Mathematics student from nations as far-flung as Australia, Belize and Hong Kong may have actually everyday call with decagons, since those nations have provided the 10-sided form in their coins. Most of us, though, only encounter decagons in math class and also when feeling an especially patriotic.

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Decagons are Polygons

Decagons belong to the household of two-dimensional shapes called polygons. Polygons acquire their surname from Greek, an interpretation "many-angled," due to the fact that all polygons have multiple interior angles. Castle close in a two-dimensional space using only straight sides.


How countless Sides does A Decagon Have?

A decagon is a 10-sided polygon, with 10 inner angles, and also 10 vertices i beg your pardon is whereby the sides meet.

Properties the Decagons

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

10 sides10 inner angles10 vertices

Decagon Angles

In numerous decagons, the sum of internal angles will be 1,440°, yet that is not an identifying home because complicated decagons will certainly not have that sum.

Regular decagons have actually two additional identifying properties:

10 exterior angles of 36°, summing to 360°10 inner angles of 144°, summing come 1,440°


Drawing a decagon bring away no skill; simply draw 10 line segments the connect, and you have it. Drawing a regular, convex decagon, however, takes real skill, due to the fact that each inner angle should be 144°, and all sides need to be same in length.

Drawing a concave decagon is actually reasonably easy: do a five-pointed star and also fill the in. That is the star that appears 50 time on our country"s flag, and the synopsis of such a pentagram (five-pointed star) is a decagon. That means you watch a most decagons anytime friend are close to our flag:


Be mindful to recognize the difference between the five-pointed pentagram (the lines overcome themselves) and the decagon (either simply the synopsis of a pentagram or a solidly fill pentagram).

Decagon Definitions


A decagon is a 10-sided polygon, through 10 interior angles, and also 10 vertices i beg your pardon is where the political parties meet.A regular decagon has actually 10 equal-length sides and equal-measure interior angles. Every angle procedures 144° and they all add up to 1,440°.An irregular decagon has sides and angles that space not all same or congruent.A convex decagon bulges outward, through no interior angle better than 180°.A concave decagon have indentations, creating interior angles higher than 180°.A simple decagon does no have any kind of sides the cross or intersect.A complex decagon has self-intersecting sides, is complex, and also highly irregular.

Regular and Irregular Decagons

All polygons deserve to be attracted as consistent (equal-length sides, equal-measure interior angles) or rarely often rare (not restricted to congruent angle or sides). The regular, convex decagon is a subtle and also elegant shape, through 10 exterior angles of 36°, 10 interior angles that 144°, and 10 vertices (intersections that sides).


Because the is a an overwhelming shape to make, the regular, convex decagon is well-known with coins (like those indigenous Australia, Belize and also Hong Kong). Rarely often rare decagons must simply have 10 political parties closing in a space, however the lengths of their sides have the right to vary greatly.

Concave and Convex Decagons

Most polygons have the right to be convex or concave. Convex decagons bulge outward, v no internal angle higher than 180°. Concave decagons have indentations, developing interior angles higher than 180°. That is why the synopsis of a five-pointed star is a concave decagon; the has 5 interior angle each of which is much greater than 180°.


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

10 sides10 vertices10 inner angles

Simple and complex Decagons

Decagons can be basic or complex. A basic decagon has actually no sides crossing themselves. A simple decagon follows all the standard "rules" that polygons.


By contrast, a facility decagon is self-intersecting. In cross its very own sides, a complex decagon set off extr interior spaces, yet it is still said to have only 10 sides, 10 inner angles, and also 10 vertices. Since it is so extremely irregular and also self-intersecting, a complicated decagon need not follow any predictable rule around interior angles or their sums.

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

Regular decagonConvex decagonSimple decagon

So, to be absolutely precise (and probably a bit fussy) you can define this number as a regular, convex, simple decagon:

Decagon Shapes

Below are number of decagons. See, without looking in ~ the identify information, if you have the right to determine whether each is continual or irregular, convex or concave, straightforward or complex. Psychic to name all the attributes, in case a shape fits more than one category.

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In order, those decagons are:

a regular, convex, simple decagona concave, simple decagona complicated decagona concave, simple decagon

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

Lesson Summary

Now that you have actually studied this lesson, you are able come recognize and also define a decagon, state the identifying nature of every decagons, and also the identifying nature of regular decagons. You are also able to determine as decagons the stars ~ above the U.S. Flag, and identify the differences between regular and irregular decagons; straightforward and facility decagons; and concave and convex decagons.