# What is the name of the quadrilateral shape described by two radii and two arcs?

What is the name of the four sided shape described by two radii and two concentric arcs? Like each black and white section taken individually on this image

Or the double score / triple score areas on a dart board.

I'm attempting to create each section individually, programmatically and need help on the best thing to google to get help actually making this.

-
I think it's a trapezium –  user9413 May 17 '11 at 12:50
@Chandru: According to Wikipedia, a non-US trapezium is a trapezoid, while a US trapezium is an irregular quadrilateral. Matt seems to be looking for a more specific term. –  user6701 May 17 '11 at 12:54
I thought a Trapezium had 4 straight edges, two of the edges on the shape I'm looking for a name for, are arcs... –  Matt Fellows May 17 '11 at 12:57
It's the difference between two circular sectors, maybe a truncated circular sector? It's not something that has a very common name anyway, I think. –  Myself May 17 '11 at 13:14
How about "annular sector" or "sector of an annulus"? Not that I ever heard that, but googling for the former brings up quite a few hits. –  t.b. May 17 '11 at 13:49

Since I'm not a native English speaker, I was a bit reluctant to post something I made up myself as an answer. But Googling a little and the comment by amWhy encouraged me (showing me that I didn't really make it up, after all):

I think the word you're looking for is

Annular Sector

in analogy with a circular sector. I think that everyone knowing what a circular sector is, should be able to figure out what an annular sector is. An alternative word would be sector of an annulus.

You can specify an annular sector by giving its center $C$, the opening angle $\theta$ and the radii $0 \lt r \lt R$ of the two circular arcs bounding it:

Both these terms, annular sector and sector of an annulus give quite a few hits in Google. For example Googling for area annular sector brings up this link where you can find the formula

$$\textbf{Area of an angular sector:} \qquad \qquad A_{r,R,\theta} = \frac{(R^{2} - r^2)}{2}\cdot \theta$$

for the area $A_{r,R,\theta}$ of an annular sector with radii $0 \lt r \lt R$ and angle $\theta$ (measured in radians). This is because an annular sector is the circular sector of radius $R$ minus the circular sector of radius $r$ whose areas are $\dfrac{R^2}{2} \cdot \theta$ and $\dfrac{r^2}{2}\cdot\theta$, respectively. If the opening angle $\theta$ is given in degrees, you can convert it to radians using the formula $$\operatorname{radians} = \operatorname{degrees} \times \frac{\pi}{180}.$$

The length of the circumference $c_{r,R,\theta}$ of an annular sector with radii $0 \lt r \lt R$ and angle $\theta$ is

$$\textbf{Circumference of an annular sector:} \qquad c_{r,R,\theta} = R \cdot (\theta + 2) + r \cdot (\theta-2)$$

where again $\theta$ is measured in radians. This is obtained by adding the lengths $R \theta$ and $r \theta$ of the two circular arcs and twice the length $(R-r)$ of the straight side.

-
The picture was created using GeoGebra. –  t.b. May 19 '11 at 6:27
On the nose! Thanks... –  Matt Fellows Sep 9 '11 at 9:32
Merci,${}$ @J.M.! –  t.b. Sep 9 '11 at 12:29
What are friends for, right? ;) –  Guess who it is. Sep 9 '11 at 12:42
@J.M. I really appreciate it. In fact, whenever you see English (or other) glitches please feel free to correct them. I'm trying to do my best but I'm sure there's a lot of room for improvement :) –  t.b. Sep 9 '11 at 13:17

A quadrilateral has straight sides, so I assume you are referring to the quadrilateral that connects the 4 relevant points.

It's a trapezoid. More specifically, it is an isosceles trapezoid

Since the circles from which you got the arcs are concentric, we know we get overlapping isosceles triangles when we take, separately, each of the two pairs of arc points together with the center of the circle. For the large isosceles triangle, this implies that the far side of the quadrilateral (relative to the center of the circle) makes angles with each of the radius that are each equal to the other. Similarly, the near side creates equal angles with the radii.

The overlapping isosceles triangles have the same central angle, so the 4 base angles are all (180-A)/2 degrees. This same configuration (ultimately that all corresponding angles where a radius intersects the near and far sides are equal) is defined whenever we have parallel lines intersecting a third line. Thus, we see that the near and far sides are parallel. This means we have a trapezoid. [Note that the bases are of different sizes to match the isosceles triangles which are definitely not congruent, so we don't have a rectangle.]

So a trapezoid with opposite base angles being congruent is an isosceles trapezoid.

-
"A quadrilateral has straight sides" would apparently contradict the fact that the two parallel sides of the "trapezoid" are circle arcs... –  Guess who it is. May 17 '11 at 15:01
Right, so we have a choice. Either the question did not mean "quadrilateral" or it didn't mean that the "sides" were curved. It used the word "described", which can be used in a general sense to mean that it marks off uniquely. I assumed then that we were talking about the quadrilateral formed by the 4 implied points. –  Jose_X May 17 '11 at 15:07
Sorry - my question was not correctly worded - I meant a four sided shape (Two of the sides are not straight, but formed by the concentric arcs of the circle to which the radii, forming the other two sides, belong) –  Matt Fellows Sep 9 '11 at 9:32