Questions: Centroids of Areas and Composite Shapes

Question 1

A rectangular steel plate (area = 20 cm², centroid at x = 5 cm) has a circular hole punched out at x = 8 cm (area of hole = 4 cm²). Where is the centroid of the remaining plate?

• A. At x = 5.0 cm — removing material doesn't change the geometric center of the original rectangle
• B. At x = 4.25 cm — the removed material on the right side shifts the centroid leftward
• C. At x = 5.75 cm — the hole at x = 8 cm pulls the centroid toward the right
• D. At x = 6.5 cm — the centroid moves to the midpoint between the plate's center and the hole

Answer: (B)

Using the negative area method: x̄ = (20×5 − 4×8) / (20 − 4) = (100 − 32) / 16 = 68/16 = 4.25 cm. The hole removed area from the right side (x = 8 is right of center at x = 5), so the remaining shape is heavier on the left, pulling the centroid leftward. Option A incorrectly treats the original centroid as unchanged. Option C inverts the effect — removing right-side material shifts the balance leftward, not rightward.

Question 2

An L-shaped bracket is decomposed into two rectangles: Rectangle A has area 8 cm² with centroid at ȳ = 6 cm; Rectangle B has area 6 cm² with centroid at ȳ = 2 cm. What is ȳ for the composite shape?

• A. ȳ = 4.0 cm — the simple average of 6 and 2
• B. ȳ = 4.29 cm — the weighted average, with Rectangle A (larger area) pulling the centroid upward
• C. ȳ = 3.71 cm — the weighted average, with Rectangle B (lower centroid) pulling the result down
• D. ȳ = 8.0 cm — the sum of the two centroid y-values

Answer: (B)

ȳ = ΣAᵢȳᵢ / ΣAᵢ = (8×6 + 6×2) / (8+6) = (48 + 12) / 14 = 60/14 ≈ 4.29 cm. Rectangle A is larger and its centroid is higher (ȳ = 6), so it pulls the composite centroid upward past the simple average of 4. Option A (simple average) ignores area weighting. Option C inverts which rectangle has more influence — the larger rectangle dominates.

Question 3

The centroid of a shape should typically lie within the physical boundary of that shape.

• T. True
• F. False

Answer: False

The centroid is a mathematical balance point, and for concave shapes or shapes with holes, it can lie entirely outside the material. A C-shaped bracket, a ring, or a hollow square tube all have centroids located in the empty interior space. This is not an error — it is the correct geometric center for those shapes. The negative-area technique works precisely because the formula does not require the centroid to be located on material.

Question 4

A cutout or hole in a composite shape can be handled by assigning it a negative area and including it in the weighted-average formula alongside the positive sub-shapes.

• T. True
• F. False

Answer: True

This is the key practical technique: you never need to calculate the irregular geometry of the remaining boundary. Treat the full solid shape as a positive area with its known centroid, and the removed piece as a shape with negative area using the centroid of the removed piece. The weighted average naturally cancels the removed material's contribution. This approach generalizes to any number of cutouts and is far less error-prone than attempting to integrate the complex remaining boundary.

Question 5

Why is the centroid formula x̄ = ΣAᵢx̄ᵢ / ΣAᵢ described as a 'weighted average'? What is being weighted, and what are the weights?

Model answer: The formula computes the average x-position of the shape, weighted by how much area each part contributes. Each sub-shape's centroid coordinate (x̄ᵢ) is multiplied by its area (Aᵢ), and the sum is divided by total area. A larger sub-shape has more 'pull' on the overall centroid than a smaller one — analogous to heavier weights on a see-saw having more influence on the balance point. The areas are the weights; the centroid coordinates are the values being averaged.

The weighted-average framing makes the behavior intuitive: adding a large sub-shape far from the current centroid pulls it strongly in that direction; adding a small sub-shape barely moves it. The negative-area technique for holes fits the same framework naturally — a negative weight pulls the centroid away from the removed material, which is exactly what physical intuition demands.