# Given the same diameter and height, which is stronger? A vertical standing cylinder or a vertical standing rectangular?

I am interested in two cases. Case 1: Both are solid. Case 2: Both are hollow with the same wall thickness.

If you mean a cylinder with the diameter $$D$$ and a square with a side $$X=D$$, The square is stronger, solid, or hollow.

If they have the same moment of inertia, I, then they are equally strong only for bending moment, but then the square post has a bit less area and a bit weaker for the axial load. let's see what size square has equal I as a cylinder with diameter D.

\begin{align} I_{cylinder} &= \dfrac{\pi D^4}{64} \\ I_{square} &= \dfrac{X^4}{12} \\ I_{square} &= I_{cylinder} \\ \dfrac{X^4}{12} &= \dfrac{\pi D^4}{64} \\ X^4 &= \dfrac{12\pi D^4}{64} \\ X &= 0.876067D \end{align}

For the above sizes, the two posts can resist the same bending moment.

However the areas are

\begin{align} A_{cylinder} &= \dfrac{\pi D^2}{4} &&= 0.78539D^2 \\ A_{square} &= X^2 &&= 0.76749D^2 \end{align}

The area of the square post with equal $$I$$ is a bit smaller.

• Thank you. Yes, I meant a cylinder with the diameter Dcm and a square with a side A = Dcm. When you stated that the square is stronger, solid or hollow, does it matter the materials used and the height? Given that a square post has smaller area than a cylinder, what does it has to do with the strength? The load is mainly from the top. Mar 9 at 4:25
• Thanks for the clarification. if the load is from the top and A=Dcm the square is of course stronger to support vertical loads and moment too. Mar 9 at 4:43
• @questioner, just to clarify: a square of side D will have a larger area than a circle of diameter D. The square will therefore be stronger than the circle.
– Wasabi
Mar 9 at 13:20
• Larger cross-sectional area leads to stronger support makes sense. Why cylinder rather than rectangular column is more often used in architecture to support ceilings? letstalkscience.ca/educational-resources/lessons/… I don't know why on the net, some people said cylinder is the strongest shape. Similarly, some said triangular column is the strongest but others said it is the weakest. Mar 9 at 14:46

From the figure below, you shall be able to judge which one is stronger. Note, for both are solid case, simply eliminate d in the formulas.

• Thank you. I am a bit confused. Does larger bending moment corresponds to stronger strength against heavy load? Is bending moment different from moment of inertia (typically represented by I) which indicates how easy or difficult to spin an object? Mar 9 at 18:15
• @questioner See the updated response. The moment of inertia can be thought as "rigidity" of a cross section, the larger, the more force it can endure, thus stronger.
– r13
Mar 9 at 19:25