What is the formula for centre of gravity?

For a system of point masses: x̄ = Σ(mᵢ·xᵢ) / Σ mᵢ and ȳ = Σ(mᵢ·yᵢ) / Σ mᵢ. Sum is over all masses; (xᵢ, yᵢ) is each mass's position; mᵢ is its mass. The result is the position where the entire system's gravity can be considered to act.

Is centre of gravity the same as centre of mass?

In a uniform gravitational field — yes, they coincide. In non-uniform gravity (e.g. very tall structures, satellites in orbital mechanics) they differ slightly because mass closer to the gravitational source is pulled harder. For all practical earth-bound engineering, CG = COM.

What is the difference between centre of gravity and centroid?

Centroid is mass-independent — purely a geometric property of an area or volume (area-weighted mean position). Centre of gravity is mass-weighted — accounts for non-uniform density. For an object made of one uniform material, centroid = CG. For an assembly of different materials (or for engineering plates with attachments), CG ≠ centroid because mass distribution differs from area distribution.

How do you find the centre of gravity of an irregular shape?

Three methods. (1) Discretise into known shapes: break into rectangles, circles, triangles; for each find centroid and area; weighted average. (2) Plumb-line method: hang the object from two different points; the CG lies on the vertical line below each suspension point — intersection is CG. (3) Balance method: rest on a knife edge in two directions; CG is directly above the balance point.

How do you balance a system using CG?

A system balances when the support is directly under the CG. Lever rule: m₁·d₁ = m₂·d₂ (mass times distance from pivot equals on both sides). For multi-mass systems use the formula above to find CG, then place the support at that point.

Calculator · Structural / Statics · Point-mass system

Centre of gravity

Add a list of point masses with their positions and the calculator finds the centre of gravity (x̄, ȳ) — the point where the entire system\'s weight effectively acts. Reviewed by a licensed PE.

Use the centre of gravity calculator

Add point masses with their (x, y) positions and mass values. Units don\'t matter as long as you\'re consistent — answer comes back in the same units. The SVG preview shows mass dots sized by m and the CG marker.

CALC.009 Composite Centroid · Built-up sections · Parallel-axis theorem

Centroid x̄

— mm

Centroid ȳ

— mm

Total area: —
Ix about overall centroid: —
Iy about overall centroid: —
Polar J = Ix + Iy: —

FORMULA · x̄ = Σ(Aᵢ·xᵢ) / Σ Aᵢ · I = Σ(Iᵢ + Aᵢ·dᵢ²) SOURCE · BEER & JOHNSTON · ROARK

Centre of gravity — at a glance

Centre of gravity formula: For point masses: x̄ = Σ(m_i·x_i) / Σ m_i, ȳ = Σ(m_i·y_i) / Σ m_i. Mass-weighted average of position.
CG vs centroid: CG is mass-weighted; centroid is area- (or volume-) weighted. They coincide only when density is uniform — otherwise CG shifts toward heavier regions.
CG vs centre of mass: In a uniform gravitational field they coincide exactly. They differ slightly for very tall structures or orbital mechanics where g varies with position.
Composite shape rule: Break the body into known sub-shapes (rectangles, circles, triangles), find each centroid and area (or mass), then x̄ = Σ(A_i·x̄_i) / Σ A_i.
Holes treated as: Subtract them: negative mass (or area). A 100×100 plate with a 20 mm hole at (30, 50) is solved as the full plate minus a small disc — the hole shifts CG away from itself.
Plumb-line method: Hang an irregular object from two different points. The CG lies on the vertical (plumb) line below each suspension point — their intersection is the CG.
Common shape CG locations: Rectangle: at geometric centre. Right triangle: 1/3 from each leg (b/3, h/3). Half-disc of radius R: 4R/(3π) ≈ 0.424R from the flat edge. Cone: h/4 from the base.

The centre of gravity formula

Eq. 01 — Centre of gravity (point masses) SI · Newtonian mechanics

\bar{x} = \frac{\sum m_{i} \cdot x_{i}}{\sum m_{i}} \qquad \bar{y} = \frac{\sum m_{i} \cdot y_{i}}{\sum m_{i}}

x̄, ȳ: centre of gravity coordinates, consistent with input
m_i: each individual mass, kg / lb / etc
x_i, y_i: each mass's position, consistent units

Worked example: balance three masses

Three masses on a horizontal plane: 2 kg at (0, 0), 5 kg at (4, 0), 3 kg at (2, 3). Find the CG.

Mass	x	y	m	m·x	m·y
1	0	0	2	0	0
2	4	0	5	20	0
3	2	3	3	6	9
Total			10	26	9

x̄ = 26 / 10 = 2.6, ȳ = 9 / 10 = 0.9. The CG sits at (2.6, 0.9). Pulled toward the heavier mass (5 kg at x=4) but raised slightly by the 3 kg mass at y=3.

CG vs centroid vs centre of mass

Term	Weighted by	Used in
Centre of gravity	Mass × local gravitational field	Statics, balance, rigid-body mechanics
Centre of mass	Mass only (uniform g assumed)	Dynamics, momentum, rotational kinetics
Centroid	Area or volume only	Beam bending, section properties

For all earth-bound engineering at human scales, gravity is uniform enough that CG = centre of mass. The distinction matters only for tall structures, satellites, and orbital mechanics.

Related concepts on this site

Frequently asked questions

What is the formula for centre of gravity?: For a system of point masses: x̄ = Σ(mᵢ·xᵢ) / Σ mᵢ and ȳ = Σ(mᵢ·yᵢ) / Σ mᵢ. Sum is over all masses; (xᵢ, yᵢ) is each mass's position; mᵢ is its mass. The result is the position where the entire system's gravity can be considered to act.
Is centre of gravity the same as centre of mass?: In a uniform gravitational field — yes, they coincide. In non-uniform gravity (e.g. very tall structures, satellites in orbital mechanics) they differ slightly because mass closer to the gravitational source is pulled harder. For all practical earth-bound engineering, CG = COM.
What is the difference between centre of gravity and centroid?: Centroid is mass-independent — purely a geometric property of an area or volume (area-weighted mean position). Centre of gravity is mass-weighted — accounts for non-uniform density. For an object made of one uniform material, centroid = CG. For an assembly of different materials (or for engineering plates with attachments), CG ≠ centroid because mass distribution differs from area distribution.
How do you find the centre of gravity of an irregular shape?: Three methods. (1) Discretise into known shapes: break into rectangles, circles, triangles; for each find centroid and area; weighted average. (2) Plumb-line method: hang the object from two different points; the CG lies on the vertical line below each suspension point — intersection is CG. (3) Balance method: rest on a knife edge in two directions; CG is directly above the balance point.
How do you balance a system using CG?: A system balances when the support is directly under the CG. Lever rule: m₁·d₁ = m₂·d₂ (mass times distance from pivot equals on both sides). For multi-mass systems use the formula above to find CG, then place the support at that point.

Sources and methodology

Halliday, D., Resnick, R., Walker, J. Fundamentals of Physics, 11th Edition. Wiley, 2018. Chapter 9 (centre of mass / centre of gravity).
Beer, F. P., Johnston, E. R. Vector Mechanics for Engineers: Statics, 12th Edition. McGraw-Hill, 2018.
Hibbeler, R. C. Engineering Mechanics: Statics, 14th Edition. Pearson, 2016.