Question

The centroid of a composite plane figure can be found by:

• A. Dividing the sum of the areas of individual shapes by the total number of shapes
• B. Adding the centroids of individual shapes and dividing by the area of the composite figure
• C. Multiplying the area of each shape by its centroid's coordinates, summing these products and dividing by the total area
• D. Taking the average of the centroids of all individual shapes

Hint:

Always multiply each shape’s centroid coordinates by its area to find the composite centroid. It ensures correct weighting and accurate results.

Answer 1

The Correct Option is C

Step 1: The centroid \( (x_c, y_c) \) of a composite plane figure is the weighted average of the centroids of the individual component shapes, where the weights are the areas of those components.
Step 2: Mathematically, the coordinates of the centroid are calculated using the formulas: \[ x_c = \frac{\sum A_i x_i}{\sum A_i}, \qquad y_c = \frac{\sum A_i y_i}{\sum A_i} \] where \( A_i \) is the area and \( (x_i, y_i) \) is the centroid of the \( i \)-th individual shape.
Step 3: This approach ensures that larger areas contribute more to the final centroid location, which is crucial for accuracy in irregular composite figures.
Why the other options are incorrect:

• (A) Merely dividing the total area by the number of shapes does not account for the shape locations or areas.
• (B) Averaging centroids without weighting by area leads to incorrect results for uneven shapes.
• (D) Similar to (B), taking a plain average ignores area contributions.