Calculate standard U.S. flag hoist, fly, union, stripe, star diameter, area, and star spacing from a known height, length, or area in four measurement units.
American Flag Dimensions Formula
Every measurement on a regulation United States flag is a fixed fraction of the hoist (the height). Once you know the hoist, you can derive every other part. The core relationships are:
Fly = Hoist * 1.9
Union Height = Hoist * 7 / 13
Union Width = Hoist * 0.76
Stripe Width = Hoist / 13
Star Diameter = Hoist * 0.0616
- Hoist is the height of the flag along the pole side. It is the base value of 1.0 that every other measurement is scaled from.
- Fly is the length of the flag from the pole. It is always 1.9 times the hoist, giving the official 10:19 ratio.
- Union Height is the vertical size of the blue canton. It covers 7 of the 13 stripes, which is approximately 0.5385 of the hoist.
- Union Width is the horizontal size of the blue canton. It runs two fifths of the fly, which works out to 0.76 of the hoist.
- Stripe Width is the thickness of each of the 13 stripes, one thirteenth of the hoist.
- Star Diameter is the width of each of the 50 stars, the tabulated factor 0.0616 of the hoist, approximately four fifths of a stripe.
The calculator lets you enter the one dimension you already know, whether that is the hoist, the fly, or the total area, and it solves for the rest. If you enter the fly it divides by 1.9 to recover the hoist; if you enter the area it takes the square root of the area divided by 1.9. Every part is then scaled from that hoist.
For the 50-star layout, nine rows alternate six and five stars. The horizontal stagger or grid step is 0.063 times the hoist; same-row centers are two steps apart, 0.126 times the hoist. Adjacent rows are 0.054 times the hoist apart. See the Executive Order 10834 attachment and authorized-size provisions. This calculator follows the standard 10:19 construction; the order also lists procurement sizes and permits authorized variations.
Standard Flag Sizes and Official Proportions
The first table lists common American flag sizes sold for poles and buildings. Many off-the-shelf sizes do not match the standard 1.9 ratio. The second table lists the tabulated ratios from Executive Order 10834; its stripe and union fractions are exact, while some decimal entries are rounded.
| Hoist x Fly (ft) | Ratio (fly/hoist) | Common use |
|---|---|---|
| 3 x 5 | 1.67 | Home and porch |
| 4 x 6 | 1.50 | Residential pole |
| 5 x 8 | 1.60 | Small commercial |
| 6 x 10 | 1.67 | Commercial pole |
| 10 x 19 | 1.90 | Exact official ratio |
| 20 x 38 | 1.90 | Large display |
| Element | Ratio to hoist |
|---|---|
| Hoist of flag (A) | 1.0 |
| Fly of flag (B) | 1.9 |
| Hoist of union (C) | 0.5385 |
| Fly of union (D) | 0.76 |
| Diameter of star (K) | 0.0616 |
| Width of stripe (L) | 0.0769 |
Example Problems
Example 1. You have a flag with a 3 ft hoist. The fly is 3 * 1.9 = 5.7 ft. The union height is 3 * 7 / 13 ≈ 1.6154 ft, and the union width is 3 * 0.76 = 2.28 ft. Each stripe is 3 / 13 = 0.2308 ft (about 2.77 in), and each star is 3 * 0.0616 = 0.1848 ft (about 2.22 in).
Example 2. You know only the fly, which is 19 ft. The hoist is 19 / 1.9 = 10 ft. From that hoist the union is 7.6 ft wide and about 5.3846 ft tall, each stripe is 0.769 ft wide, and the total area is 10 * 19 = 190 square feet.
Frequently Asked Questions
What is the official ratio of the American flag? The official ratio of hoist to fly is 10 to 19, so the length is 1.9 times the height. This was fixed by Executive Order 10834 in 1959.
Why do store bought flags not match the 1.9 ratio? Common retail sizes such as 3 by 5 or 4 by 6 feet have ratios of approximately 1.67 and 1.50. Only sizes like 10 by 19 or 20 by 38 feet hold the exact official proportion.
Does the calculator apply to all flags? Executive Order 10834 governs flags made for or by the federal government. The same proportions are the accepted standard for any flag built to spec, but a retail flag may round its dimensions to the nearest stock size.
