Polygon is defined as the shape that is made up of straight lines drawn on flat surfaces like a piece of paper. Such shapes include squares, rectangles, triangles and pentagons. Understanding shape is an important part of mathematics and it requires basic understanding of the properties of shapes along with their practical applications and real life situations too.
In geometry, the word pentagon is derived from a Greek word meaning five and consists of five sides or 5 gon. In pentagon, the sum of the internal angles is equal to 540 degree. In geometry, a pentagon is a simple shape in nature and is often self-intersecting in nature. A self-intersecting regular pentagon is known as pentagram. For a regular pentagon, it has only five lines of reflectional symmetry and rotational symmetry of order 5. Pentagons can be regular or irregular and convex or concave. A regular pentagon is one with those all the sides are equal and the angles are also equal. Its interior angles are 108 degrees and its exterior angles are 72 degrees. An irregular pentagon is a shape that does not have equal number of sides or angles and therefore does not have specified angles. A convex pentagon is the one whose vertices or points where the sides meet points outwards as opposed to a concave pentagon whose vertices points inwards. The diagonals of a convex regular shaped pentagon are in golden ratio to its sides. Its height is defined as the distance from one side to the opposite side of the vertex and its width is defined as the distance between the two farthest points that are separated by points equal to the length of the diagonal. The equation can be given as follows:
A regular pentagon can be created from just a strip of paper by tying an overhand knot into the strip and flattening the knot by pulling the ends of the paper strip. Folding one of the ends back over the pentagon reveals a pentagram. An equilateral pentagon is a polygon with five sides of equal length.
In order to find the area of a pentagon the length of the sides and the apothem needs to be known from the beginning. The distance of each side of a pentagon is represented by s. The distance between the center and a side is called as apothem and is represented by a. An apothem is a line from the center of a pentagon that is drawn to the midpoint of a side such that it becomes perpendicular to the side. While calculating the area of a regular pentagon,
1. The pentagon is divided into five triangles by drawing the five radiuses from the center to radiate out to each of the vertices.
2. After the apothem is drawn, the height of each triangle is determined. Thus the area of each triangle is calculated by A = ½ * base * height, where the base is equal to the side length of the pentagon and height = length of the apothem of the triangle.
3. Hence the area of a Pentagon = perimeter × apothem / 2
4. The small triangle is right angled and this can be used along with sine, cosine and tangent in order to find the side, radius, apothem and the 5 number of sides are related:
sin(π/n) = (Side/2) / Radius
Side = 2 × Radius × sin(π/5)
cos(π/n) = Apothem / Radius
Apothem = Radius × cos(π/5)
tan(π/n) = (Side/2) / Apothem
Side = 2 × Apothem × tan(π/5)
In case if the sides are known but the apothem is not known then it should be proceeded differently.
1. Divide the pentagon into triangles.
2. Each triangle gets subdivided into two right angled triangles.
3. The angle of the center of the pentagon is 360 (3600/10).
4. Hence tan 36 = base / height.
5. Here base = ½ of the length of the side of the pentagon.
6. Thus height or length of apothem = base length/ tan 36 = (side length/2)/ tan 36
7. In case if the height is known, then the area of the triangle = ½ * base * height = ½ * side * apothem.
8. Thus area of octagon = 5*1/2*s*a = (5*s2)/(4*tan 360) = (5s2) / (4√(5-2√5))
9. In case if the perimeter is known then, P = 5s
10. Therefore S = p/5 where p = perimeter and s = length of the side.
11. Area of the pentagon = ½* p * a where a = apothem length.
12. In case if the radius is known, then area of the pentagon is equal to,
Area = (5/2)r2sin(72º), where r is the radius.
In case of regular pentagon, the easiest way is to divide the number of geometric figures, the right angled triangles, the squares using their appropriate formulas.
Example 1: Find the area of a pentagon of side 5 cm and apothem length 3 cm.
S =5 cm
a = 3 cm
Area of a pentagon = 5/2 s*a
= 5/2 * 5 * 3 cm2
= 75/2 cm2
= 37.5 cm2
Example 2: Find the area of a pentagon of side 12 cm and apothem height 7 cm.
s = 12 cm
r = 7 cm
Therefore area of a pentagon = 5 / 2 * s *a
= 5/2 * 12 * 7 cm2
= 5∗12∗7 /2 cm2
= 420 / 2 cm2
= 210 cm2
There are various real life examples of pentagon which includes man-made structures like the Pentagon in the United States and also in nature in flowers like okra and morning glories. Other real world examples of pentagon includes home plates in baseball are often in shape of irregular pentagons. Other examples of pentagon include- the gynoecium of an apple containing five carpels, arranged in a five pointed star, star fruit with fivefold symmetry, a sea star like many echinoderms have fivefold radial symmetry, the endoskeleton of a sea urchin, brittle stars an echinoderm with pentagonal shape. Some minerals are also included under the real world example of pentagon like a Ho- Mg- Zn icosahedral quasi crystal formed as a pentagonal dodecahedron where the faces are true regular pentagons; a pyrite hedron has 12 identical pentagonal faces that are not constrained to be regular.
The Pentagon is one of the most recognizable buildings in the world as it not only symbolizes the America’s military but it was also the location of one of America’s most horrific terrorism events after the American Airlines Flight 77 slammed into the building on September 11, 2001 killing about 184 people including 64 on the plane including five hijackers and 120 pentagon employees. The entire plan for a new headquarter was called as the War Department which began in 1941 when the nation was watching out the aggressive move of Adolf Hitler in Europe and the growth of the federal workforce in Washington, D.C as America was preparing for the war. The architects and the designers involved with the building’s design came up with a unique pentagonal plan for building the odd dimension of the site. Each of the five wedges had several concentric rings of office space that are linked to each other through corridors. But after a pitched battle conservationists, who were concerned with the fact that the building would block the sweeping vistas of Washington from the cemetery, the president Franklin D. Roosevelt decided that the new headquarter should instead be held on the current site which is at the foot of the Virginia side of the 14TH Street Bridge over the Potomac. General Somervell wanted to have a headquarter with office space for about 40,000 people. So as not to obstruct views of the city across the Potomac River, it was decided that the structure would not be more than four stories high. It was also decided to build something that would require a very little steel in its construction because the rest of the steels would be used for making weapons and ships. With all these requirements, the pentagon shape is decided to be delivered as it could meet all the demands in an efficient manner. But the five sides of the Pentagon still had its detractors especially from the members of the U.S. Commission of Fine Arts who is a quasi-governmental body that is weighed in on design throughout the capital city. One of the members of the Commission argued to Roosevelt that not only the shape of the building was ugly but it can be a huge bombing target. In the end, the President gave the entire plan a ago while considering the shape of its uniqueness.