Parallel Axis Theorem for a Rectangle
Mediumparallel axis theorem
A rectangle is 4 in wide and 8 in tall. Its moment of inertia about a horizontal axis 3 in from its centroidal horizontal axis is most nearly:
- A170.7 in⁴
- B458.7 in⁴
- C288.0 in⁴
- D746.7 in⁴
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Answer: B. 458.7 in⁴
Equation: parallel-axis theorem, I = Ic + A·d², where Ic = bh³/12 for a rectangle about its own centroidal axis. Values: b = 4 in is the width, parallel to the axis. h = 8 in is the height, perpendicular to the axis, so it is the dimension that gets cubed. A = b·h = 4 × 8 = 32 in². d = 3 in is the distance between the centroidal axis xc and the new axis x′. Step 1: Ic = 4(8)³/12 = 2048/12 = 170.7 in⁴. Step 2: A·d² = 32 × 3² = 288.0 in⁴. Step 3: I = 170.7 + 288.0 = 458.7 in⁴. 170.7 in⁴ forgets the A·d² term, and 288.0 in⁴ forgets Ic.
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