# Section Properties Calculator: Area, Moments of Inertia, Torsion Constant

> A, Iy, Iz, J and section moduli of rectangles, round bars, tubes, box and I-sections. Enter the shape and dimensions of a beam cross-section — solid rectangle, solid circle, circular tube, rectangular hollow section or doubly symmetric I-section — and get the area, the second moments of area about both axes, the St. Venant torsion constant, the elastic section moduli, the torsion modulus, the radii of gyration and, with a typical steel, stainless, aluminium or timber material, the mass per metre and the bending and torsional stiffness. A drawing shows the section with its local axes, and a ready line feeds the 3-D frame model of Proofline’s truss and frame solver. Part of the Proofline math notebook.

Part of usefulweb.app (https://usefulweb.app/). Part of [Proofline](https://proofline.usefulweb.app/) (ownerApp: proofline). Results are deterministic calculations from the inputs and the listed assumptions; the API returns exactly what the web page shows.

## Use it
- Human interface: https://prooflinesectionproperties.usefulweb.app/ (interactive tool at https://prooflinesectionproperties.usefulweb.app/tool)
- JSON API: POST https://prooflinesectionproperties.usefulweb.app/api/v1/calculate with a JSON object of inputs (schema: https://prooflinesectionproperties.usefulweb.app/api/v1/tool)
- OpenAPI: https://prooflinesectionproperties.usefulweb.app/openapi.json
- MCP (Streamable HTTP): https://prooflinesectionproperties.usefulweb.app/mcp - tool `proofline_section_properties`

## Inputs
- shape (select): Shape
- b (number): Width b - Rectangle, hollow section and I-section: the width along the local y axis (flange width of an I-section).
- h (number): Depth h - Rectangle, hollow section and I-section: the depth along the local z axis.
- d (number): Diameter d - Circle and tube: the outside diameter.
- t (number): Wall thickness t - Tube and hollow section.
- tf (number): Flange thickness tf - I-section.
- tw (number): Web thickness tw - I-section.
- material (select): Material - Typical values for the mass per metre and the stiffnesses EI and GJ.
- name (text): Section name for the frame model

## Method
- Local axes as in Eurocode 3: y across the width b, z along the depth h, so Iy is the strong axis of an upright rectangle or I-section. A, Iy and Iz from the exact formulas of each shape (I-sections and box sections without root or corner radii).
- Torsion: circles and tubes J = πd⁴/32 (difference for the tube), τ = T·r/J; solid rectangles by Roark’s formulas, J = a·c³·[1/3 − 0.21·(c/a)·(1 − c⁴/(12a⁴))] and τmax = 3T/(a·c²)·[1 + 0.6095r + 0.8865r² − 1.8023r³ + 0.9100r⁴]; box sections by Bredt’s thin-wall formula J = 4Am²t/p; I-sections as thin open rectangles J = (2·b·tf³ + (h − 2tf)·tw³)/3, warping ignored.
- Section moduli W = I/(distance to the extreme fibre), torsion modulus Wt = T/τmax, radii of gyration i = √(I/A); with a material the mass per metre ρ·A and the stiffnesses E·I and G·J.
- Verified against closed forms (b·h³/12, πd⁴/64), the square-bar torsion constant 0.1406·a⁴ and τmax = T/(0.208·a³), Bredt’s formula for a thin square tube and steel-table values of IPE 200 (Iz 142 cm⁴; A and Iy within 6 % because of the fillets).

## Limits
- Nominal geometry without fillets, corner radii or tolerances; open sections ignore warping torsion. Material values are typical, not those of a certified grade. Use the manufacturer’s section tables for design.