Centroid of a plane shape and composite section online: coordinates, first moments, moments of inertia

Build a shape from simple parts: rectangle, triangle, circle, semicircle, quarter circle or a polygon by coordinates. Tick the box for holes and cut-outs. The calculator finds the area, first moments and the centroid coordinates with formulas, plus the centroidal moments of inertia Ix, Iy, Ixy by the parallel-axis theorem, and draws the shape.

More examples

How to find the centroid of a plane shape

Split a complex shape into simple parts whose centroids are known, then use the composite-body formulas. For each part take its area Aᵢ and centroid coordinates xᵢ, yᵢ in a common coordinate system:

xC = ΣAᵢ·xᵢ / ΣAᵢ = Sy / A, yC = ΣAᵢ·yᵢ / ΣAᵢ = Sx / A.

Here Sx = ΣAᵢyᵢ and Sy = ΣAᵢxᵢ are the first moments of area about the x and y axes.

Negative area method

A hole or cut-out is treated as a part with negative area: its Aᵢ enters the sums with a minus sign, and its centroid is the centroid of the removed shape. This is why a plate with a round hole has a centroid shifted away from the hole.

Centroids of simple shapes

Rectangle — at the intersection of the diagonals, A = b·h.
Triangle — at the intersection of the medians, one third of the height from the base; the coordinates are the mean of the vertices.
Circle — at the centre, A = πr².
Semicircle — on the axis of symmetry at 4r/(3π) ≈ 0.424r from the diameter, A = πr²/2.
Quarter circle — at 4r/(3π) from both straight edges, A = πr²/4.
Polygon — by the shoelace formulas: A = ½Σ(xₖyₖ₊₁ − xₖ₊₁yₖ), xC = Σ(xₖ + xₖ₊₁)(xₖyₖ₊₁ − xₖ₊₁yₖ)/(6A), and likewise yC.

Moments of inertia and the parallel-axis theorem

Moments of inertia of the parts about their own centroidal axes are known: rectangle Ix = bh³/12, circle Ix = πr⁴/4, semicircle about an axis parallel to the diameter (π/8 − 8/(9π))r⁴ ≈ 0.1098r⁴, right triangle bh³/36. For a composite shape they are transferred to the centroidal axes of the whole shape: Ix = Σ±(Ix,i + Aᵢbᵢ²), Iy = Σ±(Iy,i + Aᵢaᵢ²), Ixy = Σ±(Ixy,i + Aᵢaᵢbᵢ), where aᵢ = xᵢ − xC, bᵢ = yᵢ − yC; cut-outs take the minus sign. Principal moments: I₁,₂ = (Ix + Iy)/2 ± √(((Ix − Iy)/2)² + Ixy²). These are exactly the section properties needed in strength of materials for bending.

Example

T-section: flange 12×4 and web 4×16 (cm), axes along the bottom edge and the left edge of the flange. Press “Example” — the calculator shows the table of areas, the centroid (xC = 6, yC ≈ 12.29 cm), the moments of inertia and the drawing. Sanity check: for a symmetric shape the centroid lies on the axis of symmetry and Ixy = 0.

FAQ

How do I find the centroid of a composite shape?

Split it into simple parts, find the area and centroid of each in one coordinate system, and take the weighted mean: xC = ΣAᵢxᵢ / ΣAᵢ, yC = ΣAᵢyᵢ / ΣAᵢ.

How do I account for a hole or cut-out?

Use the negative area method: treat the removed shape as an ordinary part but take its area with a minus sign. In the calculator just tick “Cut-out”.

Where is the centroid of a semicircle and a quarter circle?

Semicircle: on the axis of symmetry at 4r/(3π) ≈ 0.424r from the diameter. Quarter circle: at 4r/(3π) from each of its two straight edges.

What is the first moment of area?

The sum of areas times distances to an axis: Sx = ΣAᵢyᵢ, Sy = ΣAᵢxᵢ. About any axis through the centroid it equals zero — that is the defining property of the centroid.

Can I use this for strength of materials?

Yes: besides the centroid it computes Ix, Iy, Ixy about centroidal axes by the parallel-axis theorem, and the principal moments I₁, I₂ with the angle α — the input data for bending calculations.