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Category Geometry & Planimetry

Shape Presets

Vertices (x, y per line) Enter one coordinate pair per line. Formats: "x, y" or "x y"

Centroid X —

Centroid Y —

Area —

Perimeter —

Vertices —

Winding —

Click canvas to add vertices. Drag to move. Right-click to delete.

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About

Miscalculating a centroid propagates errors into every downstream computation: structural load analysis shifts, CNC toolpath offsets drift, and finite-element meshes distort. This tool computes the geometric centroid C(x̅, y̅) of any simple (non-self-intersecting) polygon using the exact closed-form solution derived from the Shoelace (Gauss) area formula. It handles convex and concave polygons with n ≥ 3 vertices. The algorithm runs in O(n) time and produces machine-precision results identical to those from ISO 16792-compliant CAD software. Note: this tool assumes a planar, non-self-intersecting polygon with uniform density. For shapes with holes or variable density, decompose into sub-regions and apply the composite centroid method provided in the reference table.

Formulas

The centroid of a simple polygon with n vertices (x₀, y₀), …, (x_n−1, y_n−1) is computed from the signed area and first moments.

Signed area via the Shoelace formula:

A = 12 n−1∑i=0 (x_iy_i+1 − x_i+1y_i)

Centroid coordinates:

C_x = 16A n−1∑i=0 (x_i + x_i+1)(x_iy_i+1 − x_i+1y_i)

C_y = 16A n−1∑i=0 (y_i + y_i+1)(x_iy_i+1 − x_i+1y_i)

Where A = signed area of the polygon (positive for CCW winding). x_i, y_i = coordinates of the i-th vertex. Index n wraps to 0 (i.e., x_n = x₀). For composite shapes, the weighted centroid is C = ∑ A_i ⋅ C_i∑ A_i, where subtracted regions use negative area.

Reference Data

Shape	Centroid Location	Area Formula	Notes
Triangle	(x₁+x₂+x₃3, y₁+y₂+y₃3)	12\|x₁(y₂−y₃)+x₂(y₃−y₁)+x₃(y₁−y₂)\|	Intersection of medians
Rectangle	(w2, h2)	w × h	Geometric center
Circle	Center point	πr²	By symmetry
Semicircle	(0, 4r3π)	πr²2	Above diameter axis
Quarter Circle	(4r3π, 4r3π)	πr²4	First quadrant
Right Triangle	(b3, h3)	12bh	From right-angle vertex
Trapezoid	y̅ = h3 ⋅ a+2ba+b	(a+b)2h	a, b are parallel sides
Regular Pentagon	Geometric center	54s² ⋅ cot(π5)	All regular polygons: centroid = center
Regular Hexagon	Geometric center	3√32s²	6 equilateral triangles
Ellipse	Center point	πab	By dual symmetry
Circular Sector	2r sinθ3θ	r²θ2	θ in radians, from center along bisector
Parabolic Segment	(0, 2h5)	23bh	From base midpoint
Composite Shape	∑A_iC_i∑A_i	Sum of sub-areas	Subtract holes with negative A
I-Beam (Composite)	Weighted average of 3 rectangles	Sum of flange + web areas	Critical for structural engineering
L-Shape (Angle)	Weighted average of 2 rectangles	Sum of leg areas	Centroid lies outside shape if thin

Frequently Asked Questions

Yes. The Shoelace-based centroid formula is valid for any simple (non-self-intersecting) polygon, whether convex or concave. The signed cross-product terms correctly account for re-entrant regions. The only requirement is that edges do not cross each other. If your polygon self-intersects, decompose it into simple sub-polygons first.

The Shoelace formula produces a signed area: positive for counter-clockwise (CCW) vertex ordering and negative for clockwise (CW). The centroid formula divides by 6A, so a negative A from CW winding naturally corrects the sign of the centroid coordinates. This calculator normalizes the result regardless of input winding, but be aware that some CAD systems require consistent CCW ordering per ISO 13567.

Yes. For concave polygons (e.g., L-shapes, C-shapes, star shapes), the centroid frequently falls outside the polygon boundary. This is physically meaningful: if you cut the shape from uniform sheet material, the balance point would be at that exterior coordinate. This is why structural engineers must verify centroid location relative to support points.

Use the composite centroid method. Calculate the centroid and area of the outer boundary as a positive region, then calculate the centroid and area of the hole as a negative region. Apply the weighted formula: C = (A_outer ⋅ C_outer − A_hole ⋅ C_hole) ÷ (A_outer − A_hole). This tool handles the positive region; perform the subtraction manually for holes.

JavaScript uses IEEE 754 double-precision floating point, providing approximately 15-16 significant decimal digits. This matches or exceeds the precision of most commercial CAD systems (AutoCAD, SolidWorks) for 2D centroid calculations. Rounding errors only become relevant for polygons with more than ~10,000 vertices or coordinates exceeding 10¹².

For a 2D planar shape with uniform density in a uniform gravitational field, the geometric centroid, center of mass, and center of gravity are identical. They diverge only when: (a) density varies across the shape (use weighted integration), (b) the shape exists in a non-uniform gravitational field (relevant only at planetary scale), or (c) the body is 3D with non-uniform thickness. This calculator computes the geometric centroid assuming uniform surface density.