The Centroid Calculator calculates the center point of a shape — the point where it would balance if cut from a flat, even sheet.
For a triangle, the centroid is simply the average of its three corners. For any polygon, the calculator finds the true area centroid from the corner coordinates using the shoelace-based centroid formula. A third mode finds the average (mean) point of any set of points.
Centroid Calculator (Center of a Shape)
Find the centre (centroid) of a triangle, a polygon from its corner coordinates, or the average point of a set of points.
Formula
Triangle:
Cx = (x₁ + x₂ + x₃) ÷ 3, Cy = (y₁ + y₂ + y₃) ÷ 3
Polygon (area centroid):
Cx = Σ (xᵢ + xᵢ₊₁)(xᵢyᵢ₊₁ − xᵢ₊₁yᵢ) ÷ 6A
Cy = Σ (yᵢ + yᵢ₊₁)(xᵢyᵢ₊₁ − xᵢ₊₁yᵢ) ÷ 6A
Average of points:
x̄ = Σx ÷ n, ȳ = Σy ÷ n
Where (xᵢ, yᵢ) are corners in order, A is the polygon’s signed area from the shoelace formula, and n is the number of points.
How the Calculation Works
Step 1: The polygon is split into triangles fanning from the origin, each with its own area and centroid.
Step 2: Each triangle’s centroid is weighted by its area.
Step 3: The weighted centroids are added and divided by the total area, giving the balance point of the whole shape.
Variables Explained
| Mode | What it finds |
|---|---|
| Polygon | Center of area (geometric centroid) |
| Triangle | Intersection of the three medians |
| Average of points | Mean position; every point counts equally |
Example Calculation
L-shaped polygon
Given: (0, 0), (10, 0), (10, 4), (4, 4), (4, 10), (0, 10) cm
Check by parts: a 10 × 4 rectangle (area 40, center (5, 2)) plus a 4 × 6 rectangle (area 24, center (2, 7))
Cx = (40 × 5 + 24 × 2) ÷ 64 = 3.875; Cy = (40 × 2 + 24 × 7) ÷ 64 = 3.875
Result: centroid = (3.875, 3.875) cm; area = 64 cm²
Triangle
Given: (0, 0), (6, 0), (2, 6)
Result: centroid = ((0 + 6 + 2) ÷ 3, (0 + 0 + 6) ÷ 3) ≈ (2.67, 2)
Shape Properties
Centroid vs average of corners
For a triangle they are the same point. For other polygons they usually differ, because the area centroid depends on how the area is spread, not just where the corners are. In the L-shape above, the average of the six corners is (4.67, 4.67), not (3.875, 3.875).
Centroid outside the shape
For shapes with inward corners (such as an L or C shape), the centroid can fall outside the shape itself.
Dimensions Explained
Enter polygon corners in order around the shape. Symmetric shapes always have their centroid on the line of symmetry, which is a quick way to check a result.
Frequently Asked Questions
How do you find the centroid of a triangle?
Average the three x-coordinates and the three y-coordinates.
How do you find the centroid of a polygon?
Use the area-weighted formula with the corner coordinates, or split the polygon into rectangles and triangles and average their centroids weighted by area.
Is the centroid the same as the center of mass?
For a flat shape of uniform thickness and material, yes.
Can the centroid be outside the shape?
Yes, for shapes like an L, C or ring.
How do you find the centroid of a composite shape?
Multiply each part’s area by its centroid, add the results and divide by the total area.