Hexagon

Hexagon Hexagon Geometry is one of the most important branches of Mathematics as it deals with the study of different shapes, their dimensions, and calculations of them. In this study, we look at shapes formed by straight lines, and shapes which have curved surfaces. The 2-dimensional figures which have flat faces, which have straight lines as edges and which are closed are known as Polygons. Polygon family is a family consisting of different shapes of different number of sides. The word poly means many and gon means angle. Based on the number of sides a polygon has, we classify them into further categories. For example, polygons having 3 sides are known as Triangles, polygons having 4 sides are known as Quadrilaterals (rectangles, squares, etc), and polygon having 5 sides is known as a Pentagon and so on. A polygon which has 6 number of sides is known as a Hexagon. The word hexa means six and gon means angle. Since the polygon has 6 sides which consequently forms 6 angles, hence it is known as a Hexagon. Hexagon Definition: A polygon which has 6 number of sides (or edges) and 6 number of angles is a Hexagon. As shown in the figure on the left, hexagons have 6 vertices (or corners), 6 edges (or sides) and 6 angles. Types of Hexagons: Based on the measurements of the sides, hexagons are classified into 2 types: Regular Hexagons and Irregular Hexagons. 1) Regular Hexagons: A hexagon which has all the 6 sides equal in measure is known as a Regular Hexagon. Because it has 6 equal sides, the 6 interior angles of a hexagon are also equal. Properties: a) Regular Hexagons has 6 equal sides and 6 equal interior angles. b) As a hexagon has even number of sides, hence the opposite sides of a regular hexagon are parallel to each other. c) A line drawn from the center of the regular hexagon to any of the vertices will have the same length as the side length, as shown in the figure below. d) All the regular hexagons are convex, which means that all its 6 vertices point outward. e) The line segment joining any two non-adjacent vertices in a polygon is known as a Diagonal. The diagonals of a regular hexagon divide the hexagon into 6 equilateral triangles as shown in the figure on the right. 2)Irregular Hexagon: A hexagon which is not regular is known as the Irregular Hexagon. This implies, that an irregular hexagon has 6 sides that are not all equal in measure, or 6 interior angles that are all not equal in measure. Properties: a) Irregular Hexagons do not have 6 equal sides or 6 equal interior angles. b) The opposite sides may or may not be parallel to each other. c) An irregular hexagon can be convex in shape or concave in shape. A Convex polygon is a polygon which has all the vertices pointing outward. But in a concave polygon, one or more vertices point inward towards the center of the polygon. Because of this reason, in a concave polygon one or more interior angles is greater than 180. d) A line drawn through a concave hexagon (depending on where the line is drawn) can intersect the hexagon at more than 2 points. The below figure shows the line intersecting the hexagon at 4 points. e) In a concave hexagon, all the diagonal do not lie inside the hexagon. One or more diagonals lie outside the hexagon also, as shown in the figure below. Angles of a Hexagon: 1)Sum of all the interior angles of a Regular Hexagon: The sum of all the interior angles of any regular polygon can be calculated using the formula given below: If a regular polygon has n sides, then the sum of all its interior angles, S = (n 2) * 180 Since a hexagon has 6 sides, hence n = 6. Now Sum, S = (6 2) * 180 = 720 Therefore, Sum of all the interior angles of a regular hexagon, S = 720 2)Each Interior angle of a Regular Hexagon: The measure of each interior angle of any regular polygon can be calculated using the formula given below: If a regular polygon has n sides, then Each Interior angle = (n 2)/ n * 180 Since a hexagon has 6 sides, hence n = 6. So, Each Interior angle = (6 2) / 6 * 180 = 120 Therefore, Each interior angle of a Regular Hexagon = 120 3)Each exterior angle of a Regular Hexagon: The measure of each exterior angle of any regular polygon can be calculated using the formula given below: If a regular polygon has n sides, then Each Exterior angle = 360/n Each Exterior angle of Regular Convex Hexagon = 360/6 = 60 Therefore, Each exterior angle of a Regular Hexagon = 60 4)Diagonals of a Hexagon: Number of diagonals in a polygon of n sides = n * (n 3)/ 2 Since a hexagon has 6 sides, hence n = 6. Therefore, number of diagonals in a hexagon = 6 * (6 3)/2 = 9 diagonals. Perimeter of a Hexagon: Perimeter is the total length calculated when all the side lengths of the polygon are combined together. Perimeter of a regular or irregular polygon can be calculated by adding all the side lengths of the polygon. Perimeter of a Polygon = Sum of all its side lengths. Therefore Perimeter of a Regular Hexagon of side length s (as shown in the figure on the right) will be written as, P = s + s + s + s + s + s = 6s Example: Calculate the perimeter of a regular hexagon whose side length is 7m. Perimeter of a Regular Hexagon, P = 6 * s == Perimeter, P = 6 * 7m = 42m Example: Calculate the perimeter of the hexagon shown below. Given the side lengths of the hexagon in the figure. Perimeter of a Hexagon = Sum of all the side lengths. Therefore, Perimeter, P = 4m + 7m + 3m + 2m + 8m + 2m = 26m Area of a Hexagon: Area of any polygon is the space occupied within the boundaries or edges of the polygon. Hence, area of a hexagon is the space covered within its edges or sides. Area of a regular hexagon is different from the area of an irregular hexagon. Various procedures can be used in order to calculate its area. Let us look at the common methods used in the process. 1)Area of a Regular Hexagon: As mentioned above, diagonal of a regular hexagon divide the hexagon into 6 equal triangles, also known as 6 equilateral triangles. So if we find the area of one equilateral triangle, then the area of all the 6 triangles will be known, and then the area of the hexagon will be the triangle areas added together. Given a regular hexagon as shown in the figure above, where point C is the center of the hexagon. Triangle CPQ is an equilateral triangle, as all the angles inside triangle CPQ are equal to 60 (half of the interior angle 120). Hence all its sides are also equal. Therefore, let the side lengths of CP = PQ = CQ = s CM is the perpendicular drawn to the side PQ. Let CM = h As M becomes the midpoint of side PQ, hence MQ = s/2 (half of the side length of PQ). Now in triangle CMQ, we can apply the Pythagorean Theorem to get the relationship between the height h of the triangle, and the side length s. Hence, h2 + (s/2)2 = s2. This implies h2 + s2/4 = s2. This gives h2 = s2 s2/4 So, h2 = 3s2/4 == h = (3s2/4). Therefore, the height of the triangle CPQ, h = s* 3/2 Now, Area of triangle CPQ = 1/2 * base * height. This implies, Area A = 1/2 * s * h == A = 1/2 * s * (s * 3/2) == A = s2 * 3/4 Therefore, Area of triangle CPQ = s2 * 3/4. Now, a regular hexagon consists of 6 such congruent equilateral triangles. Hence, Area of a Regular Hexagon = 6 * s2 * 3/4 which can be further simplified as: Area of a Regular Hexagon = 3/2 * s2 * 3 Example 1: What is the area of a regular hexagon whose side length is 5m? Given that the side length, s = 5m Area of a regular hexagon, A = 3/2 * s2 * 3 Hence, Area = 3/2 * 52 * 3 = 3 which is 64.95m2 (approximately) 2)Area of an Irregular Hexagon: Since an irregular hexagon does not have equal sides or equal angles, hence we cannot use the method or formula of the regular hexagon. For an irregular hexagon, we can calculate area by using various methods. Let us look at an example below: Example: Find the area of the irregular hexagon shown in the figure below. In the given figure, we observe that the side lengths are given and the lengths of the diagonals are also given. We can see that the irregular hexagon is split into 4 triangles A, B, C and D. Since the side lengths of each triangle are given, we can use Herons formula. Herons Formula: If a triangle has side lengths as a, b and c, then s = (a + b+ c)/2 Then, Area of the triangle = [s* (s-a)* (s-b)* (s-c)] Triangle A: s = (5 + 4 + 7)/2 == s = 8 Now Area of triangle A = [s(s-a)(s-b)(s-c)] = [(8* (8 - 5) * (8 4) * (8 - 7)] Area of Triangle A = (8 * 3 * 4 * 1) = 9.8m2 Triangle B: s = (7 + 7 + 6)/2 = 10 Area of triangle B = [(10 * (10 - 7) * (10 - 7) * (10 - 6)] = (10 * 3 * 3 * 4) = 18.9m2 Similarly using Herons Formula as shown above, we get the areas of triangles C and D as well. Area of Triangle C = 8.9m2 and Area of Triangle D = 7.9m2 Now, Area of the Irregular Hexagon = Area of Triangle A + Area of Triangle B + Area of Triangle C + Area of Triangle D Area of the Hexagon = 9.8m2 + 18.9 m2 + 8.9 m2 + 7.9 m2 = 45.5m2 Hexagonal Tessellation: When a flat surface or a plane is covered by shapes that are repeated over and over again forming a periodic pattern, without any gaps or overlaps is known as Tessellation. We can find different kinds of tessellations such as tessellations of triangles, squares, rectangles etc. Regular polygons which are congruent (meaning same shape and size) form tessellations known as Regular Tessellations. There are only 3 types of Regular Tessellations, and they are of triangles, squares, and hexagons. A Hexagonal Tessellation is a tessellation formed when hexagons are arranged on a plane as shown in the figure below. This pattern for a Regular Hexagonal Tessellation is identical. In the figure below, we can see the vertex marked. At each vertex, we can observe that 3 hexagons are meeting. Since each hexagon has 6 sides, hence this kind of tessellation is named as 6.6.6 Tessellation. Hexagonal Prism: A hexagonal prism is a 3-dimensional figure consisting of 2 hexagonal bases and 6 rectangular faces. A hexagonal prism consists of 8 faces, 18 edges and 12 vertices. Because of its 8 faces, it is also known as the Octahedron. Surface Area of a Regular Hexagonal Prism: As shown in the figure below, a hexagonal prism has 2 hexagonal bases and 6 rectangular faces. Lateral area of the hexagonal prism is the sum of the areas of the 6 rectangular faces. If the height of the prism is h and the side of the base regular hexagon is s, then: Area of each rectangular face = s * h Lateral Area = Sum of the Areas of 6 Rectangular Faces Therefore, Lateral Area = 6 * s * h Surface Area of a Prism = Bases Area + Lateral Area The perpendicular from the center of the hexagon to its base side is also known as the Apothem (shown as d in the figure below). If the base side length is s, then as mentioned above Apothem or height of the hexagon is s * 3/2 Then, Area of the Hexagon = 3 * s * d = 3 * s * 3/2 * s = 3/2 * 3 * s2 Since there are 2 such base hexagons, hence Bases Area = 2 * 3/2 * 3 * s2 = 33 * s2 Surface Area = Bases Area + Lateral Area Hence, Surface Area of a Regular Hexagonal Prism = (33 * s2) + (6* s* h) (Where s is the side length of the base regular hexagon and h is the height of the prism). Example: How much is the surface area of a regular hexagonal prism if given the side length of the base regular hexagon is 4 inches, and height of the prism is 6 inches? Given that the side length of the base regular hexagon, s = 4 inches Height of the prism, h = 6 inches Surface Area of a Regular Hexagonal Prism = (33 * s2) + (6* s* h) Therefore, Surface Area = (33 * 42) + (6 * 4 * 6) = 483 + 144 = 227 square inches (approx.) Volume of a Hexagonal Prism: Volume of a Prism is the amount of space occupied within the boundaries or edges of the prism. Volume of a Hexagonal Prism = Area of the Base * Height of the Prism Therefore, Volume of a Hexagonal Prism = 3/2 *3 * s2 * h Example: Calculate the volume of a hexagonal prism whose base side length is 4 inches and height of the prism is 6 inches. Given that the side length of the base regular hexagon, s = 4 inches Height of the prism, h = 6 inches Volume of a Hexagonal Prism = 3/2 *3 * s2 * h Hence, Volume = 3/2* 3 * 42 * 6 = 249.4 cubic inches.