Home FDM 3D printing, by the numbers

Design parts that fit on the first print, and know why they fit.

Most wasted filament comes from geometry or physics nobody wrote down: bores that shrink and pull inward, print speeds that outrun the hotend’s melt zone, or slicer flow adjustments masking a mechanical extruder error. LayerMath gives you transparent calculators, printable test gauges and source-backed guides so you can verify every number with calipers.

Calculate fit & download STL gauge Check volumetric flow & E-steps

Calculators

Fit, Clearance & Shrinkage Calculator (+ STL Gauge)

Compute CAD dimensions for press, sliding, loose and print-in-place joints after polymer shrinkage and nozzle geometry, and download a printable STL test gauge.

Extrusion Flow, Volumetric Speed & E-Steps Calculator

Calculate capsule road cross-section, volumetric flow rate (mm³/s), speed limits, slicer flow ratio and Klipper/Marlin extruder steps.

Dialing in a printer? Start in this order

Step 1 · Calibration

Calibrating Extruder E-Steps and Klipper Rotation Distance

Learn how to calibrate Marlin E-steps and Klipper rotation_distance using the 100 mm extrusion test, gear ratios, and hardware-level stepper math.

Step 2 · Calibration

Slicer Flow Rate and Extrusion Multiplier Calibration Math

Understand slicer flow math using the stadium cross-section formula, why single-wall caliper tests mislead, and how to tune extrusion multiplier.

Step 3 · Design & Tolerances

Designing 3D-Printed Parts That Fit: Clearances and Tolerances

Calculate radial and diametral clearances for press, sliding, running, and print-in-place FDM joints using hole-basis design and extrusion math.

Step 4 · Calibration

Maximum Volumetric Speed Explained: Hotend Flow Rate Math

Calculate maximum volumetric speed (mm³/s) from layer height, line width, and print speed, and learn how hotend melt zones limit 3D printing flow.

More guides

3D Printing Cost Math: Filament, Electricity, and Wear · Cost & Energy
Stepper Angles, GT2 Belts, Lead Screws, and Z Layer Heights · Hardware & Motion
Filament Moisture, Hydrolysis, and Drying Temperatures · Materials
First-Layer Squish, Line Width, and Z-Offset Calibration · Calibration
Nozzle Diameter vs. Layer Height, Line Width, and Overhangs · Hardware & Motion
PLA vs PETG vs ASA: Mechanical and Thermal Trade-Offs · Materials
Retraction, Stringing, and Pressure Advance Explained · Troubleshooting
Why Vertical and Horizontal 3D-Printed Holes Print Undersized · Design & Tolerances

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What makes LayerMath different

Formulas in the open. Every calculator shows the exact equation it uses—from the Slic3r stadium cross-section to polygon chord drop and polymer thermal shrinkage—so you can check the math on paper.

Sources on every guide. We cite firmware documentation (Klipper, Marlin), slicer manuals, polymer technical data sheets and kinematics references so you can read the primary material.

No invented bench data. When a guide calls for caliper logs, power-meter readings or pull tests that have not been run yet, it marks the spot with a visible “Field test pending” box rather than making up numbers. Read more in our editorial policy or learn who runs the site.