Pentagon is a polygon with five sides and five vertices. A pentagon may exist either convex or concave, every bit depicted in the adjacent effigy. When convex, the pentagon (or whatever closed polygon in that thing) does have all its interior angles lower than 180°. A concave polygon, to the reverse, does have 1 or more than of its interior angles larger than 180°. A pentagon is regular when all its sides and interior angles are equal. Having but the sides equal is not adequate, because the pentagon can be concave with equal sides. In that case the pentagon is called equilateral. The adjacent figure illustrates the classification of pentagons, also presenting equilateral ones that are concave. Any pentagon that is not regular is called irregular.
Types of pentagons
The sum of the internal angles of a pentagon is abiding and equal to 540°. This is truthful for either regular or irregular pentagons, convex or concave. It can be hands proved by decomposing the pentagon to individual, non overlapping triangles. If we try to draw direct lines betwixt all vertices, avoiding any intersections, we divide the pentagon into iii individual triangles. There are many dissimilar ways to describe 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 next, the internal angles should measure up to 3x180°=540°.
A pentagon tin be divided into three triangles
Backdrop of regular pentagons
Symmetry
A regular pentagon has 5 axes of symmetry. Each i of them passes through a vertex of the pentagon and the middle of the opposite edge, as shown in the post-obit drawing. All axes of symmetry intersect at a mutual point, the center of the regular pentagon. This is in fact its center of gravity or centroid.
Axes of symmetry of regular pentagon
Interior angle and primal angle
By definition the interior angles of a regular pentagon are equal. It is too a common property of all pentagons that the sum of their interior angles is always 540°, every bit explained previously. Therefore, the interior angle, , of a regular pentagon should be 108°:
Five identical, isosceles, triangles are defined if we describe straight lines from the center of the regular pentagon towards each i of its vertices. The central angle, , of each triangle is:
Focusing on 1 of the five triangles, its ii 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 fundamental angles is 180°:
In other words and are supplementary.
Interior and key angle of a regular pentagon
The regular pentagon is divided into 5 identical isosceles triangles having a mutual 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 chosen cirmuscribed circle or circumcircle of the regular pentagon (indeed this is a common characteristic of all regular polygons). The center of this circle is as well the center of the pentagon, where all the symmetry axes are intersecting also. The radius of circumcircle, , is usually called circumradius.
Some other circle can likewise be drawn, that touches tangentially all five edges of the regular pentagon at the midpoints (as well a common characteristic of all regular polygons). This is the then called inscribed circle or incircle. Its heart is the same with the center of the circumcircle and it is tangent to all 5 sides of the regular pentagon. The radius of incircle, , is ordinarily called inradius.
The following figure depicts both confining circle of the regular pentagon and the inscribed one.
Circumcircle and incircle of a regular pentagon
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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 key angle and the side length. It turns out that these expressions are valid for whatsoever regular polygon, not simply the pentagon. We can obtain specific expression for the regular pentagon past setting θ = 72°. These expressions are:
Surface area and perimeter
In gild to find the area of a regular pentagon we have take into account that its full area is divided into v identical isosceles triangles. All. these triangle have one side and 2 sides , while their height, bandage 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 expanse of each triangle is then: . Therefore, the total area of the 5 triangles is establish:
An approximation of the last relationship is:
The perimeter of any N-sided regular polygon is but 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 exist fatigued 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 contrary edge. Information technology is indeed perpendicular to the opposite edge and passes through the center of the pentagon. By definition though the altitude from the heart to a vertex is the circumradius of the pentagon while the distance from the middle to an edge is the inradius . Therefore the following expression is derived:
It is possible to limited the summit 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 go the following approximations:
Width
The width is the distance betwixt two opposite vertices of the regular pentagon (the length of its diagonal). In society to detect this distance nosotros will employ the right triangle highlighted with dashed line in the figure above. The hypotenuse of the triangle is really the side length of the pentagon, which is . Also, i of the triangle angles is supplementary to the next interior angle of the pentagon. Information technology has been explained earlier, though, that the supplementary of is indeed the central bending . Therefore, we may notice the length of the triangle side:
Finally, we can make up one's mind the full width by calculation twice the length to the side length (due to symmetry the triangle to the right of the pentagon is identical to the 1 examined).
Substituting, we become an approximation of the last formula:
The diagonal of a regular pentagon is related through the golden ratio with its side
How to depict a regular pentagon
You can describe a regular pentagon given the side length , using elementary cartoon 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 middle point at the right finish of the linear segment and radius equal to the segment length.
Echo the terminal step, changing the heart-betoken at the left cease of the linear segment. Radius is the same.
Depict a line, perpendicular to the linear segment , passing through the intersection of the 2 arcs. It crosses the linear segment at its center.
Besides draw a line, perpendicular to the linear segment, passing through left terminate of the linear segment . Mark the intersection point with the round arc (the one drawn at footstep iv)
Depict another round arc, past placing i needle of the compass at the heart of the linear segment , (that was found in step 5) and the cartoon tip at the intersection marked in pace 6. Rotate the compass, until it crosses the extension of the linear segment, drawn in step 2. Mark this new intersection also.
Describe another circular arc, by placing 1 needle of the compass at the right stop of the linear segment and the cartoon tip at the intersection marked in stride vii. Rotate the compass clockwise. Mark two intersections, one with the arc, fatigued in pace iv, and the other with the line, fatigued in step 5. These are two vertices of the pentagon.
Placing the compass needle at the 2nd intersection, and the cartoon tip at the 1st one (both intersections marked in the last stride) depict a circular arc until it crosses the arc, drawn in step 3. Marking this new intersection, which is a vertex of the pentagon.
The 2 ends of the linear segment , as well every bit the iii intersections marked at steps 8 and 9 are the five vertices of the regular pentagon. Describe linear segments betwixt them to construct the last shape.
The following figure illustrates the drawing procedure pace by step.
Drawing a regular pentagon given its side length .
Note, that the described procedure is not strictly a by "ruler and compass" structure. In steps v 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 utilize of triangle in steps 5 and vi, if a strict geometric drawing by "ruler and compass" is required.
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Examples
Example 1
Determine the circumradius, the inradius and the area of a regular pentagon, with side length
We will utilize the exact analytical expressions for the circumradius and the inradius, in terms of the side length , that accept been described in the previous sections. These are:
Since the side length is given, all we accept 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 , past this formula:
Substituting we observe:
Example 2
What is the diameter of the biggest regular pentagon that tin be fitted inside a:
circle, with diameter 25''
square with side 25''
1. Fitting a regular pentagon in a circumvolve
The biggest regular pentagon to fit inside a circle should bear on the circle with all its vertices. In other words, the circle must exist the circumcircle of the pentagon and as a upshot its radius should be the circumradius:
All the same, the circumradius is related to the side length through the formula:
Therefore:
From the last equation nosotros tin calculate the required side length , if nosotros 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 credible, that the width is really, the biggest of the two dimensions. Therefore, the biggest regular pentagon, to fit inside a foursquare, should be limited by its width alone. In other words, the width of the pentagon must be equal to side of the foursquare:
However, the width of the regular pentagon is related to the side length with the formula:
Therefore:
From the terminal equation we tin can calculate the required side length , if we substitute the values of and :
Regular pentagon cheat-sheet
In the following table a curtailed listing of the principal formulas, related to the regular pentagon is included. Also some approximations that may prove handy for applied problems are listed too.
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