Centroid of Area

NCEES FE Civil spec: Statics, E. Centroid of area. FE Reference Handbook: Statics chapter, centroids and the plane-area properties table.

The centroid is the geometric center of an area: the point where the area would balance if it were a thin plate of uniform thickness. Centroids locate the neutral axis for bending, the resultant of distributed loads and the center of pressure in fluids, so this section supports a large part of the FE Civil exam.

What the NCEES spec covers

Statics section E, "Centroid of area", covers centroids of standard shapes and composite areas built from rectangles, triangles, circles and cutouts. Expect T-sections, L-angles, I-shapes and plates with holes.

Key concepts

Symmetry first. If a shape has an axis of symmetry, the centroid lies on it. A doubly symmetric shape has its centroid at the intersection of the two axes.
Composite areas. Split the shape into simple parts, find each area and centroid from a common reference, then take the area-weighted average.
Holes are treated as negative areas.
Standard locations: rectangle at h/2; triangle at h/3 from the base; semicircle at $4 r / (3 π)$ from the diameter; quarter circle at $4 r / (3 π)$ from each straight edge.

Core formulas

\overset{x}{ˉ} = \frac{\int x d A}{A} \overset{y}{ˉ} = \frac{\int y d A}{A}

\overset{x}{ˉ} = \frac{\sum A _{i} x ˉ _{i}}{\sum A _{i}} \overset{y}{ˉ} = \frac{\sum A _{i} y ˉ _{i}}{\sum A _{i}}

Where it is in the FE Reference Handbook

The Statics chapter defines centroids of masses, areas, lengths and volumes, and includes a table of plane-area properties (area, centroid location and moments of inertia) for rectangles, triangles, circles, semicircles, quarter circles and other shapes. Learn where that table is before exam day.

Worked example

A T-section has a flange 120 mm wide by 20 mm thick sitting on top of a web 20 mm wide by 100 mm tall. Find the centroid height $\overset{y}{ˉ}$ measured from the bottom of the web.

Part	Area (mm²)	ȳᵢ (mm)	Aᵢȳᵢ (mm³)
Web 20 × 100	2,000	50	100,000
Flange 120 × 20	2,400	110	264,000
Total	4,400		364,000

\overset{y}{ˉ} = \frac{364 , 000}{4 , 400} = 82.7 mm from the bottom

By symmetry, $\overset{x}{ˉ}$ is on the vertical centerline. The centroid sits in the upper part of the web, closer to the flange, because the flange has more area. A quick sanity check: the answer must lie between the two part centroids (50 and 110 mm), and it does.

Common mistakes

Measuring part centroids from different reference lines.
Adding a hole as a positive area instead of subtracting it.
Placing a triangle centroid at h/3 from the apex instead of from the base.
Double counting the overlap where a flange and web meet.
Forgetting that the answer must fall between the extreme part centroids.

You can check composite shapes visually with the centroid visualizer. The same T-section continues in the area moments of inertia lesson.

Practice Problems

Try these from the free PolyCalc FE Civil problem bank. Each one has a full worked solution.

A T-section is made of a 100 mm wide × 20 mm tall flange sitting on top of a 20 mm wide × 80 mm tall web. Measured from the bottom of the web, where is the centroid of the whole section? See the worked solution or practice the Statics & Equilibrium deck
Where is the area centroid of a uniform 4 m high rectangle, measured from its base? See the worked solution or practice the Section Properties deck
An unequal-leg angle has a vertical leg 150 mm long and a horizontal leg 100 mm long, both 10 mm thick, meeting at the bottom-left heel. Measured from the outside face of the horizontal leg, the centroid of the section is located most nearly at: See the worked solution or practice the Section Properties deck

Related Calculators and Examples

Frequently Asked Questions

How do you find the centroid of a composite area?

Split the shape into simple parts, find each part area and centroid measured from one common reference, then divide the sum of area times centroid by the total area. Treat holes as negative areas.

Where is the centroid of a triangle?

One third of the height from the base, which is two thirds of the height from the apex. For a right triangle, it is also one third of the base width from the vertical side.

Is the centroid the same as the center of gravity?

For a body of uniform density and thickness in a uniform gravity field, yes. The centroid is purely geometric, while the center of gravity depends on how weight is distributed.

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