## Theoretical background

#### Table of contents

- Definitions

- Properties of regular pentagons

- Symmetry

- Interior angle and central angle

- Circumcircle and incircle

- Area and perimeter

- Bounding box

- How to draw a regular pentagon

- Examples

- Regular pentagon cheat-sheet

- See also

### Definitions

Pentagon is a polygon with five sides and five vertices. A pentagon may be either **convex **or **concave**, as depicted in the next figure. When convex, the pentagon (or any closed polygon in that matter) does have all its interior angles lower than 180°. A concave polygon, to the contrary, does have one or more of its interior angles larger than 180°. A pentagon is **regular **when all its sides and interior angles are equal. Having only the sides equal is not adequate, because the pentagon can be concave with equal sides. In that case the pentagon is called **equilateral**. The next figure illustrates the classification of pentagons, also presenting equilateral ones that are concave. Any pentagon that is not regular is called **irregular**.

The sum of the internal angles of a pentagon is constant and equal to 540°. This is true for either regular or irregular pentagons, convex or concave. It can be easily proved by decomposing the pentagon to individual, non overlapping triangles. If we try to draw straight lines between all vertices, avoiding any intersections, we divide the pentagon into three individual triangles. There are many different ways to draw lines between the vertices, resulting in different triangles, however their count is always three. In a single triangle the sum of internal angles is 180°, therefore, for 3 triangles, positioned side by side, the internal angles should measure up to 3x180°=540°.

### Properties of regular pentagons

#### Symmetry

A regular pentagon has five axes of symmetry. Each one of them passes through a vertex of the pentagon and the middle of the opposite edge, as shown in the following drawing. All axes of symmetry intersect at a common point, the center of the regular pentagon. This is in fact its center of gravity or centroid.

#### Interior angle and central angle

By definition the interior angles of a regular pentagon are equal. It is also a common property of all pentagons that the sum of their interior angles is always 540°, as explained previously. Therefore, the interior angle, , of a regular pentagon should be 108°:

Five identical, isosceles, triangles are defined if we draw straight lines from the center of the regular pentagon towards each one of its vertices. The central angle, , of each triangle is:

Focusing on one of the five triangles, its two remaining angles are identical and equal to 54°, so that the sum of all angles in the triangle is 180°, (72°+54°+54°). That is also the half of the interior angle , (108°/2=54°). It is not coincidence that the sum of interior and central angles is 180°:

In other words and are supplementary.

The regular pentagon is divided into five identical isosceles triangles having a common vertex, the polygon center.

#### Circumcircle and incircle

It is possible to draw a circle that passes through all the five vertices of the regular pentagon . This is the so called cirmuscribed circle or **circumcircle **of the regular pentagon (indeed this is a common characteristic of all regular polygons).The center of this circle is also the center of the pentagon, where all the symmetry axes are intersecting also. The radius of circumcircle, , is usually called **circumradius**.

Another circle can also be drawn, that touches tangentially all five edges of the regular pentagon at the midpoints (also a common characteristic of all regular polygons). This is the so called inscribed circle or **incircle**. Its center is the same with the center of the circumcircle and it is tangent to all five sides of the regular pentagon. The radius of incircle, , is usually called **inradius**.

The following figure depicts both circumscribed circle of the regular pentagon and the inscribed one.

We will try to find the relationships between the side length of the regular pentagon and its circumradius and inradius .To this end, we will examine the triangle with sides the circumradius, the inradius and half the pentagon edge, as highlighted in the figure below. This is a right triangle since by definition the incircle is tangential to all sides of the polygon.

Using basic trigonometry we find:

where the central angle and the side length. It turns out that these expressions are valid for any regular polygon, not just the pentagon. We can obtain specific expression for the regular pentagon by setting θ = 72°. These expressions are:

#### Area and perimeter

In order to find the area of a regular pentagon we have take into account that its total area is divided into five identical isosceles triangles. All. these triangle have one side and two sides , while their height, cast from the vertex lying at the pentagon center, is equal to (remember that the incircle is tangential to all sides of the pentagon touching them at their midpoints). The area of each triangle is then: . Therefore, the total area of the five triangles is found:

An approximation of the last relationship is:

The perimeter of any N-sided regular polygon is simply the sum of the lengths of all sides: . Therefore, for the regular pentagon :

### Bounding box

The bounding box of a planar shape is the smallest rectangle that encloses the shape completely. For the regular pentagon the bounding box may be drawn intuitively, as shown in the next figure, but its exact dimensions need some calculations.

##### Height

The height of the regular pentagon is the distance from one of its vertices to the opposite edge. It is indeed perpendicular to the opposite edge and passes through the center of the pentagon. By definition though the distance from the center to a vertex is the circumradius of the pentagon while the distance from the center to an edge is the inradius . Therefore the following expression is derived:

It is possible to express the height in terms of the circumradius , or the inradius or the side length , using the respective analytical expressions for these quantities. The following formulas are derived:

where .

Substituting the value of to the last expressions we get the following approximations:

##### Width

The width is the distance between two opposite vertices of the regular pentagon (the length of its diagonal). In order to find this distance we will employ the right triangle highlighted with dashed line in the figure above. The hypotenuse of the triangle is actually the side length of the pentagon, which is . Also, one of the triangle angles is supplementary to the adjacent interior angle of the pentagon. It has been explained before, though, that the supplementary of is indeed the central angle . Therefore, we may find the length of the triangle side:

Finally, we can determine the total width by adding twice the length to the side length (due to symmetry the triangle to the right of the pentagon is identical to the one examined).

Substituting, we get an approximation of the last formula:

The diagonal of a regular pentagon is related through the golden ratio with its side

### How to draw a regular pentagon

You can draw a regular pentagon given the side length , using simple drawing tools. Follow the steps described below:

- First draw a linear segment with length , equal to the desired pentagon side length.
- Extend the linear segment to the left.
- Construct a circular arc, with center point at the right end of the linear segment and radius equal to the segment length.
- Repeat the last step, changing the center-point at the left end of the linear segment. Radius is the same.
- Draw a line, perpendicular to the linear segment , passing through the intersection of the two arcs. It crosses the linear segment at its middle.
- Also draw a line, perpendicular to the linear segment, passing through left end of the linear segment . Mark the intersection point with the circular arc (the one drawn at step 4)
- Draw another circular arc, by placing one needle of the compass at the middle of the linear segment , (that was found in step 5) and the drawing tip at the intersection marked in step 6. Rotate the compass, until it crosses the extension of the linear segment, drawn in step 2. Mark this new intersection too.
- Draw another circular arc, by placing one needle of the compass at the right end of the linear segment and the drawing tip at the intersection marked in step 7. Rotate the compass clockwise. Mark two intersections, one with the arc, drawn in step 4, and the other with the line, drawn in step 5. These are two vertices of the pentagon.
- Placing the compass needle at the 2
^{nd}intersection, and the drawing tip at the 1^{st}one (both intersections marked in the last step) draw a circular arc until it crosses the arc, drawn in step 3. Mark this new intersection, which is a vertex of the pentagon. - The two ends of the linear segment , as well as the three intersections marked at steps 8 and 9 are the five vertices of the regular pentagon. Draw linear segments between them to construct the final shape.

The following figure illustrates the drawing procedure step by step.

Note, that the described procedure is not strictly a by “ruler and compass” construction. In steps 5 and 6, a triangle was used in order to draw perpendicular lines from points of another line. This was selected for simplicity and in order to shorten the number of required steps. Drawing a perpendicular line, is a straightforward geometric construction, using ruler and compass alone, and one could replace the use of triangle in steps 5 and 6, if a strict geometric drawing by “ruler and compass” is required.

### Examples

##### Example 1

Determine the circumradius, the inradius and the area of a regular pentagon, with side length

We will use the exact analytical expressions for the circumradius and the inradius, in terms of the side length , that have been described in the previous sections. These are:

Since the side length is given, all we have to do is substitute its value 5'' to these expressions. The circumradius is:

,

and the inradius:

.

The area of a regular pentagon is also given in terms of the side length , by this formula:

Substituting we find:

##### Example 2

What is the diameter of the biggest regular pentagon that can be fitted inside a:

- circle, with diameter 25''
- square with side 25''

##### 1. Fitting a regular pentagon in a circle

The biggest regular pentagon to fit inside a circle should touch the circle with all its vertices. In other words, the circle must be the circumcircle of the pentagon and as a result its radius should be the circumradius:

However, the circumradius is related to the side length through the formula:

Therefore:

From the last equation we can calculate the required side length , if we substitute the values of and :

##### 2. Fitting a regular pentagon is a square

The height and the width of the regular polygon are approximated by the following expressions:

From these approximations, it is apparent, that the width is actually, the biggest of the two dimensions. Therefore, the biggest regular pentagon, to fit inside a square, should be limited by its width alone. In other words, the width of the pentagon must be equal to side of the square:

However, the width of the regular pentagon is related to the side length with the formula:

Therefore:

From the last equation we can calculate the required side length , if we substitute the values of and :

### Regular pentagon cheat-sheet

In the following table a concise list of the main formulas, related to the regular pentagon is included. Also some approximations that may prove handy for practical problems are listed too.

## Regular pentagon quick reference | |
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Circumradius: | |

Inradius: | |

Height: | |

Width: | |

Area: | |

Interior angle: | |

Central angle: | |

Approximations: |