Pentagon

The area of any regular polygon is:

${\displaystyle A={\frac {1}{2}}Pr}$

where P is the perimeter of the polygon, and r is the inradius (equivalently the apothem). Substituting the regular pentagon's values for P and r gives the formula

${\displaystyle A={\frac {1}{2}}\cdot 5t\cdot {\frac {t\tan(54^{\circ })}{2}}={\frac {5t^{2}\tan(54^{\circ })}{4}}}$

with side length t.

Like every regular convex polygon, the regular convex pentagon has an inscribed circle. The apothem, which is the radius r of the inscribed circle, of a regular pentagon is related to the side length t by

${\displaystyle r={\frac {t}{2\tan(\pi /5)}}={\frac {t}{2{\sqrt {5-{\sqrt {20}}}}}}\approx 0.6882\cdot t.}$

Like every regular convex polygon, the regular convex pentagon has a circumscribed circle. For a regular pentagon with successive vertices A, B, C, D, E, if P is any point on the circumcircle between points B and C, then PA + PD = PB + PC + PE.

For an arbitrary point in the plane of a regular pentagon with circumradius ${\displaystyle R}$, whose distances to the centroid of the regular pentagon and its five vertices are ${\displaystyle L}$ and ${\displaystyle d_{i}}$ respectively, we have [2]

${\displaystyle \textstyle \sum _{i=1}^{5}d_{i}^{2}=5(R^{2}+L^{2}),}$

${\displaystyle \textstyle \sum _{i=1}^{5}d_{i}^{4}=5((R^{2}+L^{2})^{2}+2R^{2}L^{2}),}$

${\displaystyle \textstyle \sum _{i=1}^{5}d_{i}^{6}=5((R^{2}+L^{2})^{3}+6R^{2}L^{2}(R^{2}+L^{2})),}$

${\displaystyle \textstyle \sum _{i=1}^{5}d_{i}^{8}=5((R^{2}+L^{2})^{4}+12R^{2}L^{2}(R^{2}+L^{2})^{2}+6R^{4}L^{4}).}$

If ${\displaystyle d_{i}}$ are the distances from the vertices of a regular pentagon to any point on its circumscircle, then [2]

${\displaystyle 3(\textstyle \sum _{i=1}^{5}d_{i}^{2})^{2}=10\textstyle \sum _{i=1}^{5}d_{i}^{4}.}$

The regular pentagon is constructible with compass and straightedge, as 5 is a Fermat prime. A variety of methods are known for constructing a regular pentagon. Some are discussed below.

One method to construct a regular pentagon in a given circle is described by Richmond[3] and further discussed in Cromwell's Polyhedra.[4]

The top panel shows the construction used in Richmond's method to create the side of the inscribed pentagon. The circle defining the pentagon has unit radius. Its center is located at point C and a midpoint M is marked halfway along its radius. This point is joined to the periphery vertically above the center at point D. Angle CMD is bisected, and the bisector intersects the vertical axis at point Q. A horizontal line through Q intersects the circle at point P, and chord PD is the required side of the inscribed pentagon.

To determine the length of this side, the two right triangles DCM and QCM are depicted below the circle. Using Pythagoras' theorem and two sides, the hypotenuse of the larger triangle is found as ${\displaystyle \scriptstyle {\sqrt {5}}/2}$. Side h of the smaller triangle then is found using the half-angle formula:

${\displaystyle \tan(\phi /2)={\frac {1-\cos(\phi )}{\sin(\phi )}}\ ,}$

where cosine and sine of ϕ are known from the larger triangle. The result is:

${\displaystyle h={\frac {{\sqrt {5}}-1}{4}}\ .}$

With this side known, attention turns to the lower diagram to find the side s of the regular pentagon. First, side a of the right-hand triangle is found using Pythagoras' theorem again:

${\displaystyle a^{2}=1-h^{2}\$ ;\ a={\frac {1}{2}}{\sqrt {\frac {5+{\sqrt {5}}}{2}}}\ .}

Then s is found using Pythagoras' theorem and the left-hand triangle as:

${\displaystyle s^{2}=(1-h)^{2}+a^{2}=(1-h)^{2}+1-h^{2}=1-2h+h^{2}+1-h^{2}=2-2h=2-2\left({\frac {{\sqrt {5}}-1}{4}}\right)\ }$

${\displaystyle ={\frac {5-{\sqrt {5}}}{2}}\ .}$

The side s is therefore:

${\displaystyle s={\sqrt {\frac {5-{\sqrt {5}}}{2}}}\ ,}$

a well-established result.[5] Consequently, this construction of the pentagon is valid.

Method using Carlyle circles

The Carlyle circle was invented as a geometric method to find the roots of a quadratic equation.[6] This methodology leads to a procedure for constructing a regular pentagon. The steps are as follows:[7]

1. Draw a circle in which to inscribe the pentagon and mark the center point O.
2. Draw a horizontal line through the center of the circle. Mark the left intersection with the circle as point B.
3. Construct a vertical line through the center. Mark one intersection with the circle as point A.
4. Construct the point M as the midpoint of O and B.
5. Draw a circle centered at M through the point A. Mark its intersection with the horizontal line (inside the original circle) as the point W and its intersection outside the circle as the point V.
6. Draw a circle of radius OA and center W. It intersects the original circle at two of the vertices of the pentagon.
7. Draw a circle of radius OA and center V. It intersects the original circle at two of the vertices of the pentagon.
8. The fifth vertex is the rightmost intersection of the horizontal line with the original circle.

Steps 6–8 are equivalent to the following version, shown in the animation:

6a. Construct point F as the midpoint of O and W.

7a. Construct a vertical line through F. It intersects the original circle at two of the vertices of the pentagon. The third vertex is the rightmost intersection of the horizontal line with the original circle.

8a. Construct the other two vertices using the compass and the length of the vertex found in step 7a.

1. We first note that a regular pentagon can be divided into 10 congruent triangles as shown in the Observation. Also, cos 36° = ${\displaystyle {\tfrac {1+{\sqrt {5}}}{4}}}$.†
2. In Step 1, we use four units (shown in blue) and a right angle to construct a segment of length 1+√5, specifically by creating a 1-2-√5 right triangle and then extending the hypotenuse of √5 by a length of 1. We then bisect that segment – and then bisect again – to create a segment of length ${\displaystyle {\tfrac {1+{\sqrt {5}}}{4}}}$ (shown in red.)
3. In Step 2, we construct two concentric circles centered at O with radii of length 1 and length ${\displaystyle {\tfrac {1+{\sqrt {5}}}{4}}}$. We then place P arbitrarily on the smaller circle, as shown. Constructing a line perpendicular to OP passing through P, we construct the first side of the pentagon by using the points created at the intersection of the tangent and the unit circle. Copying that length four times along the outer edge of the unit circles gives us our regular pentagon.

${\displaystyle 0=\cos 90^{\circ }}$

${\displaystyle =\cos(72^{\circ }+18^{\circ })}$

${\displaystyle =\cos 72^{\circ }\cos 18^{\circ }-\sin 72^{\circ }\sin 18^{\circ }}$ (using the angle addition formula for cosine)

${\displaystyle =(2\cos ^{2}36^{\circ }-1){\sqrt {\tfrac {1+\cos 36^{\circ }}{2}}}-2\sin 36^{\circ }\cos 36^{\circ }{\sqrt {\tfrac {1-\cos 36^{\circ }}{2}}}}$ (using double and half angle formulas)

Let u = cos 36°. First, note that 0 < u < 1 (which will help us simplify as we work). Now,

${\displaystyle {\begin{aligned}0&{}=(2u^{2}-1){\sqrt {\tfrac {1+u}{2}}}-2{\sqrt {1-u^{2}}}\cdot u{\sqrt {\tfrac {1-u}{2}}}\\2{\sqrt {1-u^{2}}}\cdot u{\sqrt {\tfrac {1-u}{2}}}&{}=(2u^{2}-1){\sqrt {\tfrac {1+u}{2}}}\\2{\sqrt {1+u}}{\sqrt {1-u}}\cdot u{\sqrt {1-u}}&{}=(2u^{2}-1){\sqrt {1+u}}\\2u(1-u)&{}=2u^{2}-1\\2u-2u^{2}&{}=2u^{2}-1\\0&{}=4u^{2}-2u-1\\u&{}={\frac {2+{\sqrt {(-2)^{2}-4(4)(-1)}}}{2(4)}}\\u&{}={\frac {1+{\sqrt {5}}}{4}}\end{aligned}}}$

This follows quickly from the knowledge that twice the sine of 18 degrees is the reciprocal golden ratio, which we know geometrically from the triangle with angles of 72,72,36 degrees. From trigonometry, we know that the cosine of twice 18 degrees is 1 minus twice the square of the sine of 18 degrees, and this reduces to the desired result with simple quadratic arithmetic.

The regular pentagon according to the golden ratio, dividing a line segment by exterior division

Pentagon at a given side length

1. Draw a segment AB whose length is the given side of the pentagon.
2. Extend the segment BA from point A about three quarters of the segment BA.
3. Draw an arc of a circle, centre point B, with the radius AB.
4. Draw an arc of a circle, centre point A, with the radius AB; there arises the intersection F.
5. Construct a perpendicular to the segment AB through the point F; there arises the intersection G.
6. Draw a line parallel to the segment FG from the point A to the circular arc about point A; there arises the intersection H.
7. Draw an arc of a circle, centre point G with the radius GH to the extension of the segment AB; there arises the intersection J.
8. Draw an arc of a circle, centre point B with the radius BJ to the perpendicular at point G; there arises the intersection D on the perpendicular, and the intersection E with the circular arc that was created about the point A.
9. Draw an arc of a circle, centre point D, with the radius BA until this circular arc cuts the other circular arc about point B; there arises the intersection C.
10. Connect the points BCDEA. This results in the pentagon.

${\displaystyle {\frac {\overline {BJ}}{\overline {AB}}}={\frac {\overline {AB}}{\overline {AJ}}}={\frac {1+{\sqrt {5}}}{2}}=\varphi \approx 1.618}$

Euclid's method for pentagon at a given circle, using of the golden triangle, animation 1 min 39 s

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.[8][9]

A direct method using degrees follows:

1. Draw a circle and choose a point to be the pentagon's (e.g. top center)
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. Draw a guideline through it and the circle's center
4. Draw lines at 54° (from the guideline) intersecting the pentagon's point
5. Where those intersect the circle, draw lines at 18° (from parallels to the guideline)
6. Join where they intersect the circle

After forming a regular convex pentagon, if one joins the non-adjacent corners (drawing the diagonals of the pentagon), one obtains a pentagram, with a smaller regular pentagon in the center. Or if one extends the sides until the non-adjacent sides meet, one obtains a larger pentagram. The accuracy of this method depends on the accuracy of the protractor used to measure the angles.

Overhand knot of a paper strip

• A regular pentagon may be created from just a strip of paper 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.
• Construct a regular hexagon on stiff paper or card. Crease along the three diameters between opposite vertices. Cut from one vertex to the center to make an equilateral triangular flap. Fix this flap underneath its neighbor to make a pentagonal pyramid. The base of the pyramid is a regular pentagon.

Symmetries of a regular pentagon. Vertices are colored by their symmetry positions. Blue mirror lines are drawn through vertices and edges. Gyration orders are given in the center.

The regular pentagon has Dih₅ symmetry, order 10. Since 5 is a prime number there is one subgroup with dihedral symmetry: Dih₁, and 2 cyclic group symmetries: Z₅, and Z₁.

These 4 symmetries can be seen in 4 distinct symmetries on the pentagon. John Conway labels these by a letter and group order.[10] Full symmetry of the regular form is r10 and no symmetry is labeled a1. The dihedral symmetries are divided depending on whether they pass through vertices (d for diagonal) or edges (p for perpendiculars), and i when reflection lines path through both edges and vertices. Cyclic symmetries in the middle column are labeled as g for their central gyration orders.

Each subgroup symmetry allows one or more degrees of freedom for irregular forms. Only the g5 subgroup has no degrees of freedom but can be seen as directed edges.

• 
  Pentagonal cross-section of okra.
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  Morning glories, like many other flowers, have a pentagonal shape.
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  The gynoecium of an apple contains five carpels, arranged in a five-pointed star
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  Starfruit is another fruit with fivefold symmetry.

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  A sea star. Many echinoderms have fivefold radial symmetry.
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  Another example of echinoderm, a sea urchin endoskeleton.
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  An illustration of brittle stars, also echinoderms with a pentagonal shape.

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  A Ho-Mg-Zn icosahedral quasicrystal formed as a pentagonal dodecahedron. The faces are true regular pentagons.
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  A pyritohedral crystal of pyrite. A pyritohedron has 12 identical pentagonal faces that are not constrained to be regular.

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  The Pentagon, headquarters of the United States Department of Defense.
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  Home plate of a baseball field