Centroid Calculator
Find the center of mass $(\bar{x}, \bar{y})$ of a region bounded by curves
$$ A = \int_{-2}^{2} (4-x^2) dx $$
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Lower Curve g(x) (Optional)
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Centroid Coordinates
Region & Centroid
Step-by-Step Solution
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By Prof. David Anderson
Professor of Mathematics | 20+ Years Experience
"The concept of the Centroid often confuses students because it changes definition based on context. In Geometry, it's a simple average. In Calculus, it's about balancing 'Moments' using integration. I created this Centroid Calculator (Center of Mass Calculator) to handle both scenarios: finding the geometric center of a triangle or calculating the centroid of a region bounded by curves."
Centroid Calculator Guide: Find Geometric Center and Center of Mass
How to Calculate the Centroid of Triangles, Polygons, and Integration Regions
The Centroid (often called the Geometric Center) is the arithmetic mean position of all the points in a shape. Think of it as the perfect balancing point. If you need to find the Center of Mass for a shape with uniform density, you are looking for the Centroid.
Finding it requires different math depending on your problem. Coordinate Geometry uses simple averages for polygons, while Calculus uses integration for curves. Our Centroid Calculator handles both methods step-by-step.
1. Geometry Method: Discrete Points
If you need to find the centroid of a triangle with vertices, the math is simple. The centroid coordinates $(\bar{x}, \bar{y})$ are just the average of the x and y values.
Centroid Formula (Triangle)
$$ (\bar{x}, \bar{y}) = \left( \frac{x_1+x_2+x_3}{3}, \frac{y_1+y_2+y_3}{3} \right) $$
Algorithm Centroid of Polygon
To calculate the centroid of a polygon, we decompose it into simpler shapes (rectangles, triangles). Our Geometric Center Calculator uses the weighted average formula:
$$ \bar{x} = \frac{\sum A_i x_i}{\sum A_i}, \quad \bar{y} = \frac{\sum A_i y_i}{\sum A_i} $$
Where $A_i$ is the area and $(x_i, y_i)$ is the centroid of each sub-shape.
2. Calculus: Centroid of a Region
In Calculus II, we calculate the Centroid of a region bounded by curves $f(x)$ and $g(x)$. This requires using the Calculus Centroid Formula involving integrals and moments.
Calculus Centroid Formulas
$$ \bar{x} = \frac{1}{A} \int_a^b x [f(x) - g(x)] \, dx $$
$$ \bar{y} = \frac{1}{A} \int_a^b \frac{1}{2} [f(x)^2 - g(x)^2] \, dx $$
$$ \bar{y} = \frac{1}{A} \int_a^b \frac{1}{2} [f(x)^2 - g(x)^2] \, dx $$
Find the Area First!
Before using the Centroid Calculator to find $\bar{x}$ or $\bar{y}$, you must first calculate the total Area ($A$) of the region:
$A = \int_a^b [f(x) - g(x)] \, dx$.
3. How to Calculate Centroids with Integration
Follow these steps to find the Center of Mass manually, or verify the results from our Centroid Integration Calculator.
Step 1
Calculate Area (A)
Set up the definite integral for the area between the curves $f(x)$ and $g(x)$. This is the denominator in the centroid formula.
Step 2
Calculate Moments ($M_y$ and $M_x$)
Find the First Moment of Area.
$M_y$ (for $\bar{x}$) $= \int x(f-g)dx$
$M_x$ (for $\bar{y}$) $= \int \frac{1}{2}(f^2-g^2)dx$
$M_y$ (for $\bar{x}$) $= \int x(f-g)dx$
$M_x$ (for $\bar{y}$) $= \int \frac{1}{2}(f^2-g^2)dx$
Step 3
Divide Moments by Area
Finally, use the Center of Mass Calculator logic: divide the moments by the total area to get the specific coordinates.
$\bar{x} = M_y / A, \quad \bar{y} = M_x / A$
4. Master Class: Centroid Examples
Type A: Centroid of a Triangle
Geometry
Find the centroid of a triangle with vertices $(0,0), (4,0), (2,3)$.
$\bar{x} = \frac{0 + 4 + 2}{3} = \frac{6}{3} = 2$
$\bar{y} = \frac{0 + 0 + 3}{3} = \frac{3}{3} = 1$
Centroid Coordinates: $(2, 1)$
$\bar{y} = \frac{0 + 0 + 3}{3} = \frac{3}{3} = 1$
Centroid Coordinates: $(2, 1)$
Type B: Centroid of Region Bounded by Curves
Calculus Integration
Use the Centroid Calculator to find the center of the region under $y = x^2$ from $x=0$ to $x=3$. ($g(x)=0$).
1. Find Area: $\int_0^3 x^2 dx = [\frac{x^3}{3}]_0^3 = 9$.
2. Calculate $M_y$: $\int_0^3 x(x^2) dx = \int x^3 dx = [\frac{x^4}{4}]_0^3 = 20.25$.
3. Calculate $M_x$: $\frac{1}{2} \int_0^3 (x^2)^2 dx = \frac{1}{2} \int x^4 dx = 24.3$.
$\bar{x} = 20.25 / 9 = 2.25$
$\bar{y} = 24.3 / 9 = 2.7$
Centroid: $(2.25, 2.7)$
$\bar{y} = 24.3 / 9 = 2.7$
Centroid: $(2.25, 2.7)$
5. Professor's FAQ
What is the difference between Centroid and Center of Mass?
The Centroid is purely geometric—it assumes the shape has uniform density. The Center of Mass Calculator takes into account varying density ($\rho(x)$). If density is constant, the Centroid and Center of Mass are the same point.
Can the centroid be outside the shape?
Yes! When you find the centroid of non-convex shapes like a boomerang, a donut, or an L-shaped bracket, the geometric center often lies in the empty space outside the material boundaries.
How do I use symmetry to find the centroid?
Always check for symmetry! If a region is symmetric about the y-axis, then $\bar{x} = 0$ automatically. If it's symmetric about a line $x=k$, then $\bar{x}=k$. This is a key trick in our Centroid Finder.
What if the region is bounded by y-functions?
If curves are given as $x=f(y)$ and $x=g(y)$, you must integrate with respect to $y$ ($dy$). The Centroid Calculator swaps the formulas: Area uses $dy$, $\bar{y}$ uses the standard moment, and $\bar{x}$ uses the $1/2$ squared moment.
References & Further Reading
- Stewart, J. (2020). Calculus: Early Transcendentals (9th ed.). Cengage Learning. (Section 8.3: Applications to Physics and Engineering).
- Hibbeler, R. C. (2016). Engineering Mechanics: Statics (14th ed.). Pearson. (Chapter 9: Center of Gravity and Centroid).
- Paul's Online Math Notes. "Center of Mass." Lamar University.
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