A Complete Step-by-Step Guide (With Mathematics)

GPS is something we use every day — but how does it actually find your location?

In this project, we build a 3D working model of GPS trilateration using simple materials like cardboard, plastic balls, and threads — no electronics required.

This guide explains everything from theory to final assembly.

📌 Project Overview

Concept Demonstrated

• 3D Coordinate Geometry
• Distance Formula in 3D
• Intersection of Spheres
• GPS Trilateration Principle

Model Type

• Physical mechanical model
• Removable distance threads
• True-scale coordinate grid
• School-level working demonstration

🧠 The Mathematics Behind GPS

GPS determines your location using distance from satellites.

If a satellite is at coordinate: (a, b, c)

And your receiver is at: (x, y, z)

The distance between them is:

√[(x − a)² + (y − b)² + (z − c)²]

Each satellite forms a sphere in 3D space.

Your location is where all spheres intersect.

📐 Model Coordinates Used

Cube Size:

40 cm × 40 cm × 40 cm

Satellites (Top Corners)

• S1 (0, 0, 40)
• S2 (40, 0, 40)
• S3 (0, 40, 40)
• S4 (40, 40, 40)

Receiver Position

(20, 20, 15)

🧮 Distance Calculation

Distance from S1 to receiver:

√(20² + 20² + (−25)²)
= √(400 + 400 + 625)
= √1425
≈ 37.7 cm

All four satellites are symmetrically placed, so all thread lengths are equal.

This makes the model visually clean and mathematically elegant.

🛠 Materials Required

Structure

• Corrugated cardboard sheets (40×40 cm)
• White chart paper
• Strong adhesive
• Packing tape

Satellites

• 4 Plastic balls (5–6 cm)
• Drill hole through center

Receiver

• 1 Plastic ball (red)
• 15 cm wooden dowel
• 4 small eye screws

Distance Threads

• 4 Nylon threads (38 cm each)
• Loop knots for removable attachment

Tools

• Ruler
• Cutter
• Drill needle
• Pencil

🏗 Step-by-Step Construction

Step 1: Build the Cube

Attach base and three walls at 90° angles. Reinforce corners.

Step 2: Attach A3 Graph Sheets

Use printed 1 cm grid sheets for precision.

Step 3: Install Receiver

• Mark (20,20) on base
• Fix 15 cm dowel vertically
• Attach red ball on top

Step 4: Mount Satellites

Glue plastic balls at top four corners.

Step 5: Attach Threads

• Pass thread through satellite hole
• Tie knot inside
• Create removable loop at other end
• Hook to receiver eye screws

🎬 How to Demonstrate at Expo

Show progressive trilateration:

1 satellite → infinite possible positions
2 satellites → circle
3 satellites → two possible points
4 satellites → one exact location

“This red ball represents the receiver.
Each thread represents the radius of a sphere.
The only point satisfying all four distance equations is this exact coordinate.”

Loosen one thread slightly — the receiver shifts.
Tighten again — it returns.

Judges love interactive models.

🎯 Why 4 Satellites?

With only 3 satellites:

• Two intersection points are possible.

The 4th satellite removes ambiguity and corrects clock error.

💡 What Makes This Project Strong

✔ Real mathematics
✔ Clean 3D coordinate visualization
✔ Interactive demonstration
✔ Low cost
✔ No electronics needed
✔ Strong conceptual clarity

📊 Educational Value

• Algebra
• 3D geometry
• Real-world navigation systems
• Engineering modeling
• Mathematical visualization

It turns textbook formulas into physical reality.

🏆 Final Thoughts

This 3D GPS model proves that advanced concepts like satellite navigation can be demonstrated using simple materials and strong mathematical understanding.

It is ideal for:

• School science expos
• Mathematics exhibitions
• STEM demonstrations
• Concept visualization workshops