What is the measure of each interior angle of a regular convex decagon

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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.

What you'll learn:

1. Decagons Are Polygons
2. Properties of Decagons
3. Decagon Angles
4. Decagon Definitions
5. Regular and Irregular Decagons
6. Concave and Convex Decagons
7. Simple and Complex Decagons
8. Decagon Shapes

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.

How Many Sides Does A Decagon Have?

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

Properties of Decagons

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

1. 10 sides
2. 10 interior angles
3. 10 vertices

Decagon Angles

In many decagons, the sum of interior angles will be 1,440°, but that is not an identifying property because complex decagons will not have that sum.

Regular decagons have two additional identifying properties:

1. 10 exterior angles of 36°, summing to 360°
2. 10 interior angles of 144°, summing to 1,440°

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°, 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).

Decagon Definitions

• 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° and they all add up to 1,440°.
• 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°.
• A concave decagon have indentations, creating interior angles greater than 180°.
• A simple decagon does not have any sides that cross or intersect.
• A complex decagon has self-intersecting sides, is complex, and highly irregular.

Regular and Irregular Decagons

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°, 10 interior angles of 144°, 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.

Concave and Convex Decagons

Most polygons can be convex or concave. Convex decagons bulge outward, with no interior angle greater than 180°. Concave decagons have indentations, creating interior angles greater than 180°. 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°.

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 1,440°. That is not the case with the next category of decagons: complex.

Simple and Complex Decagons

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]

Decagon Shapes

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:

1. a regular, convex, simple decagon
2. a concave, simple decagon
3. a complex decagon
4. a concave, simple decagon

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

Lesson Summary

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.

Next Lesson:

Ratios and Proportions

home / geometry / shape / decagon

A decagon is a polygon with 10 sides and 10 vertices. "Deca-" means "ten."

Decagon classifications

Like other polygons, a decagon can be classified as regular or irregular.

Regular decagon	Irregular dodecagon
All sides and interior angles are congruent	Sides and angles have various lengths

A decagon can also be classified as convex or concave.

A convex decagon is a polygon where no line segment between any two points on its boundary lies outside of it. None of the interior angles is greater than 180°. Think of a convex decagon as bulging outwards, like the regular decagon above.

Conversely, a concave decagon, like the irregular decagon shown above, has at least one-line segment that can be drawn between points on its boundary, but lies outside of it. Also, at least one of its interior angles is greater than 180°

A convex decagon does not have to be a regular decagon. Yet, a regular decagon is always a convex decagon.

The decagon above is convex but clearly has sides and angles that are not congruent. This decagon is irregular. Also, no part of any segment drawn between two boundary points, such as AB, lies outside of the decagon.

Angles of a decagon

Decagons can be broken into a series of triangles by diagonals drawn from its vertices. This series of triangles can be used to find the sum of the interior angles of the decagon.

Diagonals are drawn from vertex A in the convex decagon below, forming 8 triangles. Similarly, 8 triangles can also be drawn in a concave decagon. Since the sum of the angles of a triangle is 180°, the sum of the interior angles of a dodecagon is 8 × 180° = 1440°.

A regular decagon has equal interior angle measures. Since 1440°/10 = 144°, each interior angle in a regular decagon has a measure of 144°. Also, each exterior angle has a measure of 36°.

A regular decagon can be inscribed in a circle. Each vertex on the decagon lies on the circle. Also, the circle and decagon share the same center.

Page 2

home / geometry / shape / 3d

A three-dimensional space (3D) has three dimensions, such as length, width, and height (or depth). The term "3D" is commonly used to describe shapes and figures in geometry. We live in a 3D world, every object we touch, see, and use are 3D objects. The following is a few examples.

3D shapes and figures

While the dimensions of a 2D shape can be described with length and width, a 3D shape requires an additional dimension, often referred to as height or depth.

2D	3D
The dimensions of the 2D rectangle above are length and width.	The base of the 3D rectangular prism above is a rectangle that gives it two of its dimensions. The height of the prism is perpendicular to its base giving the prism its third dimension.

3D Coordinate geometry

Determining the position of a point in 3D is similar to determining the position of a point in 2D, except that there is a third axis, the z-axis, in addition to the x- and y-axes. All three axes are perpendicular to each other, as shown in the figure below.

While the x- and y-axes in 2D are conventionally the horizontal and vertical axes, respectively, in 3D their orientations can vary. As such, when determining which direction to move along any axis, the negative and positive direction must be indicated on the axes, as in the image above. A negative value indicates movement along an axis in the negative direction, and a positive value indicates movement in the positive direction. As such, the point (2, 3, 4) indicates movement in the positive direction along each axis in the coordinate space above.

See also geometric figures, 1D, 2D.