Mathematics students from countries as far-flung as Australia, Belize and Hong Kong may have everyday contact with decagons, since those countries have used the 10-sided shape in their coins. Most of us, though, only encounter decagons in mathematics class and when feeling particularly patriotic.

Decagons belong to the family of two-dimensional shapes called polygons. Polygons get their name from Greek, meaning "many-angled," because all polygons have multiple interior angles. They close in a two-dimensional space using only straight sides.

A **decagon** is a 10-sided polygon, with 10 interior angles, and 10 vertices which is where the sides meet.

For a polygon to be a decagon, it must have these identifying properties:

- 10 sides
- 10 interior angles
- 10 vertices

In many decagons, the sum of interior angles will be $\mathrm{1,440}\xb0$, but that is not an identifying property because complex decagons will not have that sum.

Regular decagons have two additional identifying properties:

- 10 exterior angles of $36\xb0$, summing to $360\xb0$
- 10 interior angles of $144\xb0$, summing to $\mathrm{1,440}\xb0$

Drawing a decagon takes no skill; simply draw 10 line segments that connect, and you have it. Drawing a regular, convex decagon, however, takes real skill, since each interior angle must be $144\xb0$, and all sides must be equal in length.

Drawing a concave decagon is actually fairly easy: make a five-pointed star and fill it in. That is the star that appears 50 times on our country's flag, and the outline of such a pentagram (five-pointed star) is a decagon. That means you see a lot of decagons anytime you are near our flag:

Be careful to know the difference between the five-pointed pentagram (the lines cross themselves) and the decagon (either just the outline of a pentagram or a solidly filled pentagram).

- A
**decagon**is a 10-sided polygon, with 10 interior angles, and 10 vertices which is where the sides meet. - A
**regular decagon**has 10 equal-length sides and equal-measure interior angles. Each angle measures $144\xb0$ and they all add up to $\mathrm{1,440}\xb0$. - An
**irregular decagon**has sides and angles that are not all equal or congruent. - A
**convex decagon**bulges outward, with no interior angle greater than $180\xb0$. - A
**concave decagon**have indentations, creating interior angles greater than $180\xb0$. - A
**simple decagon**does not have any sides that cross or intersect. - A
**complex decagon**has self-intersecting sides, is complex, and highly irregular.

All polygons can be drawn as regular (equal-length sides, equal-measure interior angles) or irregular (not restricted to congruent angles or sides). The regular, convex decagon is a subtle and elegant shape, with 10 exterior angles of $36\xb0$, 10 interior angles of $144\xb0$, and 10 vertices (intersections of sides).

Because it is a challenging shape to make, the regular, convex decagon is popular with coins (like those from Australia, Belize and Hong Kong). Irregular decagons must simply have 10 sides closing in a space, but the lengths of their sides can vary greatly.

Most polygons can be convex or concave. Convex decagons bulge outward, with no interior angle greater than $180\xb0$. Concave decagons have indentations, creating interior angles greater than $180\xb0$. That is why the outline of a five-pointed star is a concave decagon; it has five interior angles each of which is far greater than $180\xb0$.

So far all the decagons discussed -- regular, irregular, concave, and convex -- share the same properties:

- 10 sides
- 10 vertices
- 10 interior angles

They also share one more property: their interior angles always add to $\mathrm{1,440}\xb0$. That is not the case with the next category of decagons: complex.

Decagons can be simple or complex. A simple decagon has no sides crossing themselves. A simple decagon follows all the conventional "rules" of polygons.

By contrast, a complex decagon is self-intersecting. In crossing its own sides, a complex decagon sets off additional interior spaces, but it is still said to have only 10 sides, 10 interior angles, and 10 vertices. Because it is so highly irregular and self-intersecting, a complex decagon need not follow any predictable rule about interior angles or their sums.

Notice that a single decagon can fit into several categories. The classic regular decagon meets all these requirements:

- Regular decagon
- Convex decagon
- Simple decagon

So, to be absolutely accurate (and maybe a bit fussy) you can describe this figure as a regular, convex, simple decagon:

*[insert drawing of regular decagon]*

Below are several decagons. See, without looking at the identifying information, if you can determine whether each is regular or irregular, convex or concave, simple or complex. Remember to name all the attributes, in case a shape fits more than one category.

In order, those decagons are:

- a regular, convex, simple decagon
- a concave, simple decagon
- a complex decagon
- a concave, simple decagon

Did you correctly identify all of them by all their qualities? We sure hope so!

Now that you have studied this lesson, you are able to recognize and define a decagon, state the identifying properties of all decagons, and the identifying properties of regular decagons. You are also able to identify as decagons the stars on the U.S. flag, and identify the differences between regular and irregular decagons; simple and complex decagons; and concave and convex decagons.

Instructor: **Malcolm M.**

Malcolm has a Master's Degree in education and holds four teaching certificates. He has been a public school teacher for 27 years, including 15 years as a mathematics teacher.

Malcolm has a Master's Degree in education and holds four teaching certificates. He has been a public school teacher for 27 years, including 15 years as a mathematics teacher.

In geometry, a decagon is a ten-sided polygon or 10-gon.

**Number of sides:**10**Number of vertices:**10**Interior angle:**$144\xb0$; summing to $\mathrm{1,440}\xb0$**Exterior angle:**$36\xb0$; summing to $360\xb0$**Area:**½ × perimeter × apothem**Perimeter:**10 × side**Properties:**Convex, cyclic, equilateral, isogonal, isotoxal**Type:**Regular polygon

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