Centroid Calculator – Guide & Formulas
Calculate the centroid (geometric center) of triangles, rectangles, circles, and composite shapes. Get coordinates and formulas instantly.
The Centroid Calculator finds the geometric center (centroid) of common 2D shapes and composite areas. Unlike center of mass which considers mass distribution, the centroid is purely geometric—based on the shape's area distribution.
Key Takeaway
Use the free Centroid Calculator to calculate the centroid (geometric center) of triangles, rectangles, circles, and composite shapes. get coordinates and formulas instantly. Get instant results with step-by-step explanations.
How to Use the Centroid Calculator
- Select the shape type (triangle, rectangle, circle, composite, etc.).
- Enter the required dimensions or vertex coordinates.
- For composite shapes, add multiple sub-shapes with their properties.
- Review the centroid coordinates (x̄, ȳ) and the calculation steps.
The Formula
Variable Definitions
- x̄: X-coordinate of the centroid
- ȳ: Y-coordinate of the centroid
- A_i: Area of the i-th sub-shape
- x_i: X-coordinate of the centroid of the i-th sub-shape
- y_i: Y-coordinate of the centroid of the i-th sub-shape
Example: Centroid of a Triangle
Find the centroid of a triangle with vertices at (0,0), (6,0), and (3,4).
- Step 1: Recall the centroid formula for a triangle: (x̄, ȳ) = ((x₁+x₂+x₃)/3, (y₁+y₂+y₃)/3).
- Step 2: Calculate x̄: (0 + 6 + 3)/3 = 9/3 = 3.
- Step 3: Calculate ȳ: (0 + 0 + 4)/3 = 4/3 ≈ 1.33.
- Step 4: The centroid is at (3, 1.33).
- Step 5: This point divides each median in a 2:1 ratio from vertex to midpoint.
Frequently Asked Questions
What is a centroid?
The centroid is the geometric center of a 2D shape—the point where the shape would balance perfectly if it were cut from uniform-density material. It is the average of all points in the shape.
How is centroid different from center of mass?
Centroid is purely geometric (based on area distribution), while center of mass considers mass distribution. For uniform-density objects, they are identical.
What is the centroid of a triangle?
The centroid of a triangle is at the intersection of its three medians. For vertices (x₁,y₁), (x₂,y₂), (x₃,y₃), the centroid is ((x₁+x₂+x₃)/3, (y₁+y₂+y₃)/3).
Where is the centroid of a rectangle?
For a rectangle with corners at the origin and (l,w), the centroid is at (l/2, w/2)—the exact geometric center.
What is the centroid of a circle?
For a uniform circle or disk, the centroid is at the center point. For a circular arc, the centroid is at distance 2r/π from the center along the axis of symmetry.
How do I find the centroid of a composite shape?
Decompose the shape into simpler parts, find each part's area and centroid, then compute the weighted average: x̄ = Σ(A_i × x_i) / Σ(A_i).
What is the centroid of a semicircle?
For a semicircle of radius R, the centroid is at distance 4R/(3π) from the flat edge, along the axis of symmetry perpendicular to the diameter.
Can the centroid be outside the shape?
Yes, for concave shapes (like a C-shape or crescent), the centroid can lie outside the physical boundary of the shape.
What is the centroid of a trapezoid?
For a trapezoid with parallel sides a and b and height h, the centroid is at height h(a+2b)/(3(a+b)) from the base a, measured along the axis of symmetry.
How is centroid used in structural engineering?
Engineers calculate centroids of cross-sections to determine bending stress distribution, beam deflection, and structural load paths. The neutral axis passes through the centroid.
What is the centroid of a regular polygon?
For any regular polygon (square, pentagon, hexagon, etc.), the centroid is at the geometric center—the intersection of all axes of symmetry.
How does centroid relate to moment of inertia?
The moment of inertia about the centroidal axis is the minimum possible moment of inertia. The parallel axis theorem relates centroidal and non-centroidal moments.
What is the first moment of area?
The first moment of area about an axis is the integral of distance × area. The centroid is located where the first moment of area about that axis equals zero.
How do I find the centroid using integration?
For continuous shapes, use x̄ = (1/A)∫x dA and ȳ = (1/A)∫y dA, where the integrals are taken over the entire area of the shape.
What is the centroid of a triangle's medians?
The three medians of a triangle always intersect at the centroid, which is 2/3 of the way from each vertex to the midpoint of the opposite side.
Can centroid be calculated for 3D shapes?
Yes, 3D centroids follow the same principle: x̄ = Σ(V_i × x_i) / Σ(V_i), using volume instead of area. The centroid of a sphere is at its center.
How is centroid used in fluid mechanics?
The centroid determines the center of pressure on submerged surfaces. Hydrostatic force acts at a point below the centroid, called the center of pressure.
What is the centroid of a composite shape with holes?
Treat holes as negative area. Calculate the centroid of the outer shape as positive, the centroid of each hole as negative, then compute the weighted average.
How do I verify my centroid calculation?
Check that the centroid lies within the bounds of the shape (for convex shapes), verify with known formulas for simple shapes, and ensure the moment of area about the centroid axis equals zero.
What is the centroid of a quarter circle?
For a quarter circle of radius R, the centroid is at (4R/(3π), 4R/(3π)) measured from the center along both axes of symmetry.