Second Moment of Area Calculator

About the Second Moment of Area Calculator

This Second Moment of Area Calculator (also called the Area Moment of Inertia) helps you calculate I_x, I_y, and cross-sectional area for common engineering shapes with a live diagram. Second moment of area is a key property for predicting bending stress, deflection, and stiffness of beams and structural members.

Shapes Supported

• Solid sections: Rectangle, Circle, Triangle (isosceles)
• Hollow sections: Hollow Rectangle (Box), Hollow Circle
• Steel profiles: I-Beam, C-Channel, L-Angle

What the Calculator Outputs

• I_x: Second moment of area about the horizontal centroidal axis (x-axis).
• I_y: Second moment of area about the vertical centroidal axis (y-axis).
• Area (A): Cross-sectional area of the selected shape.

Formulas Used

All results are computed about centroidal axes. For composite sections (I-Beam, C-Channel, L-Angle), the calculator builds the section from rectangles, finds the centroid, then applies the parallel axis theorem.

Rectangle:
• Area: A = b·h
• I_x = b·h³/12
• I_y = h·b³/12

Hollow Rectangle (Box):
• Area: A = B·H − b·h
• I_x = (B·H³ − b·h³)/12
• I_y = (H·B³ − h·b³)/12

Circle:
• Area: A = π·r²
• I_x = π·r⁴/4
• I_y = π·r⁴/4

Hollow Circle:
• Area: A = π·(R² − r²)
• I_x = π·(R⁴ − r⁴)/4
• I_y = π·(R⁴ − r⁴)/4

Triangle (Isosceles):
• Area: A = b·h/2
• I_x = b·h³/36
• I_y = h·b³/48

I-Beam (Symmetric):
• Area: A = 2(b·t_f) + a·(H − 2t_f)
• I_x = (a·h³)/12 + (b/12)·(H³ − h³)
• I_y = (h·a³)/12 + 2·(t_f·b³)/12

C-Channel:
• Area: A = Σ(w·h)
• I_x = Σ(I_x,local + A·Δy²)
• I_y = Σ(I_y,local + A·Δx²)

L-Angle:
• Area: A = Σ(sign·w·h)
• I_x = Σ(sign·(I_x,local + A·Δy²))
• I_y = Σ(sign·(I_y,local + A·Δx²))

How to Use the Calculator

1. Select a cross-section (e.g., I-Beam, C-Channel, L-Angle).
2. Choose your display units (mm, cm, in) and enter the dimensions.
3. View the live cross-section visualisation and read the updated I_x, I_y, and A results instantly.

Notes on the Diagram

• The axes shown pass through the centroid of the cross-section.
• Composite sections are evaluated by rectangle composition and centroidal transformations.
• Use consistent units for dimensions—outputs automatically follow the selected display unit.

Common Engineering Uses

• Beam design: estimate deflection and bending performance for floors, frames, and supports.
• Structural checks: compare section stiffness when choosing between shapes (e.g., box vs channel).
• Optimization: evaluate how changing flange/web thickness affects stiffness and weight.
• Learning & verification: validate hand calculations for mechanics of materials problems.