Octagon

An octagon (or octagon Greek ὀκτάγωνον oktágōnon ) is a polygon with eight vertices and eight sides. You can, as all polygons that are not triangles, divided into convex, concave and turned-over octagons. To the convex octagons includes the regular octagon in which all sides are of equal length and all angles equal. This is discussed in more detail.

Construct may be a regular octagon, as to all pages to construct mid-perpendicular to a square, and connecting the intersection points of the perpendicular bisectors to the radius of the corners.

Formulas

Area calculation

Decompose the regular octagon into 8 isosceles triangles. The unique angles of a triangle is 360 ° / 8 = 45 °. The same two angles of the triangle be 67.5 °. The height halved the isosceles triangle. It is created by drawing the height a right triangle with angles of 67.5 °, 22.5 ° and 90 °. The following solutions assume that this right triangle, whereby:

  • A is the side of the octagon
  • A ' is half the side length of the octagon
  • Ri is the radius of the inscribed circle
  • Ru is the radius of the circumscribed circle
  • A is the area of the octagon
  • A ' is the area of the right triangle

Given the radius of the inner circle ri: The desired leg ( opposite leg to the acute angle ) can be determined by the tangent of 22.5 °:

The area of the right triangle is obtained by

The isosceles triangle has twice the area of the right triangle, the octagon eight times the area of the isosceles triangle:

Formula 1:

Let the side length a of the octagon: Analogous to the above consideration, can the radius r of the inscribed circle with the tangent of 22.5 ° determined, a ' is the half of a:

Formula 2:

The area of the right triangle is obtained by

Substituting A 'in the formula for the area ( see formula 1) a, one obtains

Given the radius of the circumscribed circle ru: The ratio of a 'to R corresponds to the sine of the acute angle:

The radius of the inscribed circle is (see formula 2)

The area of the right triangle is obtained by

Substituting A 'in the formula for the area ( see formula 1) a, one obtains

Or with the addition theorems for trigonometric functions

General formulas for regular n-gons From the above approaches, the following formulas for n-gons can be derived:

For a given radius of the inscribed circle is true:

For a given side length a of the n -gon is true:

Convex set Octagon (disambiguation)
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