Triangle Area Calculator
Calculate triangle area, perimeter, angles, altitudes, circumradius, inradius, and classification. Supports base-height, Heron's formula, and SAS methods with unit selection and a complete 15-prop..
Cross-Sectional Area Calculator
Calculate cross-sectional area, perimeter, moment of inertia, and section modulus for circles, rectangles, triangles, pipes, ellipses, and I-beams.
Common Presets
mm
Cross-Sectional Area⧉
7,853.98 mm²
Total area of the cross-section
Perimeter⧉
314.16 mm
Total boundary length of the cross-section
Moment of Inertia (Ix)⧉
4,908,738.52 mm⁴
Second moment of area about horizontal centroidal axis
Section Modulus (Sx)⧉
98,174.77 mm³
Ix divided by distance to extreme fibre — governs bending stress
Area in cm²⧉
78.54 cm²
Cross-sectional area converted to cm²
Area in in²⧉
12.17 in²
Cross-sectional area converted to square inches
Property Comparison
Area (mm²)
7,853.98
Perimeter
314.16
Ix (×10⁻³)
4,908.74
Sx (×10⁻¹)
9,817.48
Shape Formulas Reference
| Shape | Area | Ix |
|---|---|---|
| Circle | πr² | πr⁴/4 |
| Rectangle | bh | bh³/12 |
| Triangle | bh/2 | bh³/36 |
| Hollow Circle | π(R²−r²) | π(R⁴−r⁴)/4 |
| Ellipse | πab | πab³/4 |
| I-Beam | 2bf·tf + (h−2tf)·tw | bf·h³/12 − (bf−tw)(h−2tf)³/12 |
Standard Pipe Sizes (NPS)
| NPS | OD (mm) | Wall (mm) | Area (mm²) |
|---|---|---|---|
| ½″ | 21.3 | 2.77 | 161.25 |
| ¾″ | 26.7 | 2.87 | 214.86 |
| 1″ | 33.4 | 3.38 | 318.77 |
| 1½″ | 48.3 | 3.68 | 515.85 |
| 2″ | 60.3 | 3.91 | 692.67 |
| 3″ | 88.9 | 5.49 | 1,438.60 |
| 4″ | 114.3 | 6.02 | 2,047.83 |
| 6″ | 168.3 | 7.11 | 3,600.46 |
Planning notes, formulas, and examples
About the Cross-Sectional Area Calculator
The cross-sectional area of a structural member is one of the most fundamental quantities in engineering design. It determines how much load a beam, pipe, or column can carry, how heat conducts through a material, and how fluid flows through a conduit.
This calculator supports six common cross-section shapes: solid circle, rectangle, triangle, hollow circle (pipe), ellipse, and I-beam. For each shape you enter the relevant dimensions and receive the area, perimeter, second moment of area (moment of inertia Ix), and elastic section modulus (Sx). These properties are critical for stress analysis, deflection calculations, and material selection.
Engineers use Ix when calculating beam deflection via the formula δ = PL³/(48EI), and Sx when checking bending stress σ = M/S. Pipe designers need the cross-sectional area to determine flow velocity and pressure drop. The calculator includes preset buttons for standard NPS pipe sizes and common wide-flange beam profiles, plus a reference table of standard pipe dimensions.
Whether you are sizing a structural beam, selecting a conduit, or solving a homework problem, the page keeps those section properties attached to the same chosen shape and dimensions.
When This Page Helps
Use this when you need area and section properties for a beam, pipe, plate, or conduit without switching between separate formulas for each profile. It is useful in structural and flow calculations because the chosen shape, dimensions, area, inertia, and section modulus stay connected.
How to Use the Inputs
- Choose a cross-section shape from the dropdown (circle, rectangle, triangle, hollow circle, ellipse, or I-beam).
- Enter the required dimensions for that shape in millimetres.
- View the calculated area, perimeter, moment of inertia, and section modulus.
- Use the preset buttons for standard pipe or beam sizes.
- Check the bar chart for a visual comparison of properties.
- Refer to the formula table for the equations used.
Formula used
Circle: A = πr², Ix = πr⁴/4
Rectangle: A = bh, Ix = bh³/12
Triangle: A = bh/2, Ix = bh³/36
Hollow Circle: A = π(R²−r²), Ix = π(R⁴−r⁴)/4
Ellipse: A = πab, Ix = πab³/4
I-Beam: A = 2bf·tf + (h−2tf)·tw, Ix = bf·h³/12 − (bf−tw)(h−2tf)³/12Example Calculation
Result: For shape=5, radius=10, width=15, the tool returns the solved cross outputs shown in the result cards.
This example uses a realistic input set from the calculator workflow. After entry, the calculator applies the built-in cross formulas and reports derived values, checks, and classifications automatically.
Tips & Best Practices
- For structural beams, higher Ix means less deflection under the same load.
- Section modulus Sx determines the maximum bending stress: σ = M / Sx.
- Hollow sections are weight-efficient — they distribute material far from the neutral axis.
- I-beams are optimized for bending; flanges resist bending while the web resists shear.
- Always check that the inner diameter is less than the outer diameter for hollow sections.
When To Use This Calculator
Calculate cross-sectional area, perimeter, moment of inertia, and section modulus for circles, rectangles, triangles, pipes, ellipses, and I-beams. Use it when you need a repeatable calculation in the math / geometry category and want the setup, result, and supporting values kept together. This is especially helpful when small input changes, unit choices, or rounding decisions can change the final number.
How To Check The Result
Start by confirming that the inputs match the formula shown on the page. Then compare the main output with the worked example and any secondary values shown by the calculator. If the result will be used in another calculation, keep extra precision until the final step and record the assumptions beside the number.
Practical Notes
Treat the result as a calculation aid rather than a substitute for context. For schoolwork, include the formula and substitution steps. For planning, technical, financial, or health-related decisions, verify important numbers against primary records, current rules, or a qualified professional before acting on them.
Sources & Methodology
Last updated:
Frequently Asked Questions
Cross-sectional area is used to calculate stress (σ = F/A), flow rate (Q = Av), heat transfer, and material requirements for structural members.
Ix is the second moment of area about the horizontal centroidal axis. It measures resistance to bending — higher Ix means less deflection.
Section modulus Sx = Ix / c, where c is the distance from the centroid to the extreme fibre. It directly relates bending moment to maximum stress.
Pipes place material farther from the neutral axis, increasing Ix per unit area. This gives better bending and torsional resistance for the same mass.
The calculator includes Schedule 40 pipe data from ½″ to 6″ NPS with outside diameter, wall thickness, and calculated area.
For an arbitrary shape, decompose it into standard sub-shapes and sum the areas. Use the parallel axis theorem for composite Ix calculations.
More in this topic
Ellipse Area Calculator
Calculate ellipse area from semi-major and semi-minor axes. Also shows circumference (Ramanujan approximation), eccentricity, foci distance, linear eccentricity, directrix, and flattening.
Crescent (Lune) Area Calculator
Calculate the area of a crescent or lune formed by two overlapping circles. Enter radii and distance between centers to find crescent area, overlap area, and union area.