The United Nations recognizes 193 national flags of its member states. Each flag is unique, giving the full collection a great variety of symbols and colors. Despite all the diversity, almost all of them — 190, in fact — are the same basic rectangular shape. Only three stand out from the others and refuse to fit in a rectangular box. Can you identify them?

The first example of a non-rectangular flag comes with a qualification. It might actually be rectangular. More on that later.

The flag of Vatican City dates to June 7, 1929. Pope Pius XI signed the Lateran Pacts, designating Vatican City as a sovereign state. The flag is divided vertically with white and yellow. On the white side is an emblem of two keys crossed, one silver and one gold symbolizing the keys given to St. Peter. Above the keys is the Papal Tiara.

Any internet search will tell you that Vatican City’s flag is square. That’s the way it is generally depicted. In this official letter to the German nunciature, however, the Vatican’s representative insisted the flag is rectangular. We are sending this to the Commonplace Fun Facts Fact-Checking Department to see if further light can be shed on this.

Variations of Switzerland’s flag, with its white cross and red background, have been around since the Napoleonic era. Its current form was specified in the 2017 flag law (SR 232.21): “the Swiss flag shows a Swiss cross on a square background.” The Swiss cross is defined as “a white, upright, free-standing cross depicted against a red background, whose arms, which are all of equal size, are one-sixth longer than they are wide.”

In keeping with the country’s long tradition of neutrality, the square shape, simple cross, and two colors make it easy to reproduce and difficult to fight about.

In a class all by itself is the flag of Nepal. It is not only non-rectangular, it is the only non-quadrilateral flag in the world.

It consists of two single pennons (or pennants), known as a double-pennon. Crimson red symbolizes bravery, as well as its national flower, the rhododendron. The blue border represents peace. Prior to 1962, the flag’s emblems, both the sun and the crescent moon, had human faces, but they were removed to modernize the flag. Within the pennons is a crescent moon with eight rays and a sun with sixteen rays.

The dimensions for the flag are specific and would make a good final exam for a geometry class. They are set forth in Schedule 1 of Nepal’s constitution, reprinted below, with links provided to give a visual representation of the specifications.

Schedule 1
Method of Making the National Flag of Nepal

(A) Method of Making the shape inside the Border
(1) On the lower portion of a crimson cloth draw a line AB of the required length from left to right.
(2) From A draw a line AC perpendicular to AB making AC equal to AB plus one third AB. From AC mark off D making the line AD equal to line AB. Join BD.
(3) From BD mark off E making BE equal to AB.
(4) Touching E draw a line FG, starting from the point F on line AC, parallel to AB to the right hand-side. Mark off FG equal to AB.
(5) Join CG.

(B) Method of making the Moon
(6) From AB mark off AH making AH equal to one-fourth of line AB and starting from H draw a line HI parallel to line AC touching line CG at point I.
(7) Bisect CF at J and draw a line JK parallel to AB touching CG at point K.
(8) Let L be the point where lines JK and HI cut one another.
(9) Join JG.
(10) Let M be the point where line JG and HI cut one another.
(11) With center M and with a distance shortest from M to BD mark off N on the lower portion of line HI.
(12) Touching M and starting from O, a point on AC, draw a line from left to right parallel to AB.
(13) With center L and radius LN draw a semi-circle on the lower portion and let P and Q be the points where it touches the line OM respectively.
(14) With the center M and radius MQ draw a semi-circle on the lower portion touching P and Q.
(15) With center N and radius NM draw an arc touching PNQ at R and S. Join RS. Let T be the point where RS and HI cut one another.
(16) With center T and radius TS draw a semi-circle on the upper portion of PNQ touching at two points.
(17) With center T and radius TM draw an arc on the upper portion of PNQ touching at two points.
(18) Eight equal and similar triangles of the moon are to be made in the space lying inside the semi-circle of No (16) and outside the arc of No (17) of his Schedule.

(C) Method of Making the Sun
(19) Bisect line AF at U, and draw a line UV parallel to AB line touching line BE at V.
(20) With center W, the point where HI and UN cut one another and radius MN draw a circle.
(21) With center W and radius LN draw a circle.
(22) Twelve equal and similar triangles of the sun are to be made in the space enclosed by the circle of No (20) and No (21) with the two apexes of two triangles touching line HI.

(D) Method of Making the Border
(23) The width of the border will be equal to the width of TN. This will be of deep blue color and will be provided on all the sides of the flag. However, on the given angles of the flag the external angles will be equal to the internal angles.
(24) The above mentioned border will be provided if the flag is to be used with a rope. On the other hand, if it is to be hoisted on a pole, the hole on the border on the side AC can be extended according to requirements.
Explanation:- The lines HI, RS, FE, ED, JG, OQ, JK and UV are imaginary. Similarly, the external and internal circles of the sun and the other arcs except the crescent moon are imaginary. These are not shown on the flag.