In geometry, a pentagon is any five-sided polygon. However, the term is commonly used to mean a regular pentagon, where all sides are equal and all angles are equal (to 108°). Its Schläfli symbol is {5}.

The area of a regular pentagon with side length a is given by $A\; =\; \backslash frac\{5a^2\}\{4\}\backslash cot\; \backslash frac\{\backslash pi\}\{5\}\; =\; \backslash frac\; \{a^2\}\{4\}\; \backslash sqrt\{25+10\backslash sqrt\{5\}\}\; \backslash approx\; 1.72048\; a^2$

Constructing a pentagon

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A regular pentagon is constructible using a compass and straightedge, either by inscribing one in a given circle or constructing one on a given edge. This process was described by Euclid in his Elements circa 300 BC.

One method to construct a regular pentagon in a given circle is as follows:

1. Draw a circle in which to inscribe the pentagon and mark the center point O. (This is the green circle in the diagram to the right).
2. Choose a point A on the circle that will serve as one vertex of the pentagon. Draw a line through O and A.
3. Construct a line perpendicular to the line OA passing through O. Mark its intersection with one side of the circle as the point B.
4. Construct the point C as the midpoint of O and B.
5. Draw a circle centered at C through the point A. Mark its intersection with the line OB (inside the original circle) as the point D.
6. Draw a circle centered at A through the point D. Mark its intersections with the original (green) circle as the points E and F.
7. Draw a circle centered at E through the point A. Mark its other intersection with the original circle as the point G.
8. Draw a circle centered at F through the point A. Mark its other intersection with the original circle as the point H.
9. Construct the regular pentagon AEGHF.

If you join the non-adjacent corners (drawing the diagonals of the pentagon), you obtain a pentagram, with a smaller regular pentagon in the center. Or if you extend the sides until the non-adjacent ones meet, you obtain a larger pentagram.

An alternative method of construction is illustrated in the animation: Pentagon construct.gif

Some relevant trigonometric values

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$\backslash sin\; \backslash frac\{\backslash pi\}\{10\}\; =\; \backslash sin\; 18^\backslash circ\; =\; \backslash frac\{\backslash sqrt\; 5\; -\; 1\}\{4\}$

$\backslash cos\; \backslash frac\{\backslash pi\}\{10\}\; =\; \backslash cos\; 18^\backslash circ\; =\; \backslash frac\{\backslash sqrt\{2(5\; +\; \backslash sqrt\; 5)\}\}\{4\}$

$\backslash tan\; \backslash frac\{\backslash pi\}\{10\}\; =\; \backslash tan\; 18^\backslash circ\; =\; \backslash frac\{\backslash sqrt\{5(5\; -\; 2\; \backslash sqrt\; 5)\}\}\{5\}$

$\backslash cot\; \backslash frac\{\backslash pi\}\{10\}\; =\; \backslash cot\; 18^\backslash circ\; =\; \backslash sqrt\{5\; +\; 2\; \backslash sqrt\; 5\}$

$\backslash sin\; \backslash frac\{\backslash pi\}\{5\}\; =\; \backslash sin\; 36^\backslash circ\; =\; \backslash frac\{\backslash sqrt\{2(5\; -\; \backslash sqrt\; 5)\}\; \}\{4\}$

$\backslash cos\; \backslash frac\{\backslash pi\}\{5\}\; =\; \backslash cos\; 36^\backslash circ\; =\; \backslash frac\{\backslash sqrt\; 5+1\}\{4\}$

$\backslash tan\; \backslash frac\{\backslash pi\}\{5\}\; =\; \backslash tan\; 36^\backslash circ\; =\; \backslash sqrt\{5\; -\; 2\backslash sqrt\; 5\}$

$\backslash cot\; \backslash frac\{\backslash pi\}\{5\}\; =\; \backslash cot\; 36^\backslash circ\; =\; \backslash frac\{\backslash sqrt\{5(5\; +\; 2\backslash sqrt\; 5)\}\}\{5\}$

External links

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• Pentagons & Pentagrams new facts about pentagons and pentagrams by Antonio Gutierrez from Geometry Step by Step from the Land of the Incas. Key concept: Menelaus Theorem.

• Quasi crystal inspired pentagon tile by Alexander Braun.

Polygons

Femkant | Fünfeck | Pentágono | Pentagone (figure) | Segi lima | Fimmhyrningur | Pentagono (geometria) | מחומש | Vijfhoek | 五角形 | Pentagon | Pięciokąt | Pentágono | Пятиугольник | Viisikulmio | Pentagon | รูปห้าเหลี่ยม | 正五边形