Description

Create perfectly fitted envelopes for any paper size using a mathematically derived model.

This calculator is based on a 45° diagonal projection method, allowing you to generate a square envelope blank from any rectangular insert. See Appendix A for the mathematical derivation. A 45° rotation is used because it produces equal horizontal and vertical projections of the insert. This symmetry allows a single square sheet to fold evenly around a rectangular insert while keeping the score lines equidistant from the insert on all four sides. Other rotation angles do not provide this symmetry and would require different paper dimensions and scoring positions. Unlike the printed lookup tables supplied with many commercial scoring boards, including the Vaessen Creative Score Easy, this approach is not limited to predefined sizes and gives you full control over fit and clearance. See Appendix B for an explanation of why the method works.

Mathematical Derivation:

• L: side length of the square paper (mm)
• S₁: first score line position (mm)
• S₂: second score line position (mm)
• x: width of the finished insert (mm)
• y: length of the finished insert (mm)
• z: padding allowance (mm)

Note: All calculations assume that the insert is rotated exactly 45° within the square sheet and that the padding is uniform on all four sides.

Input Variables:

The calculator requires three inputs:

• x — the width of the finished insert (mm)
• y — the length of the finished insert (mm)
• z — the desired clearance (padding) between the insert and the score line (mm)

Padding (z) controls how tight or loose the envelope fits.

Recommended defaults:

• 10 mm → tight fit
• 15 mm → standard (default)
• 20 mm → loose / bulky inserts

Calculated Variables:

Using these values, the calculator determines:

• L — the required square paper size
• S₁ — the first scoring position
• S₂ — the second scoring position

This method is especially useful for:

• ISO paper sizes
• JIS paper sizes
• folded letters and brochures
• greeting cards
• invitations
• photographs
• engineering drawings
• archival document sleeves
• custom paper dimensions
• precision paper crafting

Design Rationale

Commercial envelope scoring boards are convenient for common paper sizes but are inherently limited by printed lookup tables. As soon as a project involves a folded document, a custom-sized photograph, a greeting card, or a non-standard paper format such as JIS B-series paper, those tables no longer provide a suitable solution.

This calculator replaces predefined lookup tables with a mathematical model. By calculating the dimensions directly from the insert size and the desired padding, it can generate a precisely fitted envelope for virtually any rectangular insert.

Calculator

Enter the finished dimensions of the insert below. The calculator determines the required square paper size, the score line positions and a visual representation of the resulting envelope layout.

Precalculated Values For Folded Sheets

A series

Paper / Fold	Width (x) in mm	Length (y) in mm	Padding (z)	Square paper size	Score 1	Score 2	Rounded cut size	Rounded score marks
A4 half-fold (A5)	210	148.5	10 mm	273.5 mm × 273.5 mm	158.5 mm	115.0 mm	273 mm × 273 mm	158 mm and 115 mm
A4 half-fold (A5)	210	148.5	15 mm	283.5 mm × 283.5 mm	163.5 mm	120.0 mm	283 mm × 283 mm	163 mm and 120 mm
A4 half-fold (A5)	210	148.5	20 mm	293.5 mm × 293.5 mm	168.5 mm	125.0 mm	293 mm × 293 mm	168 mm and 125 mm
A4 tri-fold	210	99	10 mm	238.5 mm × 238.5 mm	158.5 mm	80.0 mm	238 mm × 238 mm	158 mm and 80 mm
A4 tri-fold	210	99	15 mm	248.5 mm × 248.5 mm	163.5 mm	85.0 mm	248 mm × 248 mm	163 mm and 85 mm
A4 tri-fold	210	99	20 mm	258.5 mm × 258.5 mm	168.5 mm	90.0 mm	258 mm × 258 mm	168 mm and 90 mm
A5 half-fold (A6)	148	105	10 mm	198.9 mm × 198.9 mm	114.7 mm	84.2 mm	199 mm × 199 mm	115 mm and 84 mm
A5 half-fold (A6)	148	105	15 mm	208.9 mm × 208.9 mm	119.7 mm	89.2 mm	209 mm × 209 mm	120 mm and 89 mm
A5 half-fold (A6)	148	105	20 mm	218.9 mm × 218.9 mm	124.7 mm	94.2 mm	219 mm × 219 mm	125 mm and 94 mm
A6 half-fold (A7)	105	74	10 mm	146.6 mm × 146.6 mm	84.2 mm	62.3 mm	147 mm × 147 mm	84 mm and 62 mm
A6 half-fold (A7)	105	74	15 mm	156.6 mm × 156.6 mm	89.2 mm	67.3 mm	157 mm × 157 mm	89 mm and 67 mm
A6 half-fold (A7)	105	74	20 mm	166.6 mm × 166.6 mm	94.2 mm	72.3 mm	167 mm × 167 mm	94 mm and 72 mm

B series — ISO

Paper / Fold	Width (x) in mm	Length (y) in mm	Padding (z)	Square paper size	Score 1	Score 2	Rounded cut size	Rounded score marks
B5 ISO half-fold (B6)	176	125	10 mm	232.8 mm × 232.8 mm	134.5 mm	98.4 mm	233 mm × 233 mm	134 mm and 98 mm
B5 ISO half-fold (B6)	176	125	15 mm	242.8 mm × 242.8 mm	139.5 mm	103.4 mm	243 mm × 243 mm	139 mm and 103 mm
B5 ISO half-fold (B6)	176	125	20 mm	252.8 mm × 252.8 mm	144.5 mm	108.4 mm	253 mm × 253 mm	144 mm and 108 mm
B6 ISO half-fold (B7)	125	88	10 mm	170.6 mm × 170.6 mm	98.4 mm	72.2 mm	171 mm × 171 mm	98 mm and 72 mm
B6 ISO half-fold (B7)	125	88	15 mm	180.6 mm × 180.6 mm	103.4 mm	77.2 mm	181 mm × 181 mm	103 mm and 77 mm
B6 ISO half-fold (B7)	125	88	20 mm	190.6 mm × 190.6 mm	108.4 mm	82.2 mm	191 mm × 191 mm	108 mm and 82 mm

B series — JIS

Paper / Fold	Width (x) in mm	Length (y) in mm	Padding (z)	Square paper size	Score 1	Score 2	Rounded cut size	Rounded score marks
B5 JIS half-fold (B6)	182	128.5	10 mm	239.6 mm × 239.6 mm	138.7 mm	100.9 mm	240 mm × 240 mm	139 mm and 101 mm
B5 JIS half-fold (B6)	182	128.5	15 mm	249.6 mm × 249.6 mm	143.7 mm	105.9 mm	250 mm × 250 mm	144 mm and 106 mm
B5 JIS half-fold (B6)	182	128.5	20 mm	259.6 mm × 259.6 mm	148.7 mm	110.9 mm	260 mm × 260 mm	149 mm and 111 mm
B6 JIS half-fold (B7)	128	91	10 mm	174.9 mm × 174.9 mm	100.5 mm	74.3 mm	175 mm × 175 mm	101 mm and 74 mm
B6 JIS half-fold (B7)	128	91	15 mm	184.9 mm × 184.9 mm	105.5 mm	79.3 mm	185 mm × 185 mm	106 mm and 79 mm
B6 JIS half-fold (B7)	128	91	20 mm	194.9 mm × 194.9 mm	110.5 mm	84.3 mm	195 mm × 195 mm	111 mm and 84 mm

How to Make a Custom Envelope for JIS B5 Folded in Half

Finished insert size

A JIS B5 sheet is 182 mm × 257 mm. Folded in half, the insert becomes:

182 mm × 128.5 mm

Using 15 mm padding, the calculator gives:

Measurement	Exact	Practical (Rounded)
Square paper size	249.6 mm × 249.6 mm	250 mm × 250 mm
Score 1	143.7 mm	144 mm
Score 2	105.9 mm	106 mm

Tools and materials

You will need:

• One JIS B4 donor sheet
• One JIS B5 sheet folded in half
• Paper cutter
• Vaessen Creative Score Easy scoring board
  • Diagonal scoring guide
  • Folding knife or bone folder
• Vaessen Creative Corner Punch
• Optional 10 mm Corner Punch
• Paper glue

Practical reference

For a JIS B5 sheet folded in half with 15 mm padding, use:

Cut size: 250 mm × 250 mm
Score marks: 144 mm and 106 mm
Score pattern: 144 → 106 → 144 → 106

Practical Tip

Before cutting expensive cardstock, test the calculated dimensions using ordinary printer paper. This allows the fit to be verified and the padding adjusted if a tighter or looser envelope is desired.

Step 1: Prepare the workspace

Set out the paper cutter, scoring board, diagonal guide, folding tool, punch, glue and paper.

Step 2: Cut the donor sheet

Start with a JIS B4 donor sheet.

Cut it to:

250 mm × 250 mm

The calculated size is 249.6 mm, so 250 mm is the practical cut size.

Step 3: Set up the scoring board

Place the diagonal scoring guide into the Vaessen Creative Score Easy board.

Step 4: Make the first score

Place the square sheet on the scoring board with one corner aligned to the diagonal guide.

Score at:

144 mm

Step 5: Rotate and make the second score

Rotate the paper 90 degrees.

Score at:

106 mm

Step 6: Complete all four score lines

Continue rotating the paper and alternate the score marks:

144 mm → 106 mm → 144 mm → 106 mm

When finished, the sheet should have four score lines.

Step 7: Check the score pattern

The score lines should form an X-style structure where the folds meet.

Step 8: Punch out the corners

Use the corner punch to remove the small corner sections where the folds meet. This reduces bulk and helps the envelope fold cleanly.

Step 9: Test the insert fit

Place the folded JIS B5 sheet into the future envelope before gluing. Confirm that it fits comfortably.

Step 10: Optional: round the flap corner

Use a 10 mm Corner Punch if you want a cleaner finished flap.

Step 11: Fold the envelope

Fold along all score lines. Bring the side flaps inward first, then fold the bottom flap up.

Step 12: Apply glue

Apply a small amount of glue to the smaller side flap or overlap area. Avoid using too much glue.

Step 13: Press the glued flap

Fold the glued flap into place and press firmly. Let the glue set.

Step 14: Insert the folded sheet

Place the folded JIS B5 sheet into the finished envelope.

Step 15: Close the final flap

Fold the top flap down to close the envelope.

Step 16: Finished envelope

The finished envelope should fit the folded JIS B5 insert neatly with 15 mm padding.

Step 17: Store the tools

Store the diagonal guide and folding knife in the back of the Score Easy board.

The diagonal guide can be supplemented with an insert showing the Mathematical Derivation:

Vaessen Creative Score Easy Insert A4

Vaessen Creative Score Easy Insert US Letter

Engineering Note

Unlike printed scoring-board tables, the calculator derives every dimension mathematically. The equations are deterministic, meaning the same inputs always produce the same calculated results. Because the calculation consists of a fixed sequence of arithmetic operations, calculated results are produced almost instantaneously regardless of the dimensions entered.

Appendix A: Mathematical Foundation

Historical Note

The mathematical basis of this calculator originates in classical Euclidean geometry and the properties of rotating a rectangle by 45°. When a rectangle is rotated by this angle, its width and height project equally onto the horizontal and vertical axes by a factor of 1/√2 (approximately 0.70710678). This relationship follows directly from the trigonometric identities:$$\cos 45^\circ = \sin 45^\circ = \frac{1}{\sqrt{2}}$$These geometric principles have been understood for centuries and form the foundation of technical drawing, engineering, architecture and descriptive geometry. These same geometric principles are also used in computer graphics, computer-aided design (CAD), drafting and technical illustration whenever objects are projected or rotated within a two-dimensional plane. While the underlying mathematics is well established, it has rarely been explained in the context of envelope scoring boards or paper crafting. This calculator applies those principles to determine the required square paper size and score line positions for virtually any rectangular insert using a deterministic mathematical model rather than fixed lookup tables.

Numerical Stability

The calculator uses only first-order arithmetic operations (addition, subtraction, multiplication and division) together with the mathematical constant √2. No iterative approximation or numerical optimisation is required. Consequently, the calculated values are stable, repeatable and more precise than the manufacturing tolerances of most scoring boards and paper cutters.

Appendix B: Why the Calculator Works

Most commercial scoring boards provide printed lookup tables containing dimensions for only a limited selection of common paper sizes, such as the ISO A-series. These tables are simply pre-calculated solutions for specific paper dimensions and cannot accommodate arbitrary insert sizes.

This calculator replaces those fixed tables with a general mathematical solution. By applying the geometric properties of a rectangle rotated by 45°, it calculates the required square paper size and score line locations for any rectangular insert entered by the user.

Because the calculator depends only on the insert width (x), insert length (y) and the selected padding (z), it can calculate envelope dimensions for virtually any rectangular insert. Because the calculator derives the dimensions directly from the input measurements, no lookup tables or predefined templates are required. This makes the approach scalable, repeatable and suitable for both common and custom paper sizes. Unlike printed lookup tables, which are limited to predefined paper sizes, this method performs the same sequence of arithmetic operations for every calculation. In computer science, this is known as constant-time complexity, meaning the amount of computational work does not increase as the input dimensions become larger. Whether the insert measures 50 × 50 mm or 500 × 700 mm, the calculator performs the same fixed series of additions, divisions and multiplications, producing calculated results almost instantaneously on modern hardware.

Why millimetres?

Millimetres were chosen because they provide practical precision while matching the graduations found on most modern paper cutters, rulers and scoring boards. If measurements are taken in another unit, they can be converted to millimetres before using the calculator.

Engineering assumptions

The mathematical model assumes:

• a rectangular insert,
• a 45° rotation,
• uniform paper thickness,
• accurate cutting and scoring,
• a constant padding allowance.

Appendix C: Licence and Attribution

The geometric principles implemented by this calculator are based on well-established concepts from classical Euclidean geometry, elementary trigonometry and the mathematics of 45° orthogonal projection. These mathematical relationships are part of the public domain and have been widely documented in mathematics, engineering and technical drawing for centuries.

This implementation does not claim originality for the underlying mathematics. Its contribution is the application of established geometric principles to create a practical calculator, accompanying documentation and visualisation tools for envelope construction.

The original implementation, user interface, documentation, examples and explanatory material presented on this site are original works created for this project.

References

• Euclid. Elements.
• ISO 216:2007 — Writing Paper and Certain Classes of Printed Matter.
• JIS P 0138 — Japanese Standard Paper Sizes.
• Weisstein, Eric W. "Rotation Matrix." MathWorld.
• Weisstein, Eric W. "Orthogonal Projection." MathWorld.