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Apothem of a hexagon Graphs of side, s; apothem, a; and area, A of regular polygons of n sides and circumradius 1, with the base, b of a rectangle with the same area. The green line shows the case n = 6. The apothem (sometimes abbreviated as apo [1]) of a regular polygon is a line
The area of a regular polygon is half its perimeter times the apothem. As the number of sides of the regular polygon increases, the polygon tends to a circle, and the apothem tends to the radius. This suggests that the area of a disk is third the circumference of its bounding circle times the radius. [3]
The apothem is half the cotangent of /, and the area of each of the 14 small triangles is one-fourth of the apothem. The area of a regular heptagon inscribed in a circle of radius R is 7 R 2 2 sin 2 π 7 , {\displaystyle {\tfrac {7R^{2}}{2}}\sin {\tfrac {2\pi }{7}},} while the area of the circle itself is π R 2 ; {\displaystyle \pi R^{2 ...
In geometry, a polygon (/ ˈ p ɒ l ɪ ɡ ɒ n /) is a plane figure made up of line segments connected to form a closed polygonal chain. The segments of a closed polygonal chain are called its edges or sides. The points where two edges meet are the polygon's vertices or corners. An n-gon is a polygon with n sides; for example, a triangle is a 3 ...
All vertices of a regular polygon lie on a common circle (the circumscribed circle); i.e., they are concyclic points. That is, a regular polygon is a cyclic polygon . Together with the property of equal-length sides, this implies that every regular polygon also has an inscribed circle or incircle that is tangent to every side at the midpoint.
Perimeter#Formulas – Path that surrounds an area; List of second moments of area; List of surface-area-to-volume ratios – Surface area per unit volume; List of surface area formulas – Measure of a two-dimensional surface; List of trigonometric identities; List of volume formulas – Quantity of three-dimensional space
The arc length, from the familiar geometry of a circle, is s = θ R {\displaystyle s={\theta }R} The area a of the circular segment is equal to the area of the circular sector minus the area of the triangular portion (using the double angle formula to get an equation in terms of θ {\displaystyle \theta } ):
The common length of the sides equals the radius of the circumscribed circle or circumcircle, which equals times the apothem (radius of the inscribed circle). All internal angles are 120 degrees . A regular hexagon has six rotational symmetries ( rotational symmetry of order six ) and six reflection symmetries ( six lines of symmetry ), making ...