Coordinate Area Formulas

HSG.GPE.B.7 — Lesson 2 of 2

Expressing Geometric Properties with Equations

Learning Objectives

Continuing from Lesson 1. By the end of this lesson you will also be able to:

3. Compute triangle area using the coordinate area formula
4. Apply the Shoelace formula to any simple polygon
5. Solve real-world problems using coordinate measurement

Why Is an Area Formula Named After Shoelaces?

The Shoelace formula gets its name from the crisscross pattern of multiplications in its table layout.

Each product pairs coordinates from two adjacent rows — forward products and backward products — crossing back and forth like laces threading through eyelets.

Triangle Coordinate Area Formula

For vertices , , :

• No need to identify a base or height
• Works for any triangle, any orientation
• Absolute value ensures a positive result

Why the Absolute Value?

The formula produces a signed area:

• Counterclockwise vertex order → positive signed area
• Clockwise vertex order → negative signed area

Area is always positive. The absolute value removes the sign regardless of traversal direction.

Verify: Coordinate Formula on Known Triangle

Triangle , , — bounding-rectangle area from Lesson 1: 9 sq units

The Shoelace Formula: Visual Overview

Forward products (sum = 32) minus backward products (sum = 3), halved — gives area = 14.5 sq units.

Shoelace Formula for Any Simple Polygon

• Vertices listed consecutively around the boundary
• Close the polygon:
• Works for triangles, quadrilaterals, pentagons — any simple polygon

Setting Up the Shoelace Table

1. List vertices consecutively around the boundary (clockwise or counterclockwise)
2. Repeat the first vertex at the bottom to close the polygon
3. Forward products: each (current row's , next row's )
4. Backward products: each (next row's , current row's )
5. Area

Shoelace Computation: A(0,0) B(4,0) C(5,3) D(1,4)

• Forward products:
• Backward products:

Check-In: Apply the Shoelace Formula

Quadrilateral: , , ,

1. Set up the table:
2. Compute each forward product ()
3. Compute each backward product ()
4. Area forward sum backward sum

Check-In Answer

Forward ; Backward ; Area

Same Formulas, Real-World Scale

In the real world, coordinates come from:

• Survey data — GPS or physical measurement equipment
• Architectural drawings — floor plans on coordinate grids
• GIS systems — coordinate data for land parcels

The mathematics is identical — just with larger numbers and meaningful units.

Verify: Triangle Decomposition of ABCD

• : , , →
• : , , →

Three Methods: Same Triangle, Same Area

• Base-height: choose one side as base, compute perpendicular height
• Bounding rectangle: enclose in box, subtract corner triangles
• Coordinate formula: plug vertices into

M3: Vertices Must Be Listed in Order

Shoelace requires consecutive boundary traversal.

• Walk around the polygon's perimeter, visiting each vertex once, in sequence
• Wrong: — skips across, creates a self-intersecting shape
• Correct: — traces the boundary consecutively

Prevention: plot the vertices, then trace and list in order.

M4: Absolute Value Is Required

The Shoelace formula produces a signed area — the sign indicates only traversal direction.

App 1: Perimeter of a Garden

Vertices (meters): , , ,

Perimeter m

App 2: Area of a Triangular Lot

Vertices (feet): , ,

App 3: Area of an Irregular Pond

Pentagon (meters): , , , ,

• Forward sum:
• Backward sum:

Exact vs Approximate Answers

Guided Practice: City Block

Quadrilateral: , , ,

1. Is this axis-aligned or tilted? How can you tell?
2. Compute the perimeter — which sides are axis-aligned?
3. Compute the area — which method is most efficient?

Work through all three before checking.

Guided Practice Answer

Axis-aligned rectangle (opposite vertices share coordinate values)

• , , , → Perimeter linear units
• Width Height square units
• Shoelace check: Forward , Backward ; Area ✓

Lesson Summary

1. Triangle coordinate formula — direct from vertices;
2. Shoelace formula — any polygon; set up table, compute forward and backward products
3. All methods agree — coordinate formula, Shoelace, bounding rectangle, base-height
4. M3: List vertices in order — trace the boundary before setting up the table
5. M4: Always take — area is never negative

Coming Up: Geometric Measurement and Dimension

HSG.GMD.A builds on coordinate geometry:

• Volume formulas — from area cross-sections and Cavalieri's principle
• Surface area — coordinate measurement extends to 3D
• HSG.MG.A — applying these tools to real-world geometric models

The coordinate tools from HSG.GPE.B.7 appear throughout both units.