Use this calculator to compute Centroid values with step-by-step solutions.

How do I use this calculator?

Enter the required values; results and step-by-step derivation update automatically.

Where is Centroid used?

2D Geometry, calculus, physics, engineering, and related disciplines.

What formulas does this use?

All standard centroid formulas are applied and shown step-by-step in the results.

Can I get step-by-step solutions?

Yes. The calculator shows a full step-by-step derivation when results are displayed.

How accurate are the results?

Results use standard floating-point arithmetic (~15 significant digits). Suitable for education and professional use.

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GEOMETRY2D GeometryMathematics Calculator

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Centroid

The centroid is the geometric center—the balance point of a shape. Triangle: average of 3 vertices. Rectangle: (L/2, W/2). For uniform density, centroid = center of mass.

Concept Fundamentals

C = ((x₁+x₂+x₃)/3, (y₁+y₂+y₃)/3)

Triangle

C = (L/2, W/2)

Rectangle

C = (Σx/n, Σy/n)

Polygon

Center of mass

Balance

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Triangle centroid lies at intersection of medians—2/3 from vertex to opposite side. For uniform density, centroid = center of mass. L-shapes and T-sections use composite centroid formulas.

Key quantities

C = ((x₁+x₂+x₃)/3, (y₁+y₂+y₃)/3)

Triangle

Key relation

C = (L/2, W/2)

Rectangle

Key relation

C = (Σx/n, Σy/n)

Polygon

Key relation

Center of mass

Balance

Key relation

Ready to run the numbers?

Why: Centroids are used in structural engineering (load center), physics (center of mass), and design (balance point). Medians of a triangle intersect at centroid.

How: Triangle: centroid = (x₁+x₂+x₃)/3, (y₁+y₂+y₃)/3. Rectangle: (L/2, W/2). Polygon: average of vertices. Composite: weighted area average.

Triangle centroid lies at intersection of medians—2/3 from vertex to opposite side.For uniform density, centroid = center of mass.

Sources:Wolfram MathWorld - CentroidKhan Academy - Geometry

Run the calculator when you are ready.

Centroid CalculatorEnter triangle vertices, rectangle dimensions, or polygon vertices

📐

GEOMETRY2D Centroids

Centroid — Geometric Center

Find centroid of triangles (3 vertices), rectangles (length×width), or composite shapes (polygon vertices).

Area →Square →

📐 Real-World Examples — Click to Load

Calculation Settings

Shape Mode

Unit

Vertex 1 (x, y)

Vertex 2 (x, y)

Vertex 3 (x, y)

For educational and informational purposes only. Verify with a qualified professional.

🧮 Fascinating Math Facts

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Centroid = geometric center—balance point of a shape.

— Definition

△

Triangle medians intersect at the centroid.

— Property

📋 Key Takeaways

• The centroid is the geometric center — balance point of a shape
• Triangle: centroid = average of 3 vertices; lies at intersection of medians
• Rectangle: centroid = (L/2, W/2) — center of the rectangle
• Composite: for polygons, centroid = average of vertex coordinates
• For uniform density, centroid = center of mass

💡 Did You Know?

📐Triangle centroid divides each median in 2:1 ratio (closer to vertex)Source: Geometry

🏗️Engineers use centroids for beam bending — neutral axis passes through centroidSource: Structural Eng

🎯Center of a target is the centroid of the circular regionSource: Sports

🪑A chair balances when its center of mass (centroid) is over the baseSource: Physics

📏L-shapes and T-sections have centroids that can lie outside the shapeSource: Engineering

🔷For composite shapes, decompose into simple shapes and use weighted averageSource: Mechanics

📖 Formulas Explained

Triangle Centroid

Average of the three vertices. Also the intersection of medians.

C = \left(\frac{x_1+x_2+x_3}{3}, \frac{y_1+y_2+y_3}{3}\right)

Rectangle Centroid

Center of the rectangle — midpoint of the diagonal.

C = \left(\frac{L}{2}, \frac{W}{2}\right)

Polygon Centroid

Arithmetic mean of all vertices for simple polygons.

C = \left(\frac{\sum x_i}{n}, \frac{\sum y_i}{n}\right)

🎯 Expert Tips

💡 Vertex Order

For polygons, enter vertices in order (clockwise or counter-clockwise).

💡 L-Shape

Decompose into rectangles: centroid = weighted average by area.

💡 Right Triangle

Centroid of right triangle (0,0),(a,0),(0,b) is (a/3, b/3).

💡 Symmetry

If a shape has a line of symmetry, the centroid lies on it.

⚖️ Comparison Table

Feature	This Calculator	Basic	Manual
Triangle, rectangle, composite	✅	⚠️ Limited	❌
Step-by-step solutions	✅	❌	✅
Charts & visualization	✅	❌	❌
Copy & share	✅	❌	❌
7 examples	✅	❌	❌

📊 Quick Facts

Triangle Vertices

Rectangle Params

n≥3

Polygon Vertices

2:1

Median Ratio

❓ FAQ

Centroid vs center of mass?

For uniform density they coincide. Center of mass depends on density distribution.

Can centroid be outside the shape?

Yes — for L-shapes, crescents, and other concave shapes.

Does vertex order matter for polygons?

For the arithmetic mean formula, no. For signed-area formula, yes.

How to find centroid of L-shape?

Decompose into rectangles, find each centroid and area, then weighted average.

What is the median ratio in triangles?

Centroid divides each median so the vertex-side segment is 2/3 of the median.

Rectangle with corners (0,0) and (L,W)?

Centroid is (L/2, W/2).

How many points for composite?

At least 3 vertices. Add points in order around the shape.

Units for coordinates?

Use consistent units (m, cm, etc.). Result is in the same units.

📚 Sources

Khan Academy - Centroids ↗

Geometry courses

Updated: 2026-01

Wolfram MathWorld - Centroid ↗

Mathematical definitions

Engineering Toolbox ↗

Centroid tables

Updated: 2025-12

⚠️ Disclaimer: Polygon centroid uses arithmetic mean of vertices. For area-weighted centroid of complex shapes, use decomposition.

👈 START HERE

⬅️Jump in and explore the concept!

Centroid

Centroid — Geometric Center

📐 Real-World Examples — Click to Load

Calculation Settings

🧮 Fascinating Math Facts

📋 Key Takeaways

💡 Did You Know?

📖 Formulas Explained

Triangle Centroid

Rectangle Centroid

Polygon Centroid

🎯 Expert Tips

💡 Vertex Order

💡 L-Shape

💡 Right Triangle

💡 Symmetry

⚖️ Comparison Table

📊 Quick Facts

❓ FAQ

Centroid vs center of mass?

Can centroid be outside the shape?

Does vertex order matter for polygons?

How to find centroid of L-shape?

What is the median ratio in triangles?

Rectangle with corners (0,0) and (L,W)?

How many points for composite?

Units for coordinates?

📚 Sources

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We Value Your Privacy

Centroid

Centroid — Geometric Center

📐 Real-World Examples — Click to Load

Calculation Settings

🧮 Fascinating Math Facts

📋 Key Takeaways

💡 Did You Know?

📖 Formulas Explained

Triangle Centroid

Rectangle Centroid

Polygon Centroid

🎯 Expert Tips

💡 Vertex Order

💡 L-Shape

💡 Right Triangle

💡 Symmetry

⚖️ Comparison Table

📊 Quick Facts

❓ FAQ

Centroid vs center of mass?

Can centroid be outside the shape?

Does vertex order matter for polygons?

How to find centroid of L-shape?

What is the median ratio in triangles?

Rectangle with corners (0,0) and (L,W)?

How many points for composite?

Units for coordinates?

📚 Sources

Related Mathematics Calculators

Related Calculators

Area Calculator

Chord Length Calculator

Crescent Area Calculator

Generic Rectangle Calculator

Rectangle Area Calculator

Geometry2 D Conic Sections Calculator

We Value Your Privacy