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Center of area of ​​geometric figures

A click on the respective picture leads to the calculation of the centroids:

Centroid triangle
Triangle
Centroid parallelogramm
Parallelogram
Centroid trapezoid
Trapezoid
Centroid circle section
Circle section
Centroid circle segment
Circle segment
Centroid circular ring piece
Circular ring section
Centroid parabolic surfaces
Parabolic surfaces
Centroid parabolic section
Parabolic section
Centroid elliptical section
Elliptical section

General information and exercises for calculating centroids can be found under the respective keyword.

Centers of area in the plane

Triangle

Centroid triangle
Centroid triangle

\[ y_S = \frac{h}{3}\]

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Parallelogram

Centroid parallelogramm
Centroid parallelogram

\[ y_S = \frac{h}{2}\]

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Trapezoid

Centroid trapezoid
Centroid trapezoid

\[ y_S = \frac{h}{3} \cdot \frac{a+2\cdot b}{a+b}\]

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Circle section

Centroid circle section
Centroid circle section

\[ y_S = \frac{2 \cdot r \cdot sin \alpha}{3 \cdot \alpha}\]

\[ y_S = \frac{2 \cdot r \cdot l}{3 \cdot b}\]

For the semicircle it is

\[ y_S = \frac{4 \cdot r }{3 \cdot \pi}\]

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Circle segment

Centroid circle segment
Centroid circle segment

\[ y_S = \frac{2}{3} \cdot \frac{r \cdot sin^3 \alpha}{\alpha -- sin \alpha \cdot cos \alpha} \]

For the semicircle it is

\[ y_S = \frac{4 \cdot r }{3 \cdot \pi}\]

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Circular ring section

Centroid circular ring piece
Centroid circular ring section

\[ y_S = \frac{2}{3} \cdot \frac{\left( r_a^3-r_i^3 \right) \cdot sin \alpha}{\left( r_a^2-r_i^2\right) \cdot \alpha} \]

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Parabolic surfaces

Centroid parabolic surfaces
Centroid parabolic surfaces

Centroid S1

\[ x_{S1} = \frac{3 \cdot a}{8} \]

\[ y_{S1} = \frac{2 \cdot h}{5} \]

Centroid S2

\[ x_{S2} = \frac{3 \cdot a}{4} \]

\[ y_{S2} = \frac{3 \cdot h}{10} \]

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Parabolic section

Centroid parabolic section
Centroid parabolic section

\[ y_S = \frac{2 \cdot h}{5} \]

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Elliptical section

Centroid elliptical section
Centroid elliptical section

\[ y_S = \frac{2 }{3} \cdot \frac{b \cdot sin^3 \alpha}{\alpha -- sin \alpha \cdot cos \alpha} \]

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