The centroid of an area is the coordinates of the geometric centre. It is calculated from the area’s geometry and is represented in coordinate form (xc, yc). Centroids of basic shapes can be intuitive, such as the centre of a circle. Centroids of basic shapes are widely known and published. These can be used to find the centroid of composite areas or areas made up of basic shapes. Centroids of complex or arbitrary shapes can be found using the integration method of calculus.

- Skill level:
- Easy

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### Things you need

- Graph paper
- Pencil
- Calculator (optional)

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## Instructions

- 1
Draw the figure to scale on graph paper. Identify the basic shapes that make up the figure and indicate these shapes on the drawing using dotted lines at the boundaries. The basic shapes are rectangle, triangle, circle, trapezoid, circle, quarter-circular area, semicircular area, quarter-elliptical area, semielliptical area, semi-parabolic area, parabolic area, parabolic spandrel, general spandrel and circular sector. Label each basic shape with numbers starting with 1(1, 2, 3, ... , n,). Label all the important dimensions.

- 2
Define an X axis and Y axis on your graph paper. A convenient location of the graph origin point (0,0) is the lower left-hand corner of the figure. Let the centroid coordinates of the whole figure be defined as (xc, yc). Let the area of each basic shape be defined as An. Let n be the number used to label the basic shapes in Step 1.

- 3
Locate and mark the centroids of each basic shape using the equations found in the resources. Determine the coordinates of each centroid location relative to the origin of the graph. Let the centroid coordinates for each basic shape be defined as (xn, yn).

- 4
Calculate the X coordinate of the figure’s centroid (xc) using the following equation:

xc = ?xnAn ??An

? indicates finite summation. The help clarify, this equation can be represented as follows:

xc = (x1A1 + x2A2 + x3A3 +...+ xnAn) ?(A1 + A2 + A3 +...+ An)

- 5
Calculate the Y coordinate of the figure’s centroid (yc) using the following equation:

yc = ?ynAn ??An

The help clarify, this equation can be represented as follows:

yc = (y1A1 + y2A2 + y3A3 +...+ ynAn) ?(A1 + A2 + A3 +...+ An)

- 6
Check your math. Mark the centroid of the figure (xc, yc) on your sketch. It should lie near the centre of the area.

#### Tips and warnings

- Use the link in Resources or an engineering reference manual to obtain basic shape centroid equations.
- The areas of voids are considered negative areas in the above equations. An example of a void is a hole in a circle.