Explore sixteen formula topics, worked examples and five interactive demonstrations.
Understand the numbers behind the stone.
Sixteen practical formula topics, worked examples and five live demonstrations. Read the method, try your own values, then open the corresponding calculator for a complete worksheet.
Every example is educational. Area planning, physical placement and procurement assumptions answer different questions.
Usable Slab Area, Area-Only Slab Counts and Availability
Reserve margins before computing a whole-slab area lower bound.
Lu = L − left − right; Wu = W − top − bottom; U = Lu × Wu; count = ceil(R ÷ U)
Variables and units
R
Required area (m²)
U
Usable area per slab (m²)
margins
Independent edge reservations (mm)
When to use it
Use for an area-only procurement check before physical nesting.
Worked example
300 × 180 cm slab; 10 mm on each edge; demand 27.33 m²
Usable dimensions: 298 × 178 cm.
Usable area = 5.3044 m²; reserved = 5.4 − 5.3044 = 0.0956 m².
ceil(27.33 ÷ 5.3044) = 6; estimated usable area 31.8264 m²; surplus 4.4964 m².
Without margins: ceil(27.33 ÷ 5.4) = 6. Five available full slabs provide 27 m²: 0.33 m² shortfall.
Answer: 6 whole slabs as an area-only lower bound
Common mistakes
Using full area after reserving margins or rounding the quotient down.
Precision and rounding
The engine uses exact integer square micrometres and integer ceiling, not a rounded division.
Known limitations
Area sufficiency does not guarantee nesting, verified stock, recoverable surplus or actual waste.
Try it: Slab estimation
Values stay in this page’s memory. Changing units reinterprets the entered measurement. Match source and target categories for conversions.
Full slab area: 5.4 m²
Usable area: 5.3044 m²
Reserved margin area: 0.0956 m²
Whole-slab ceiling: 6
Estimated usable area: 31.8264 m²
Theoretical surplus: 4.4964 m²
Available usable area: 26.522 m²
Availability shortfall: 0.808 m²
Availability surplus: 0 m²
Area-only estimate. Actual cut-piece sizes, arrangement, saw kerf, defects and fabrication constraints may require more slabs. Theoretical surplus is not verified reusable material or actual waste.
Budget independent allowances on original net demand.
p = cutting% + breakage% + contingency%; extra = N × p ÷ 100; budget = N + extra
Variables and units
N
Original net area (m²)
p
Combined user-selected allowance (%)
When to use it
Use for an explicit procurement assumption, separate from layout geometry.
Worked example
100 m² with 5%, 3%, 2%; second example 27.33 m² with 5%
100 × 5% = 5; 100 × 3% = 3; 100 × 2% = 2 m².
Combined 10%; budget 100 + 10 = 110 m².
27.33 × 5% = 1.3665 m²; exact budget 28.6965 m².
Optional upward ordering rounding to 0.01 m² gives 28.70 m² separately.
Answer: 110 m²; second exact budget 28.6965 m²
Common mistakes
Compounding percentages or treating allowances as measured physical waste.
Precision and rounding
Exact rational areas preserve fractional square micrometres. Each percentage is on a 0.01% grid; ordering rounding is separate.
Known limitations
No default loss rate is claimed; geometric unused area must not be automatically added.
Try it: Waste allowances
Values stay in this page’s memory. Changing units reinterprets the entered measurement. Match source and target categories for conversions.
Net area: 100 m²
cuttingAllowance: 5% of original net area = 5 m²
breakageAllowance: 3% of original net area = 3 m²
contingencyAllowance: 2% of original net area = 2 m²
Combined allowance: 10% = 10 m²
Exact budgeted area: 110 m²
Planning estimate based on user-entered allowances. Not measured fabrication waste or verified reusable offcut area. Do not automatically add geometric unoccupied slab area to these allowances.
Saw Kerf, Geometric Free Regions and Potential Offcut Screening
Understand separation constraints and geometric candidate screening without inventing a physical waste formula.
For each pair: right(A) + kerf ≤ left(B), or reverse, or bottom(A) + kerf ≤ top(B), or reverse; candidates must meet dimension thresholds
Variables and units
A, B
Two finished-piece rectangles being checked (rectangles)
left, right, top, bottom
Axis coordinates of a finished rectangle within the slab (one normalized length unit)
kerf
Required axis separation between finished rectangles (mm)
free region
Disjoint rectangle in usable area outside placed rectangles (mm dimensions)
thresholds
Minimum potential-offcut length and width (mm)
When to use it
Use to interpret a validated cutting plan and potential offcut map.
Worked example
Two 800 × 600 mm pieces, 3 mm kerf; potential-offcut thresholds 300 × 300 mm
Piece A starts x=0; B can start x=803 mm with 3 mm horizontal clearance if both fit the usable slab.
A geometric 400 × 350 mm free rectangle passes 300 × 300 mm screening; 400 × 250 mm does not.
The engine partitions the complement of finished rectangles and validates containment, non-overlap and exact area reconciliation.
Answer: Pairwise clearance and candidate screening require geometric validation.
Common mistakes
Multiplying kerf by piece count as a universal waste formula.
Precision and rounding
Integer micrometre geometry validates spacing; bounded partition enumeration can be incomplete and must not publish truncated candidates as complete.
Known limitations
Partition is not a unique physical leftover shape. Cut sequence, separability, defects, vein direction, blade paths and reuse suitability are unverified. Optimizer is heuristic; no fabrication instructions.
Reading is available without JavaScript. Live examples, enhanced search and clipboard actions require JavaScript. No measurement values are saved, added to URLs or sent to a server.