

Section 2

Find the Perimeter of a Rectangle on the Coordinate Plane

Property

The perimeter of a rectangle is the sum of its four side lengths. For a rectangle with horizontal and vertical sides, the lengths can be found using the coordinates of its vertices. The perimeter $P$ is calculated by adding the lengths of the two horizontal sides and the two vertical sides.

P = \text{side}_1 + \text{side}_2 + \text{side}_3 + \text{side}_4

Examples

A rectangle has vertices at $(-2, 3)$ , $(4, 3)$ , $(4, -1)$ , and $(-2, -1)$ . The horizontal sides have length $|4 - (-2)| = 6$ units. The vertical sides have length $|3 - (-1)| = 4$ units. The perimeter is $P = 6 + 6 + 4 + 4 = 20$ units.
A square has vertices at $(1, 1)$ , $(5, 1)$ , $(5, 5)$ , and $(1, 5)$ . All sides have length $|5 - 1| = 4$ units. The perimeter is $P = 4 + 4 + 4 + 4 = 16$ units.

Explanation

To find the perimeter of a rectangle on a coordinate plane, first determine the lengths of its horizontal and vertical sides. Use the absolute value of the difference in the x-coordinates for horizontal lengths and the y-coordinates for vertical lengths. Since a rectangle has two pairs of equal-length sides, you will calculate one horizontal length and one vertical length. Finally, add all four side lengths together to find the total perimeter of the rectangle.

Section 3

Find the Perimeter of a Polygon on the Coordinate Plane

Property

The perimeter, $P$ , of a polygon on the coordinate plane is the sum of the lengths of all its sides. For a polygon with $n$ sides of lengths $s_1, s_2, ..., s_n$ , the perimeter is calculated as:

P = s_1 + s_2 + ... + s_n

The lengths of horizontal and vertical sides are found by calculating the distances between their vertices.

Examples

Find the perimeter of a polygon with vertices $A(-2, 1)$ , $B(3, 1)$ , $C(3, -3)$ , and $D(-2, -3)$ .

The side lengths are $AB = |3 - (-2)| = 5$ , $BC = |1 - (-3)| = 4$ , $CD = |3 - (-2)| = 5$ , and $DA = |-3 - 1| = 4$ .
The perimeter is $P = 5 + 4 + 5 + 4 = 18$ units.

Find the perimeter of a polygon with vertices $P(1, 4)$ , $Q(5, 4)$ , $R(5, 2)$ , $S(3, 2)$ , $T(3, 1)$ , and $U(1, 1)$ .

The side lengths are $PQ=4$ , $QR=2$ , $RS=2$ , $ST=1$ , $TU=2$ , $UP=3$ .
The perimeter is $P = 4 + 2 + 2 + 1 + 2 + 3 = 14$ units.

Lesson overview

Expand to review the lesson summary and core properties.

Expand

Section 1

Finding a Missing Vertex of a Polygon

Property

Examples

Section 2

Find the Perimeter of a Rectangle on the Coordinate Plane

Property

P = \text{side}_1 + \text{side}_2 + \text{side}_3 + \text{side}_4

Examples

A rectangle has vertices at $(-2, 3)$ , $(4, 3)$ , $(4, -1)$ , and $(-2, -1)$ . The horizontal sides have length $|4 - (-2)| = 6$ units. The vertical sides have length $|3 - (-1)| = 4$ units. The perimeter is $P = 6 + 6 + 4 + 4 = 20$ units.
A square has vertices at $(1, 1)$ , $(5, 1)$ , $(5, 5)$ , and $(1, 5)$ . All sides have length $|5 - 1| = 4$ units. The perimeter is $P = 4 + 4 + 4 + 4 = 16$ units.

Explanation

Section 3

Find the Perimeter of a Polygon on the Coordinate Plane

Property

The perimeter, $P$ , of a polygon on the coordinate plane is the sum of the lengths of all its sides. For a polygon with $n$ sides of lengths $s_1, s_2, ..., s_n$ , the perimeter is calculated as:

P = s_1 + s_2 + ... + s_n

The lengths of horizontal and vertical sides are found by calculating the distances between their vertices.

Examples

Find the perimeter of a polygon with vertices $A(-2, 1)$ , $B(3, 1)$ , $C(3, -3)$ , and $D(-2, -3)$ .

The side lengths are $AB = |3 - (-2)| = 5$ , $BC = |1 - (-3)| = 4$ , $CD = |3 - (-2)| = 5$ , and $DA = |-3 - 1| = 4$ .
The perimeter is $P = 5 + 4 + 5 + 4 = 18$ units.

Find the perimeter of a polygon with vertices $P(1, 4)$ , $Q(5, 4)$ , $R(5, 2)$ , $S(3, 2)$ , $T(3, 1)$ , and $U(1, 1)$ .

The side lengths are $PQ=4$ , $QR=2$ , $RS=2$ , $ST=1$ , $TU=2$ , $UP=3$ .
The perimeter is $P = 4 + 2 + 2 + 1 + 2 + 3 = 14$ units.

Overview

Lesson overview

Review the lesson summary and core properties without leaving the current page.

Overview

Section 1

Finding a Missing Vertex of a Polygon

Property

Examples

Section 2

Find the Perimeter of a Rectangle on the Coordinate Plane

Property

P = \text{side}_1 + \text{side}_2 + \text{side}_3 + \text{side}_4

Examples

A rectangle has vertices at $(-2, 3)$ , $(4, 3)$ , $(4, -1)$ , and $(-2, -1)$ . The horizontal sides have length $|4 - (-2)| = 6$ units. The vertical sides have length $|3 - (-1)| = 4$ units. The perimeter is $P = 6 + 6 + 4 + 4 = 20$ units.
A square has vertices at $(1, 1)$ , $(5, 1)$ , $(5, 5)$ , and $(1, 5)$ . All sides have length $|5 - 1| = 4$ units. The perimeter is $P = 4 + 4 + 4 + 4 = 16$ units.

Explanation

Section 3

Find the Perimeter of a Polygon on the Coordinate Plane

Property

The perimeter, $P$ , of a polygon on the coordinate plane is the sum of the lengths of all its sides. For a polygon with $n$ sides of lengths $s_1, s_2, ..., s_n$ , the perimeter is calculated as:

P = s_1 + s_2 + ... + s_n

The lengths of horizontal and vertical sides are found by calculating the distances between their vertices.

Examples

Find the perimeter of a polygon with vertices $A(-2, 1)$ , $B(3, 1)$ , $C(3, -3)$ , and $D(-2, -3)$ .

The side lengths are $AB = |3 - (-2)| = 5$ , $BC = |1 - (-3)| = 4$ , $CD = |3 - (-2)| = 5$ , and $DA = |-3 - 1| = 4$ .
The perimeter is $P = 5 + 4 + 5 + 4 = 18$ units.

Find the perimeter of a polygon with vertices $P(1, 4)$ , $Q(5, 4)$ , $R(5, 2)$ , $S(3, 2)$ , $T(3, 1)$ , and $U(1, 1)$ .

The side lengths are $PQ=4$ , $QR=2$ , $RS=2$ , $ST=1$ , $TU=2$ , $UP=3$ .
The perimeter is $P = 4 + 2 + 2 + 1 + 2 + 3 = 14$ units.