The C series paper sizes, standardized in ISO 269 (primarily for envelopes), are defined as the geometric mean between corresponding sizes in the A series and B series. This ensures envelopes perfectly fit A-series sheets while accounting for paper thickness and folding. Here’s the precise definition:

①. C0 Starting Point:
The largest size, C0, is derived as the geometric mean of A0 and B0:
Width(C0) = √(Width(A0) × Width(B0))
Height(C0) = √(Height(A0) × Height(B0))
②. Constant Aspect Ratio:
All C sizes maintain the aspect ratio 1:√2 (≈1:1.4142), identical to A and B series.
③. Geometric Mean Relationship:
C𝑛 = Geometric Mean(A𝑛, B𝑛)
Width(C𝑛) = √(Width(A𝑛) × Width(B𝑛))
Height(C𝑛) = √(Height(A𝑛) × Height(B𝑛))

①. C0 Calculation (using A0 = 841×1189 mm, B0 = 1000×1414 mm):
Width(C0) = √(841 × 1000) ≈ √841,000 ≈ 917 mm
Height(C0) = √(1189 × 1414) ≈ √1,681,246 ≈ 1297 mm
C0 = 917 mm × 1297 mm
②. Smaller Sizes (via Halving):
Halve the longer side to generate subsequent sizes (same as A/B series):
C1: Halve C0 height → 917 × 648 mm
C2: Halve C1 width → 648 × 458 mm
(Exact dimensions use geometric mean then rounded to mm).
③. Practical Example (C4):
A4 = 210 × 297 mm, B4 = 250 × 353 mm
Width(C4) = √(210 × 250) = √52,500 ≈ 229 mm
Height(C4) = √(297 × 353) = √104,841 ≈ 324 mm
C4 = 229 mm × 324 mm

①. Envelope Sizing: C series is optimized for envelopes to hold A-series paper:
A flat sheet of size A𝑛 fits perfectly in C𝑛.
A sheet folded once (e.g., A3 folded to A4) fits in C(𝑛-1).
Example: C4 holds an unfolded A4 sheet. C5 holds A4 folded once (A5 size) or an unfolded A5 sheet.
②. Paper Thickness Allowance: The geometric mean provides ~3% extra space, accommodating paper thickness and folds without forcing or gaping.
③. Nesting Compatibility: C-series envelopes themselves fit neatly into larger B-series envelopes (e.g., C4 envelope fits inside B4 envelope).
④. Tolerances: Follows ISO 216 (typically ±1–1.5 mm).

Universal Compatibility: Standardizes envelopes globally for A-series documents.
Efficient Design: Eliminates guesswork—each C𝑛 snugly fits A𝑛 with room for thickness.
Scalability: Maintains the √2 ratio, allowing consistent scaling between envelope sizes.

Note: The C series is occasionally used for postcards (C6) or notepads, but its core purpose remains envelopes. Like A/B series, it originated from Georg Lichtenberg’s 1786 principles and was formalized in ISO standards later.
The C series completes the ISO paper ecosystem: A for sheets, B for intermediate needs, C for encapsulation—all sharing the same mathematical harmony.