01 / 21

Coordinates to Find
Perimeter & Area

Use the coordinate plane and the distance formula to compute perimeters of polygons and areas of triangles and rectangles.

Geometry

What is Coordinate Geometry?

Coordinate geometry lets us study shapes by placing them on the coordinate plane. Every vertex of a polygon becomes an ordered pair \((x, y)\), transforming geometric questions into algebraic calculations.

1 2 3 4 5 6 1 2 3 4 5 x y \((1, 1)\) \((5, 1)\) \((5, 5)\) \((1, 5)\)

The Distance Formula

The distance between two points \((x_1, y_1)\) and \((x_2, y_2)\) comes directly from the Pythagorean Theorem:

\( (x_1, y_1) \) \( (x_2, y_2) \) \( x_2 - x_1 \) \( y_2 - y_1 \) d

The horizontal and vertical legs form a right triangle. The hypotenuse is the distance d.

Why Coordinates Matter

Precision
Exact numerical answers: no measuring errors, no rounding from a protractor or ruler.
Any Shape
Works for triangles, rectangles, and irregular polygons: just list the vertices.
Real-World
GPS coordinates, computer graphics, architecture, and surveying all use this approach.

Computing Perimeter

The perimeter of any polygon is the sum of all its side lengths. With coordinates, compute each side using the distance formula, then add:

For a rectangle aligned with the axes, this simplifies since opposite sides are equal: \(P = 2(\text{length} + \text{width})\).

Computing Area

Rectangle
If sides are parallel to the axes: \(A = \text{length} \times \text{width}\). Use the distance formula to find each dimension.
Triangle
Use the coordinate area formula:
\(A = \tfrac{1}{2}|x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2)|\)

Perimeter vs. Area

Perimeter Area
Measures Total boundary length Interior surface
Units Linear (cm, m) Square (cm², m²)
Method Sum of all side lengths Formula specific to shape
Key Formula Tool Distance formula for each side Shoelace / \(\text{base} \times \text{height}\)

Step-by-Step Approach

1
Plot the vertices. Write down each ordered pair \((x, y)\) and sketch on a coordinate plane.
2
Identify consecutive vertices. List the sides by connecting vertices in order, returning to the first.
3
Apply the distance formula to each pair of consecutive vertices to find each side length.
4
Sum for perimeter. Add all computed side lengths.
5
Use the area formula. Apply the Shoelace formula (triangle) or \(\text{length} \times \text{width}\) (rectangle).

Example 1: Rectangle Perimeter

Find the perimeter of a rectangle with vertices \(A(1, 2)\), \(B(7, 2)\), \(C(7, 5)\), and \(D(1, 5)\).

1 2 3 4 5 6 7 1 2 3 4 5 \(6\) \(3\) \(6\) \(3\) A(1,2) B(7,2) C(7,5) D(1,5)

Example 2: Rectangle Area

Using the same rectangle with vertices \(A(1, 2)\), \(B(7, 2)\), \(C(7, 5)\), \(D(1, 5)\), find its area.

For axis-aligned rectangles, area is simply the product of the horizontal and vertical distances between opposite vertices.

Example 3: Triangle Perimeter

Find the perimeter of \(\triangle PQR\) with \(P(0, 0)\), \(Q(6, 0)\), and \(R(3, 4)\).

\(PQ = 6\) \(QR = 5\) \(PR = 5\) P(0,0) Q(6,0) R(3,4)

Example 4: Triangle Area

Find the area of \(\triangle PQR\) with \(P(0, 0)\), \(Q(6, 0)\), \(R(3, 4)\) using the coordinate formula.

Verification: base \(PQ = 6\), height from \(R\) to base = \(4\). So \(A = \tfrac{1}{2}(6)(4) = 12\). ✓

Example 5: Non-Axis-Aligned Triangle

Find the perimeter and area of a triangle with vertices \(A(1, 1)\), \(B(4, 5)\), \(C(7, 2)\).

1 2 3 4 5 6 7 1 2 3 4 5 A(1,1) B(4,5) C(7,2) \(5\) \(3\sqrt{2}\) \(\sqrt{37}\)

The Shoelace Formula

For any polygon with vertices listed in order, the Shoelace formula generalizes the triangle area formula:

The name comes from the criss-cross pattern of multiplications: multiply diagonally down, then diagonally up, subtract, and take half the absolute value.

Common Mistakes

✕ Forgetting Absolute Value
The Shoelace formula can yield a negative number. Always take the absolute value before halving.
✕ Wrong Vertex Order
Vertices must be listed consecutively (clockwise or counterclockwise). Jumping across the polygon gives wrong side lengths.
✕ Forgetting to Close
For perimeter, you must include the side from the last vertex back to the first. Missing it leaves the perimeter short.

Summary of Formulas

What to Find Formula When to Use
Side Length \(\sqrt{(\Delta x)^2 + (\Delta y)^2}\) Any two points
Perimeter Sum of all side lengths Any polygon
Rectangle Area \(\text{length} \times \text{width}\) Axis-aligned rectangle
Triangle Area \(\tfrac{1}{2}|x_1(y_2-y_3)+x_2(y_3-y_1)+x_3(y_1-y_2)|\) Any triangle
General Polygon Area Shoelace formula Any simple polygon

Question 1 of 5

What is the distance between the points \((2, 3)\) and \((6, 6)\)?

Question 2 of 5

A rectangle has vertices at \((0,0)\), \((8,0)\), \((8,3)\), and \((0,3)\). What is its perimeter?

Question 3 of 5

Find the area of a triangle with vertices \(A(0,0)\), \(B(8,0)\), and \(C(4,6)\).

Question 4 of 5

The perimeter of \(\triangle DEF\) with \(D(0,0)\), \(E(3,4)\), and \(F(6,0)\) is:

Question 5 of 5

A rectangle has vertices \((2,1)\), \((2,5)\), \((9,5)\), and \((9,1)\). What is its area?