The United States flag is one of the few flags in the world whose every internal dimension is fixed by a published set of ratios. Once you know the hoist — the short side, the one along the pole — every other measurement follows by multiplication. This flag proportions calculator applies those ratios to whatever size you are working at, and shows the letter labels used in the source document so you can check the output against the original.
Arb Digital publishes this as part of a wider set of measurement and scaling tools, because the flag question comes up in sign making, print, embroidery, event staging and classroom projects, and the answer is almost always given as a table of standard sizes rather than as a rule you can apply to a non-standard one. The rule is more useful than the table.
What This Flag Proportions Calculator Does
Enter one measurement and it returns the full set: overall hoist and fly, the hoist and fly of the union, the width of each stripe, the diameter of each star, the vertical spacing between star rows, and the horizontal spacing between star columns. It also gives the flag area, which is what actually drives fabric or vinyl cost.
Three starting points are available. If you know the hoist, everything scales from it directly. If you know the fly — because you are matching an existing flag or a fixed banner width — the hoist is derived by dividing by 1.9 first. If you have a space to fill, enter its height and width and the tool sizes the largest correctly proportioned flag that fits, which is almost always limited by one dimension and not the other.
The tool is a scaling aid. It does not address flag etiquette, display rules or the many conventions that govern how and when a flag is flown, all of which are separate matters set out in law and custom.
How to Use It
- Choose your starting point. Hoist is the usual one. Fly is useful when matching an existing flag, and fit mode when you have a wall, window or stage opening to work within.
- Enter the measurement and pick a unit. Feet, inches, centimetres or metres — every output comes back in the same unit, because the ratios are dimensionless.
- In fit mode, enter both the available height and width. The flag is sized to whichever limit binds first.
- Set the decimal places. Small flags need three or four; a large outdoor flag rarely needs more than two.
- Read the full dimension list under the results for the star spacing values, which are the ones people usually get wrong.
The Formula: The Executive Order 10834 Ratios
The governing document is Executive Order 10834, The Flag of the United States, signed on 21 August 1959 after Hawaii's admission brought the star count to fifty. The order itself refers to an attachment that carries the dimensional table, printed at 3 CFR, 1959–1963 Comp., page 368. Every dimension in that table is expressed as a multiple of the hoist, which is defined as 1.0.
The ratios are: hoist of flag A = 1.0; fly of flag B = 1.9; hoist of the union C = 0.5385, which is seven thirteenths; fly of the union D = 0.76; vertical star spacing E = F = 0.054; horizontal star spacing G = H = 0.063; star diameter K = 0.0616; and stripe width L = 0.0769, which is one thirteenth.
Two of those ratios are worth checking against each other, because doing so confirms you have read the table correctly. The union is nine rows of stars deep: one E gap at the top, eight F gaps between the nine rows, and one E gap at the bottom. That is ten times 0.054, or 0.540, against a union hoist of 0.5385 — agreement to within a rounding step. Across the union there are eleven star columns: one G gap at each edge and ten H gaps between columns, which is twelve times 0.063, or 0.756, against a union fly of 0.76. Again, agreement to a rounding step. If your reading of the table does not survive that check, you have mixed up the letters.
Why 1.9 and Not 1.5 or 2.0
The 1 to 1.9 ratio is unusual. Most national flags use 2:3, 1:2 or 3:5, and the American ratio sits awkwardly between the last two. It is a legacy of the way the flag's internal geometry works out rather than a design choice made for its own sake: thirteen stripes of equal width across the hoist, a union spanning seven of them, and a star field that has to hold rows of six and five stars alternately with sensible margins.
The practical consequence is that most flags you can buy are not at the official ratio. A 3 by 5 foot flag, by far the most common domestic size, has a ratio of 1:1.667. A 4 by 6 is 1:1.5. Only certain sizes — 5 by 9.5, 8 by 15.2, 10 by 19 — hit 1:1.9 exactly. Executive Order 10834 also lists a schedule of authorised sizes for federal use, from 20 by 38 feet down to 1.32 by 2.5 feet, and those all sit at the official ratio.
This matters if you are reproducing a flag graphically. Take the official ratios and squeeze them into a 3:5 rectangle and the union will be wrong, the stripes will be the wrong width, or both. Decide first whether you are drawing an official-proportion flag or a commercial-size one, because they are not the same object.
The Stars Are the Part People Get Wrong
Fifty stars sit in nine horizontal rows, alternating six and five: six, five, six, five, six, five, six, five, six. That is five rows of six and four rows of five, which is thirty plus twenty. Rows of six use every second column position starting at the first; rows of five use every second position starting at the second. This is why the horizontal spacing H is half of a column pitch rather than a full one — there are eleven notional column positions across the union, and no row uses all of them.
Star diameter K is the diameter of the circumscribed circle, not the width across two points or the height of the star. A five-pointed star drawn to a 0.0616 circumscribed circle is noticeably smaller than one drawn to a 0.0616 point-to-point width, and the difference is very visible at large sizes. The stars also each have one point straight up.
Stripes, Union and the Thirteen-Part Grid
The whole flag is built on a thirteen-part division of the hoist. Each stripe is exactly one thirteenth, or 0.0769. The union spans exactly seven of those stripes, which is 7 ÷ 13 = 0.5385. The union therefore covers four red and three white stripes at the hoist end, and the seventh stripe from the top is the last one it touches. Below the union all thirteen stripes run the full fly length.
That grid is the reason a stripe-width error propagates. Get the stripe width slightly wrong and the union no longer aligns with a stripe boundary, which is instantly visible. If you are drawing rather than sewing, set up a thirteen-row grid across the hoist first and place everything on it, rather than measuring each element independently from an edge.
The statutory basis for the thirteen stripes themselves sits in 4 U.S.C. § 1, which specifies thirteen horizontal stripes, alternate red and white, with a union of white stars on a blue field. The star count is amended as states are admitted; the proportions come from the executive order.
Scaling, Area and Material
Flag area is simply hoist × fly, which at the official ratio is 1.9 × hoist². Doubling the hoist therefore quadruples the material, a fact that catches people out when quoting for a large flag. The tool reports area alongside the linear dimensions for exactly this reason. If you need to move that figure between unit systems, the area converter handles square feet to square metres, and the length converter does the same for the linear dimensions.
For the ratio itself, the aspect ratio calculator reduces any two dimensions to their simplest ratio — that is a general-purpose tool for screens, images and print, whereas this page applies one specific legally published ratio set to nine named parts of a single flag. The rectangle area calculator and the square footage calculator cover the plain area arithmetic if you are working out mounting or wall coverage, and the golden ratio calculator is there if you are curious how 1:1.9 compares with the proportions used in design more generally.
Arb Digital builds free calculators that answer a real question properly and earn search traffic for doing it. Browse what we have published, or tell us what your audience keeps asking.
Browse Free Tools Get in TouchCommon Mistakes to Avoid
- Assuming a shop-bought flag is at official proportions — 3 by 5 and 4 by 6 foot flags are not 1:1.9 and never were.
- Treating star diameter as point-to-point width — K is the circumscribed circle, and using it as a width makes every star too large.
- Measuring elements from the flag edge independently — build a thirteen-part grid across the hoist and place everything on it instead.
- Forgetting that the union spans seven stripes exactly — if it lands mid-stripe, a ratio has been misapplied somewhere upstream.
- Scaling area linearly — doubling the hoist quadruples the fabric, not doubles it.
Related Free Tools From Arb Digital
Reduce any two dimensions with the aspect ratio calculator, convert measurements with the length converter and the area converter, and work out coverage with the rectangle area calculator or the square footage calculator. More sit in the free online tools hub.
Frequently Asked Questions
Executive Order 10834 sets the hoist at 1.0 and the fly at 1.9, with the union 0.5385 by 0.76, stripes 0.0769 each, stars 0.0616 in diameter, vertical star spacing 0.054 and horizontal star spacing 0.063. Every figure is a multiple of the hoist.
Because 5 divided by 3 is about 1.667, not 1.9. Common retail sizes were settled by convention and convenience rather than by the executive order, which lists its own schedule of authorised federal sizes.
One thirteenth of the hoist, or 0.0769 times it. On a flag with a 5 foot hoist that is 0.3845 feet, which is about 4.6 inches.
In nine rows alternating six and five stars: five rows of six and four rows of five. Rows of six occupy alternate column positions starting at the first, and rows of five start at the second.
No. The ratio K = 0.0616 gives the diameter of the circle that circumscribes the star, so all five points touch that circle. Using it as a point-to-point width produces stars that are visibly too big.
No. These ratios are specific to the United States flag as set out in Executive Order 10834. Other national flags have their own published specifications with entirely different proportions.
No. It handles dimensions only. Rules on how, when and where a flag may be displayed are a separate matter governed by law and long-standing custom.
