In
geometry, a
pentagon is
any five-sided
polygon. A pentagon may be
simple or self-intersecting. The
internal
angles in a
simple pentagon total
540°. A
pentagram is an example of a
self-intersecting pentagon.
Regular pentagons
The term
pentagonal is commonly used to mean a
regular convex pentagon, where all sides are equal
and all interior angles are equal (to 108°). Its
Schläfli symbol is {5}. The
chords of this pentagon are in
golden ratio to its sides.
The area of a regular convex pentagon with side length
t
is given by
- A = \frac{{t^2 \sqrt {25 + 10\sqrt 5 } }}{4} = \frac{5t^2
\tan(54^\circ)}{4} \approx 1.720477401 t^2.
A
pentagram or pentangle
is a
regular star pentagon. Its
Schläfli symbol is {5/2}. Its sides
form the diagonals of a regular convex pentagon - in this
arrangement the
sides of the two
pentagons are in the
golden
ratio.
When a regular pentagon is
inscribed in a circle with radius
R, its edge length
t is given by the
expression
- t = R\ {\sqrt { \frac {5-\sqrt{5}}{2}} } = 2R\sin 36^\circ =
2R\sin\frac{\pi}{5} \approx 1.17557050458 R.
Derivation of the Area formula
The area of any regular polygon is:
- A = \frac{1}{2}Pa
where P is the perimeter of the polygon, and a is the
apothem. We can then substitute the respective
values for P and a, which makes the formula:
- A = \frac{1}{2} \times \frac{5t}{1} \times
\frac{t\tan(54^\circ)}{2}
with t as the given side length. Then we can then rearrange the
formula as:
- A = \frac{1}{2} \times \frac{5t^2\tan(54^\circ)}{2}
and then, we combine the two terms to get the final formula, which
is:
- A = \frac{5t^2\tan(54^\circ)}{4}.
Derivation of the Diagonal Length formula
The diagonals of a regular pentagon (hereby represented by D) can
be calculated using the following formula:
- D = T \times \frac {1+ \sqrt {5} }{2}
where T = the side length of the pentagon, itself.
Construction
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:
Construction of a regular
pentagon
An alternative method is this:
Constructing a pentagon
- 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).
- Choose a point A on the circle that will serve as one
vertex of the pentagon. Draw a line through O and
A.
- Construct a line perpendicular to the line OA passing
through O. Mark its intersection with one side of the
circle as the point B.
- Construct the point C as the midpoint of O
and B.
- 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.
- Draw a circle centered at A through the point
D. Mark its intersections with the original (green) circle
as the points E and F.
- Draw a circle centered at E through the point
A. Mark its other intersection with the original circle as
the point G.
- Draw a circle centered at F through the point
A. Mark its other intersection with the original circle as
the point H.
- Construct the regular pentagon AEGHF.
A direct method using degrees follows:
- Draw a circle and choose a point to be the pentagon's (e.g. top
center)
- Draw a guideline through it and the circle's center
- Draw lines @ 54 degrees (from the guideline) intersecting the
pentagon's point
- Where those intersect the circle, draw lines @ 18 degrees (from
parallels to the guideline)
- Join where they intersect the circle
After forming a regular convex pentagon, 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.
A simple method of creating a regular pentagon from just a strip of
paper is by tying an
overhand knot
into the strip and carefully flattening the knot by pulling the
ends of the paper strip. Folding one of the ends back over the
pentagon will reveal a
pentagram when
backlit.
Pentagons in nature
Plants
Image:BhindiCutUp.jpg|Pentagonal cross-section of
okra.Image:Morning_Glory_Flower.jpg|
Morning glories, like many other flowers,
have a pentagonal shape.Image:Sterappel dwarsdrsn.jpg|The
gynoecium of an
apple
contains five carpels, arranged in a
five-pointed starImage:Carambola
cut.jpg|
Starfruit is another fruit with
fivefold symmetry.
Animals
Image:Cervena_morska_hviezdica.jpg|A
sea
star. Many
echinoderms have fivefold
radial symmetry.Image:Haeckel_Ophiodea.jpg|An illustration of
brittle stars, also echinoderms with a
pentagonal shape.
See also
External links