Engineering question: Where is the geometric center of a shape, and how can we calculate it for irregular parts?
Follow the modules in order. Pause the videos to sketch and calculate instead of only watching.
Additional composite-shape video and independent practice can be completed after class.
Imagine balancing a uniform cardboard shape on the tip of a pencil. Where should the pencil touch?
A centroid is the geometric center of a shape. Think of averaging the position of every tiny piece of its area. It is located using coordinates, not by guessing the middle of a bounding box.
For a thin plate of uniform thickness and density in a uniform gravitational field, its area centroid also gives the horizontal location of its center of gravity.
Locating centroids helps us reason about balance, mass distribution, structural cross-sections and where the weight of an object acts. In later lessons, this connects to beam bending and structural stiffness.
Remember: The centroid is a location. It does not by itself tell us how strong or stiff a part is.
Based on geometry: area, line or volume.
Based on how mass is distributed.
Where the resultant gravitational force acts.
You cut an L-shaped plate from plywood. Will its balance point be at half the total width and half the total height? Sketch your prediction, then revisit it after the worked example.
Not necessarily. An L-shaped plate is uneven: there is more material near the upright and top flange. Its centroid is an area-weighted location, not necessarily the center of the surrounding rectangle.
Always choose an origin O and measure both centroid coordinates from the same origin. The yellow dot is the centroid.
For the 120 mm × 60 mm plate: x̄ = 60 mm, ȳ = 30 mm. A uniform rectangle has two lines of symmetry; the centroid is at their intersection.
For this particular orientation, measured from the right-angle corner: x̄ = 40 mm, ȳ = 20 mm. Measure one-third from each perpendicular side, not from the sloping edge.
Use the first video as a broad introduction. The second explains the composite-area method in depth. Captions and playback speed are available through YouTube. If embedded playback is blocked, open the direct link.
Less Boring Lectures. Watch especially the sections on the centroid of an area, triangles and composite bodies.
While watching: Write one sentence explaining why a centroid is an area-weighted average.
Open introductory video on YouTube ↗Jeff Hanson. Shows how to organize component areas and coordinates; later demonstrates holes and the negative-area method.
While watching: Copy a table header: Shape | Area | xᵢ | yᵢ | Axᵢ | Ayᵢ.
Open composite-shape video on YouTube ↗Optional extension: Jeff Hanson — How to Break Down Composite Shapes, focused on picking regions and an origin.
We will solve the exact type of problem you will see in the practice packet. Study the origin, colored regions, area table and final reasonableness check.
The upright is 20 × 80 mm. The separate upper extension to its right is 40 × 20 mm. They touch, but do not overlap, so their areas can be added.
Use the lower-left corner as origin O. Each component rectangle has its own centroid, measured from that same origin.
| Region | A (mm²) | xᵢ (mm) | yᵢ (mm) | Axᵢ (mm³) | Ayᵢ (mm³) |
|---|---|---|---|---|---|
| Web 1 | 1,600 | 10 | 40 | 16,000 | 64,000 |
| Top 2 | 800 | 40 | 70 | 32,000 | 56,000 |
| Total | 2,400 | — | — | 48,000 | 120,000 |
Step 3 · Solve: x̄ = 48,000 / 2,400 = 20 mm. ȳ = 120,000 / 2,400 = 50 mm.
Step 4 · Check: The centroid moves toward the web because the web contributes twice as much area as the upper extension. It is not at the bounding-box center (30, 40).
Area is measured in mm², and each centroid coordinate is measured in mm. Multiplying gives mm³ (a first moment of area). Dividing by total area in mm² produces a coordinate in mm.
Move the slider to change only the top extension's width. The vertical web stays 20 × 80 mm and the top extension stays 20 mm tall. Watch the yellow centroid shift.
Prediction: When you widen the top extension, should the centroid move right, left, up or down?
Each part contributes to the total based on its area. Adding material high on the right pulls both centroid coordinates toward that region.
As the top extension gets wider, its area becomes a larger fraction of the total. The overall centroid shifts right and upward toward the centroid of the added area.
Removed material is negative area. Subtract its area and its first moments from the uncut plate.
Sanity check: Removing material from the upper-right makes the centroid move left and down.
Draw the shape, label your origin, calculate the coordinates, then expand the solution. Work in your 2.2 practice packet or engineering notebook.
A: Find the centroid of a uniform rectangle 90 mm wide and 40 mm tall, measured from its lower-left corner.
B: Find the centroid of a right triangle with base 90 mm and height 60 mm, measured from its right-angle corner.
A: (45, 20) mm, half of each rectangle dimension.
B: (30, 20) mm, one-third of each perpendicular dimension from the right angle.
A vertical web 30 mm wide and 60 mm tall supports a centered horizontal flange 100 mm wide and 20 mm tall. The flange sits immediately above the web, making the overall shape 80 mm high. Origin is the lower-left of the overall 100 mm width.
Can you predict x̄ from symmetry, then calculate ȳ?
Web: A₁ = 1,800 mm², center (50, 30). Flange: A₂ = 2,000 mm², center (50, 70). x̄ = 50 mm by symmetry. ȳ = [1,800(30) + 2,000(70)] / 3,800 = 51.05 mm.
You take a uniform rectangular plate and remove a small square from its upper-left corner. Predict the direction the centroid moves.
It moves right and down: removing material from the upper-left leaves relatively more area on the right and bottom.
A student splits an L-shaped plate into two overlapping rectangles and adds their areas without correction. What will happen?
The overlap is counted twice, so the calculated centroid may be incorrect. Divide the part into nonoverlapping regions, or subtract the overlap exactly once.
These free, publicly available engineering resources provide additional explanations. You do not need calculus for today's problems.
Open Statics reference explaining the different centers and when they coincide.
Engineering LibreTexts — 7.4 Centroids ↗Compare rectangular and triangular centroids, including drawings with coordinate axes.
Mechanics Map — 2D shape reference ↗Explore the deeper relationship between area, symmetry and centers of mass.
OpenStax — Moments and Centers of Mass ↗Complete your POE 2.2 Centroids & Center of Gravity classwork and homework worksheet (or equivalent notebook pages if assigned by your teacher). Submit your own calculations and sketches; the videos and on-page solutions are study aids.
Due date: Check the Daily Agenda for the current deadline. Your teacher will identify where to submit the completed packet.
