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Concept
Decagon
Summary
In geometry, a decagon (from the Greek δέκα déka and γωνία gonía, "ten angles") is a ten-sided polygon or 10-gon. The total sum of the interior angles of a simple decagon is 1440°.
A regular decagon has all sides of equal length and each internal angle will always be equal to 144°. Its Schläfli symbol is {10} and can also be constructed as a truncated pentagon, t{5}, a quasiregular decagon alternating two types of edges.
The picture shows a regular decagon with side length and radius of the circumscribed circle.
The triangle has two equally long legs with length and a base with length
The circle around with radius intersects in a point (not designated in the picture).
Now the triangle is a isosceles triangle with vertex and with base angles .
Therefore . So and hence is also a isosceles triangle with vertex . The length of its legs is , so the length of is .
The isosceles triangles and have equal angles of 36° at the vertex, and so they're similar, hence:
Multiplication with the denominators leads to the quadratic equation:
This equation for the side length has one positive solution:
So the regular decagon can be constructed with ruler and compass.
Further conclusions
and the base height of (i.e. the length of ) is and the triangle has the area: .
The area of a regular decagon of side length a is given by:
In terms of the apothem r (see also inscribed figure), the area is:
In terms of the circumradius R, the area is:
An alternative formula is where d is the distance between parallel sides, or the height when the decagon stands on one side as base, or the diameter of the decagon's inscribed circle.
By simple trigonometry,
and it can be written algebraically as
A regular decagon has 10 sides and is equilateral. It has 35 diagonals
As 10 = 2 × 5, a power of two times a Fermat prime, it follows that a regular decagon is constructible using compass and straightedge, or by an edge-bisection of a regular pentagon.
An alternative (but similar) method is as follows:
Construct a pentagon in a circle by one of the methods shown in constructing a pentagon.
Official source
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