Statics Quiz: Centroid Composite Areas

6 questions · exam conditions

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Centroid Composite AreasQuestion 1 of 6

An H-shaped cross-section consists of three identical rectangles (each 6 in × 1.5 in). Two rectangles are positioned horizontally as flanges: one with its bottom edge at y = 0 and another with its bottom edge at y = 4.5 in. A third rectangle is positioned vertically as a web, with its left edge at x = 2.25 in, connecting the flanges. If measured from the left edge of the lower flange, what is the $x$ -coordinate of the composite centroid?

3.00 in

2.85 in

3.15 in

2.67 in

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Statics Quiz

Statics Quiz: Centroid Composite Areas

Practice Centroid Composite Areas in Statics with focused quiz questions that help you check what you know, review explanations, and build confidence with test-style prompts.

What this quiz covers

This quiz focuses on Centroid Composite Areas, giving you a quick way to practice the rules, question types, and explanations that matter most for Statics.

How to use this quiz

Try each quiz question before looking at the correct answer. Use the explanations to review missed ideas, then come back to similar questions until the pattern feels familiar.

All questions

Question 1

3.00 in (correct answer)
2.85 in
3.15 in
2.67 in

Explanation: All three rectangles have area = 9 in². Lower flange: centroid at (3, 0.75). Upper flange: centroid at (3, 5.25). Web (oriented vertically): extends from x = 2.25 to x = 3.75, centroid at (3, 3). Since all three components have their centroids at x = 3 in, the composite centroid is at x = 3.00 in. Choice B incorrectly assumes web centroid shifts due to connection points. Choice C adds unnecessary eccentricity for web orientation. Choice D uses incorrect web positioning calculations.

Question 2

A composite area consists of a rectangle (4 in × 6 in) with a circular hole (diameter 2 in) centered 1.5 in from the left edge and 2 in from the bottom edge. If the bottom-left corner of the rectangle is at the origin, what is the x-coordinate of the centroid of the composite area?

2.13 in (correct answer)
2.00 in
1.87 in
2.26 in
1.74 in

Explanation: When you encounter composite area centroids, you're dealing with shapes that have been combined or have holes removed. The key principle is using the method of composite areas: treat each component separately, then combine using weighted averages. For this problem, you have a rectangle minus a circular hole. First, find the centroid of the full rectangle (4 in × 6 in). Its centroid is at the geometric center: x = 2.0 in from the left edge. The area is 24 in². Next, locate the circular hole (diameter 2 in, radius 1 in). Its center is 1.5 in from the left edge, and its area is

\pi r^2 = \pi(1)^2 = \pi

in². Now apply the composite area formula:

\bar{x} = \frac{A_1\bar{x_1} - A_2\bar{x_2}}{A_1 - A_2}

Where subscript 1 represents the rectangle and subscript 2 represents the hole (subtracted).

\bar{x} = \frac{24(2.0) - \pi(1.5)}{24 - \pi} = \frac{48 - 1.5\pi}{24 - \pi} = \frac{48 - 4.71}{24 - 3.14} = \frac{43.29}{20.86} = 2.13 \text{ in}

Answer A (2.13 in) is correct. Answer B (2.00 in) represents the centroid of the full rectangle without accounting for the hole. Answer C (1.87 in) likely results from incorrectly adding the hole's contribution instead of subtracting it. Answer D (2.26 in) probably comes from calculation errors in the composite formula. Remember: for composite areas, always subtract both the area and the moment of removed sections, and double-check your positive/negative signs in the calculations.

Question 3

A symmetrical I-beam cross-section has identical flanges (6 in × 1 in) at top and bottom, connected by a web (1 in × 4 in). A circular hole (diameter 2 in) is drilled through the web center. What is the y-coordinate of the centroid measured from the bottom of the lower flange?

3.00 in (correct answer)
2.85 in
3.15 in
2.75 in
3.25 in

Explanation: When finding the centroid of a composite shape with holes, you need to treat the hole as a negative area and use the principle of moments. For symmetric I-beams, the centroid should lie at the geometric center due to symmetry about both axes. Let's establish coordinates with the bottom of the lower flange as y = 0. The I-beam consists of: lower flange (6×1 in), web (1×4 in), and upper flange (6×1 in), giving a total height of 6 inches. Without any holes, the centroid would be at y = 3.0 in due to perfect symmetry. Now consider the circular hole (diameter 2 in, radius 1 in) drilled through the web center. Since the hole is positioned at the geometric center of the cross-section (at y = 3.0 in), removing this area doesn't shift the centroid vertically. The moments about any horizontal axis remain balanced because you're removing area symmetrically about the original centroidal axis. Using the composite area formula:

\bar{y} = \frac{\sum A_i y_i}{A_{total}}

, where the hole contributes negative area and moment, the calculation confirms the centroid remains at y = 3.00 in. Answer A (3.00 in) is correct. Answers B (2.85 in) and D (2.75 in) suggest incorrectly shifting the centroid downward, perhaps from forgetting the hole's symmetric position. Answer C (3.15 in) implies an upward shift, which would only occur if the hole were below the centerline. Remember: when holes or cutouts are positioned at the existing centroid of a symmetric shape, they don't shift the centroidal location—they only reduce the total area.

Question 4

A steel bracket is formed by cutting a triangular notch from the upper-right corner of a rectangular plate (10 in × 6 in). The triangular cutout is a right triangle with legs of 4 in and 3 in, positioned so that the right angle vertex is at the upper-right corner of the original rectangle. What is the distance from the lower-left corner of the original rectangle to the centroid of the remaining area?

6.89 in (correct answer)
7.12 in
6.45 in
6.73 in

Explanation: Original rectangle: A₀ = 60 in², centroid at (5, 3). Triangle cutout: A₁ = 6 in², with vertices at (10,6), (6,6), (10,3), centroid at (8.67, 5). Remaining area = 54 in². Centroid: x̄ = (60×5 - 6×8.67)/54 = 4.04 in, ȳ = (60×3 - 6×5)/54 = 2.78 in. Distance = √(4.04² + 2.78²) = 6.89 in. Choice B uses incorrect triangle centroid calculation. Choice C forgets to subtract triangle area from denominator. Choice D uses geometric center of remaining perimeter instead of area centroid.

Question 5

A composite cross-section for a structural beam consists of a wide-flange shape approximated by three rectangles: a web (0.5 in × 12 in) and two identical flanges (6 in × 1 in each). The flanges are positioned at the top and bottom of the web, sharing common edges. During fabrication, the top flange is accidentally welded with a 0.2 in vertical offset above its intended position. What is the $y$ -coordinate of the centroid measured from the bottom of the web?

6.12 in (correct answer)
6.00 in
5.88 in
6.25 in

Explanation: Web: A₁ = 6 in², centroid at y₁ = 6 in. Bottom flange: A₂ = 6 in², centroid at y₂ = 0.5 in. Top flange (with offset): A₃ = 6 in², intended at y = 11.5 in but shifted to y₃ = 12.7 in. Composite centroid: ȳ = (6×6 + 6×0.5 + 6×12.7)/18 = (36 + 3 + 76.2)/18 = 6.12 in. Choice B ignores the fabrication offset. Choice C incorrectly applies offset as downward shift. Choice D doubles the offset effect in calculations.

Question 6

A T-shaped cross-section is formed by welding a rectangular flange (8 in × 2 in) to the top of a rectangular web (2 in × 6 in). The web and flange share a common centerline. If the bottom of the web is used as the reference datum (y = 0), and considering that the flange thickness causes the actual contact between flange and web to occur at y = 6 in, what is the $x$ -coordinate of the centroid measured from the left edge of the flange?

4.00 in (correct answer)
3.85 in
4.15 in
3.67 in

Explanation: Both the flange (8×2) and web (2×6) are symmetric about their centerlines, which coincide. The flange extends from x = 0 to x = 8 in, with centroid at x = 4 in. The web extends from x = 3 to x = 5 in, with centroid at x = 4 in. Since both components have the same x-coordinate for their centroids, the composite centroid is at x = 4.00 in regardless of the y-positions. Choice B incorrectly weights by distance from datum. Choice C adds an unnecessary eccentricity factor. Choice D uses wrong web width in calculations.