Calculator · Engineering Mechanics
Centroid Calculator
The area and geometric centroid of one shape or of a composite built from several. A cutout is a component with negative signed area — the same algebra, never a special formula.
Components
- ShapePositionMaterial & massMaterialMass model
Rectangle · solid · no mass defined
1 of 16 components.
Diagram
- Solid component — filled outline
- Cutout — dashed outline, negative area
- · component centroid
- C — geometric centroid
- G — centre of mass
- Dashed through C — centroidal x̄ / ȳ axes
- 1 / 2 — principal axes through C
- a — custom axis through A, long-dashed
Results
| Total signed area A | 200 cm² |
|---|---|
| Geometric centroid x̄ | 0 m |
| Geometric centroid ȳ | 0 m |
| Centroid C | (0, 0) m |
This is the geometric centroid of the area. It coincides with the centre of mass only when the density is the same everywhere.
Diagnostics
Technical reference
x̄ = ΣAi·xi / ΣAi
ȳ = ΣAi·yi / ΣAi
- Ai
- signed area of component i (negative for a cutout)[m²]
- xi, yi
- centroid of component i[m]
- x̄, ȳ
- composite centroid[m]
Every component contributes ΣAi·xi with its own sign, so a cutout subtracts from both the area and the moment sums — there is no separate hole formula.
The centroid is a purely geometric location. It coincides with the centre of mass for a uniformly dense body. With non-uniform density they generally differ, although special symmetric mass distributions can still place them at the same point.
Assumptions
- · Planar geometry in a global x-y frame; +x right, +y up.
- · A cutout contributes a negative signed area; the centroid uses the same weighted sum.
- · Components are combined algebraically: overlapping solids are counted independently and containment is not checked.
- · A net signed area of zero leaves the centroid undefined — no value is invented.
Continue with this section
The same components, with inertia, principal axes and mass properties.