Area of a Hexagon

Area of a Hexagon Area of a Hexagon Geometry is a branch of Mathematics which deals with studying shapes of various dimensions and analyzing their properties. In this study of shapes, the polygon family is one of the most commonly studied figures. Polygons are closed 2-dimesional flat objects which have straight sides. The polygon consisting of 6 sides and 6 angles is known as the Hexagon. Any closed flat 2-dimensional structure with 6 straight sides will be referred to as a Hexagon. The word hex means 6 and gon refers to angle. As shown in the diagram below, a hexagon has 6 sides or edges, 6 angles and 6 vertices. Area of a hexagon is the space occupied within the boundaries (or sides) of the hexagon. Using the side and angle measurements, we can find the area of a hexagon. Hexagons can be commonly observed in different forms in our beautiful nature around. The below figure shows us the shaded portion inside the boundaries of the hexagon which is called the Area of a Hexagon. Types of Hexagons: Hexagons are classified primarily into 2 types: Regular Hexagons and Irregular Hexagons. 1) Regular Hexagon: A hexagon which has all of its 6 sides equal in measure is known as the Regular Hexagon. Along with its 6 equal sides, it also has all its 6 angles equal in measure. Therefore when a regular hexagon is divided into triangles by joining its diagonals, 6 equilateral triangles are formed as shown in the figure below. (Note: Diagonal is a line segment formed by joining any two non-adjacent vertices as shown in the figure on the right). Area of a regular hexagon can be easily calculated by considering the 6 equilateral triangles formed inside. 2) Irregular Hexagon: A hexagon which does not have 6 equal sides is known as the Irregular Hexagon. This type of hexagon does not have 6 equal angles as well. If the vertices of an irregular hexagon point outward, then it is known as a Convex Irregular Hexagon, and if the hexagons vertices (or at least 1 vertex) points inward, then it is known as a Concave Irregular Hexagon as shown in the figure below. Since the sides and angles measurements are unequal, hence we have to use different strategies to find the area of an irregular hexagon. Area of a Hexagon: Area of a hexagon is the space occupied by within the sides of the hexagon. The method to calculate area of a regular hexagon differs from the method to calculate the area of an irregular hexagon. Area of a Regular Hexagon: A regular hexagon has all the 6 sides and 6 angles equal in measure. When diagonals passing through the center of the hexagon are drawn, 6 equilateral triangles of equal size are formed (as shown in the figure below). If the area of one equilateral triangle is calculated then we can easily calculate the area of the given regular hexagon. If a convex regular polygon has n sides, then the sum of all its interior angles, S = (n 2) * 180 Therefore, sum of all the interior angles of a convex regular hexagon S = (6 2) * 180 = 720 If a convex regular polygon has n sides, then Each Interior angle = (n 2)/ n * 180 Therefore, each interior angle of a convex regular hexagon = (6 2)/6 * 180 = 120 Given a regular convex hexagon as shown in the figure above, where point C is the center of the hexagon. Triangle ABC is an equilateral triangle, as all the angles inside triangle ABC are equal to 60 (half of the interior angle 120). Hence all its sides are also equal. Therefore, let the side lengths of AB = BC = CA = s CM is the perpendicular drawn to the side AB. Let CM = h As M becomes the midpoint of side AB, hence MB = s/2 (half of the side length of AB). Now in triangle CMB, 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 ABC , h = s* 3/2 Now, Area of triangle ABC = 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 ABC = 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 8cm? Given that the side length, s = 8cm Area of a regular hexagon, A = 3/2 * s2 * 3 Hence, Area = 3/2 * 82 * 3 = 963 which is 166.3cm2 (approximately) Example 2: If the area of a regular hexagon is 12 square feet, then what is the side length of the hexagon? Given Area of a regular hexagon, A = 12 square feet and this can be further simplified as 23 ft2. Area of a regular hexagon, A = 3/2 * s2 * 3. Therefore, 3/2 * s2 * 3 = 12. This implies s2 = 2/3 * 12/3. This gives: s2 = 2/3 * 23/3 (123 can be also written as 23). Hence, s2 = 4/3. This implies s = (4/3) = 2/3 = 1.15 (approximately) Therefore, the side length of the regular hexagon is 1.15ft (approximately) Area of an Irregular Hexagon: Finding the area of an irregular hexagon is not the same as finding the area of a regular hexagon. Area of a regular hexagon can be easily calculated using the formula we have above, but for an irregular hexagon we use various methods according to the given information to find its area. Let us look at an example below where the information is given in the figure. 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 = (10+ 6 + 10)/2 == s = 13 Now Area of triangle A = [s(s-a)(s-b)(s-c)] = [(13* (13 -10) * (13 6) * (13 -10)] Area of Triangle A = (13 * 3 * 7 * 3) = 28.6m2 Triangle B: s = (10 + 11 + 7)/2 = 14 Area of triangle B = [(14 * (14 10) * (14 7) * (14 11)] = (14 * 4 * 7 * 3) = 34.3m2 Similarly using Herons Formula as shown above, we get the areas of triangles C and D as well. Area of Triangle C = 16.9m2 and Area of Triangle D = 19.9m2 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 = 28.6 + 34.3 + 16.9 + 19.9 = 99.7m2 Perimeter of a Hexagon: Perimeter of a hexagon (regular or irregular) can be easily calculated by adding all the side lengths of the given hexagon. Perimeter of a Regular Hexagon is equal to sum of all its side lengths. Therefore, Perimeter of a Regular Hexagon of side length s (as shown in the figure on the right) = s + s + s + s + s + s This gives us Perimeter of Regular Hexagon = 6s Perimeter of an Irregular Hexagon is also sum of all its side lengths. If a, b, c, d, e, and f are the side lengths of an irregular hexagon (as shown in the figure on the right),then Perimeter of an Irregular Hexagon = a + b + c + d + e + f