Math Core

Maps, video game screens and building plans all use coordinates to say exactly where things are. When the corners of a shape are points on the coordinate plane, you can find its side lengths, perimeter and area without a ruler, just by looking at the coordinates.

Drawing a polygon from its vertices

The corners of a polygon are called its vertices (one corner is a vertex). To draw a polygon on the coordinate plane, plot each vertex, then connect them in order and connect the last one back to the first.

Rectangle ABCD with vertices A(-3, 2), B(4, 2), C(4, -2) and D(-3, -2).Open in grapher →

Finding the length of a side

Look at side $AB$ . Both endpoints have the same $y$ -coordinate (2), so the side is horizontal. Its length is the distance between the $x$ -coordinates, $-3$ and $4$ .

You can count the grid squares: from $-3$ to $0$ is 3 units, and from $0$ to $4$ is 4 more. So $AB = 3 + 4 = 7$ units.

Side $BC$ is vertical, because both points have the same $x$ -coordinate (4). Its $y$ -coordinates are $2$ and $-2$ , which are on opposite sides of the $x$ -axis. So $BC = 2 + 2 = 4$ units.

Distance along a grid line

When two points share the same $x$ -coordinate or the same $y$ -coordinate, find the distance between the other coordinates:

Same sign (same side of 0): subtract the absolute values.
Opposite signs (different sides of 0): add the absolute values.

For example, the distance from $(5, -1)$ to $(5, -7)$ is $7 - 1 = 6$ units, because $-1$ and $-7$ are both negative. The distance from $(-6, 3)$ to $(2, 3)$ is $6 + 2 = 8$ units, because $-6$ and $2$ are on opposite sides of 0.

Common mistake

Distance is never negative. If you subtract $4 - (-3)$ you get 7, but if you just subtract $4 - 3 = 1$ you have lost the part of the side that is left of the $y$ -axis. When the signs are different, add the absolute values. Counting squares on the grid is a good check.

Perimeter and area

Once you know the side lengths, use the formulas you already know.

Worked example: Rectangle ABCD

Find the perimeter and area of rectangle $ABCD$ above.

From the work above, the length is $AB = 7$ units and the width is $BC = 4$ units.

Perimeter: $2 \times 7 + 2 \times 4 = 14 + 8 = 22$ units.

Area: $7 \times 4 = 28$ square units.

Triangle PQR. The vertical segment from R down to side PQ is the height.Open in grapher →

Worked example: The area of a triangle

Find the area of triangle $PQR$ with vertices $P(-2, -1)$ , $Q(6, -1)$ and $R(1, 4)$ .

Side $PQ$ is horizontal (both $y$ -coordinates are $-1$ ), so use it as the base. Its length is $2 + 6 = 8$ units, since $-2$ and $6$ have opposite signs.

The height is the vertical distance from $R$ down to the base: from $y = 4$ to $y = -1$ is $4 + 1 = 5$ units.

A = \frac{1}{2} \times 8 \times 5 = 20 \text{ square units}

Finding a missing vertex

Worked example: Completing a rectangle

Three vertices of a rectangle are $(-4, -1)$ , $(2, -1)$ and $(2, 3)$ . What is the fourth vertex?

The first two points share $y = -1$ , so they form the bottom side. The point $(2, 3)$ is straight above $(2, -1)$ , so it is the top-right corner. The missing corner must be straight above $(-4, -1)$ and level with $(2, 3)$ .

So it has the $x$ -coordinate $-4$ and the $y$ -coordinate $3$ . The fourth vertex is $(-4, 3)$ .

Tip

Sketch it! Even a quick drawing on scrap paper shows you which sides are horizontal and vertical, and lets you count squares to check your side lengths.