Finding the Centroid of a Composite Shape

Introduction

Have you ever wondered how engineers and architects determine the balance point of irregular objects? Whether designing bridges, robots, or even everyday household items, knowing where the centroid lies is essential. In this WebQuest, you will investigate how to estimate the centroid of complex shapes by breaking them down into simpler geometric figures. Through exploration, computation, and teamwork, you will learn how mathematics helps us understand the physical world.

Task

Your task is to estimate the centroid of a composite shape created using at least three basic shapes (triangles, rectangles, circles, semi‑circles, quarter‑circles, ellipses, etc.).

You will:

Construct your own composite shape (digital or hand‑drawn).
Decompose it into simple geometric parts.
Compute the centroid of each part.
Combine the results to estimate the centroid of the entire shape.
Present your findings in a short report or slide presentation that includes:
- A labeled drawing of the composite shape
- Decomposed parts with coordinates
- Computations
- Final estimated centroid
- Brief reflection on accuracy and difficulty

Process

Step 1: Learn the Concept

Review how centroids of simple shapes are determined. Use the resources provided below.

Step 2: Create Your Composite Shape

Design a shape made of:

At least three basic shapes
Shapes must touch or overlap logically
Choose a coordinate origin for reference

Step 3: Decompose the Shape

Break your composite figure into its individual components.

Label each shape clearly (e.g., Shape A, Shape B, Shape C).

Step 4: Compute Component Centroids

For each individual part:

Identify centroid coordinates (xᵢ, yᵢ)
Determine the area (Aᵢ)

Step 5: Compute Composite Centroid

Use the formula: ∑Aixi/∑Ai and ∑Aiyi/∑Ai

Step 6: Create Your Output

Your final presentation must include:

The full composite diagram
Decomposed shapes
Calculation table
Final centroid highlighted
Reflection (What was challenging? What did you learn?)

Step 7: Submit and Share

Submit the link to your WebQuest and share your findings during class.

Resources

Centroid Formulas for Basic Shapes

Rectangles – centroid at center
Triangles – centroid at average of vertices
Circles – centroid at center
Semi‑circles – centroid located at 4r3π\frac{4r}{3\pi}3π4r from the flat side
Quarter circles – centroid at 4r3π\frac{4r}{3\pi}3π4r in both directions
Ellipses – centroid at center

Helpful Websites

Optional Tools

GeoGebra
Desmos Geometry
Graph paper PDF generators
Online centroid calculators (for checking only)

Evaluation

Criteria	Excellent (4)	Good (3)	Fair (2)	Needs Improvement (1)
Composite Shape Design	Creative, clear, uses 3 or more shapes	Clear, uses at least 3 shapes	Uses only 2 shapes	Shape unclear or minimal
Decomposition	All parts labeled and logical	Mostly logical with minor issues	Some parts missing	Incorrect or unclear
Calculations	Accurate, complete, well-organized	Mostly accurate with few errors	Several errors present	Little to no correct computation
Final Centroid	Accurate and clearly shown	Mostly correct	Some inaccuracies	Missing or incorrect
Presentation	Clear, neat, well explained	Understandable	Lacks clarity or detail	Incomplete
Reflection	Insightful and thoughtful	Adequate	Minimal	Missing

Conclusion

By completing this WebQuest, you have explored how engineers and designers break down complex shapes in order to analyze them mathematically. Estimating the centroid of a composite shape requires understanding area, geometry, and precision in computation. Reflect on your method and consider how changing the shape or decomposition might affect your results. The ability to deconstruct a complex problem into manageable parts is a powerful mathematical skill you will use across many subjects.