Centroid & Centre of Gravity Notes

Last reviewed 16 Sept 2026 · 6 min read

Definitions

Centre of gravity (CG) — the point through which the resultant of the weights of all particles of a body acts, whatever the orientation of the body.
Centre of mass — the point at which the whole mass can be considered concentrated. In a uniform gravitational field it coincides with the CG.
Centroid — the geometric centre of a line, area or volume. For a homogeneous body the CG and the centroid coincide.

Centroids matter everywhere in structures: the neutral axis of a beam passes through the centroid; loads resolve through centroids; eccentricity of a column load is measured from it.

Centroid by the method of moments

FormulaCentroid coordinates

\overset{x}{ˉ} = \frac{\int x d A}{\int d A}, \overset{y}{ˉ} = \frac{\int y d A}{\int d A}

For composite areas made of simple parts:

\overset{x}{ˉ} = \frac{\sum A _{i} x _{i}}{\sum A _{i}}, \overset{y}{ˉ} = \frac{\sum A _{i} y _{i}}{\sum A _{i}}

For lines replace $A$ by length $L$ ; for volumes by volume $V$ ; for bodies of different materials by weight $W$ .

Axis of symmetry: the centroid lies on every axis of symmetry. If an area has two axes of symmetry, the centroid is at their intersection.

Cut-outs (holes): treat the removed part as a negative area.

Standard centroids of plane areas

Area	Area	Centroid position
Rectangle $b \times h$	$bh$	$h /2$ from base
Triangle, height $h$	$bh /2$	$h /3$ from base ( $2 h /3$ from apex) — intersection of medians
Circle, radius $r$	$π r^{2}$	Centre
Semicircle	$π r^{2} /2$	$\frac{4 r}{3 π} \approx 0.424 r$ from the diameter
Quarter circle (quadrant)	$π r^{2} /4$	$\frac{4 r}{3 π}$ from each straight edge
Circular sector, angle $2 α$	$α r^{2}$	$\frac{2 r sin α}{3 α}$ from the centre
Semi-ellipse (semi-axes $a$ , $b$ , cut along $a$ )	$π ab /2$	$\frac{4 b}{3 π}$ from the major axis
Parabolic spandrel ( $y = k x^{2}$ , under curve from vertex to $x = a$ )	$ah /3$	$\overset{x}{ˉ} = 3 a /4$ , $\overset{y}{ˉ} = 3 h /10$
Semi-parabolic area (between curve and $y$ -axis)	$2 ah /3$	$\overset{x}{ˉ} = 3 a /8$ , $\overset{y}{ˉ} = 3 h /5$
Trapezium, parallel sides $a$ (bottom), $b$ (top), height $h$	$(a + b) h /2$	$\frac{h}{3} \cdot \frac{a + 2 b}{a + b}$ from side $a$

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Definitions

Centroid by the method of moments

Standard centroids of plane areas

This chapter is in the syllabus of