Moment of inertia of composite areas practice

Find Ȳ, then I about the centroidal axis with the parallel-axis theorem, part by part.

Educational practice only. Not for designing or checking real structures.

CentroidsSections

Set

Method, signs and how to type answers
  1. Find Ȳ with the centroid table: A, ȳ and A·ȳ for each part, then Ȳ = ΣA·ȳ ÷ ΣA.
  2. For each part, dy = ȳ − Ȳ is the distance from the whole area’s centroid to the part’s centroid.
  3. Īx is the part’s own moment of inertia about its own horizontal centroidal axis, from the table of shapes.
  4. The parallel-axis theorem moves it to x′: Īx + A·dy². A hole’s Ī and A·d² are negative.
  5. Add both columns: Ix′ = ΣĪx + ΣA·dy², and kx′ = √(Ix′ ÷ ΣA).

      Answers are saved on this device as you type.

      Questions

      What is the parallel-axis theorem?

      I = Ī + A·d². The moment of inertia about any axis equals the moment of inertia about a parallel axis through the area’s own centroid, plus the area times the squared distance between the two axes.

      Is d measured from the base or from the centroid?

      From the centroid of the whole area: d = ȳ − Ȳ for each part. Using ȳ from the base gives the moment of inertia about the base instead, a common slip.

      What is the moment of inertia of a semicircle about its centroid?

      About the axis through its centroid parallel to the diameter, (π/8 − 8/(9π))r⁴, about 0.1098r⁴. The often-quoted πr⁴/8 is about the diameter itself.

      How do you subtract a hole?

      Give the hole a negative area. Its own Ī and its A·d² then come out negative, and the sums subtract them. The hole’s d is still the distance to its centroid.

      What is the radius of gyration?

      k = √(I ÷ A): if the whole area were gathered into a thin strip parallel to the axis, at a distance k from it, it would have the same moment of inertia. It has units of length.

      Why are the answers in 10⁶ mm⁴?

      Moments of inertia in mm⁴ run to millions, so textbooks write them as, say, 6.13 × 10⁶ mm⁴. Here you type 6.13 in those columns. Inch problems use in⁴ directly.

      How do I give my class the same problems?

      Press Share with class and send the link: everyone gets the same set code and shapes. Print makes up to 30 different copies, each with its own answer key.

      Sources, checked 2 November 2026: Baker and Haynes, Engineering Statics: Open and Interactive, 7.5 Centroids using composite parts (holes as negative areas), Table C.1.1 Centroids of common shapes, Table C.2.1 Moments of inertia of common shapes and 10.4 Moment of inertia of composite shapes. The same formulas are on the inside covers of Hibbeler and of Beer and Johnston.