Hong Life CalculatorsUnderstanding the centroid calculator — concept and principle
This calculator finds the centre of a set of points in the plane, optionally with a weight (mass) for each point. It shows the total weight, the two moments and the average distance of the points from the centre. Use it for centre-of-mass problems, balancing loads, averaging locations on a map grid or checking homework.
How it is calculated
Total weight m = Σmᵢ
Moment about the y-axis Mᵧ = Σmᵢxᵢ; moment about the x-axis Mₓ = Σmᵢyᵢ
Centre: x̄ = Mᵧ ÷ m, ȳ = Mₓ ÷ m
- Each line is "x, y" or "x, y, weight"; a missing weight counts as 1.
- With all weights 1, the centre is simply the average of the coordinates.
The principle
The centre of mass is the balance point. A point mass pulls with a turning effect (moment) equal to its mass times its distance from an axis. The centre is the single point where all the mass could be concentrated to give the same total moments about both axes. With equal weights it is the plain average of the coordinates; for three points it is the centroid of the triangle they form. For a filled polygon with four or more corners, however, the average of the corners is not the centroid of the area.
Worked example
Default input: points (0, 0), (8, 0), (8, 6), (2, 6), all with weight 1.
- Number of points 4, total weight m = 4.
- Mᵧ = 0 + 8 + 8 + 2 = 18; Mₓ = 0 + 0 + 6 + 6 = 12.
- x̄ = 18 ÷ 4 = 4.5; ȳ = 12 ÷ 4 = 3; centre (4.5, 3).
- Distances to the centre: 5.408, 4.610, 4.610 and 3.905; their mean is 4.63325.
How to read the result
The centre lies inside the cloud of points and is pulled toward heavier or more crowded points. If you double one point's weight, the centre moves part of the way toward it. The mean distance gives a sense of how spread out the points are around the centre.
Common mistakes
- Using this for the centroid of a filled shape; the corner average differs from the area centroid for most polygons.
- Swapping the moments: Mᵧ uses x and Mₓ uses y.
- Forgetting to divide by the total weight rather than the number of points when weights differ.
- Mixing units between points.
Frequently asked questions
How do you find the centroid of points?
Average the x-coordinates and the y-coordinates, weighting each point by its mass if the masses differ.
Is the centroid of a triangle the average of its vertices?
Yes, for a triangle the vertex average equals the centroid of its area.
What is a moment in this context?
Mass times the signed distance from an axis.
Sources: OpenStax, Calculus Volume 2 (2016), §2.6 Moments and Centers of Mass