A hexagon is a closed 2D shape that is made up of straight lines. It is a two-dimensional shape with six sides, six vertices, and six edges. The name is divided into hex, which means six, and gonia, which means corners.
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Hexagon is a two-dimensional geometrical shape that is made of six sides, having the same or different dimensions of length. Some real-life examples of the hexagon are a hexagonal floor tile, pencil, clock, a honeycomb, etc. A hexagon is either regular(with 6 equal side lengths and angles) or irregular(with 6 unequal side lengths and angles).
Hexagons can be classified based on their side lengths and internal angles. Considering the sides and angles of a hexagon, the types of the hexagon are,Regular Hexagon: A regular hexagon is one that has equal sides and angles. All the internal angles of a regular hexagon are 120°. The exterior angles measure 60°. The sum of the interior angles of a regular hexagon is 6 times 120°, which is equal to 720°. The sum of the exterior angles is equal to 6 times 60°, which is equal to 360°.Irregular Hexagon: An irregular hexagon has sides and angles of different measurements. All the internal angles are not equal to 120°. But, the sum of all interior angles is the same, i.e 720 degrees.Convex Hexagon: A convex hexagon is one in which all the interior angles measure less than 180°. Convex hexagons can be regular or irregular, which means they can have equal or unequal side lengths and angles. All the vertices of the convex hexagon are pointed outwards.Concave Hexagon: A concave hexagon is one in which at least one of the interior angles is greater than 180°. There is at least one vertex that points inwards.
A hexagon is a flat two-dimensional shape with six sides. It may or may not have equal sides and angles. Based on these facts, the important properties of a hexagon are as follows.It has six sides, six edges, and six verticesAll the side lengths are equal or unequal in measurementAll the internal angles are equal to 120° in a regular hexagonThe sum of the internal angles is always equal to 720°All the external angles are equal to 60° in a regular hexagonSum of the exterior angles is equal to 360° in a hexagonA regular hexagon is also a convex hexagon since all its internal angles are less than 180°A regular hexagon can be split into six equilateral trianglesA regular hexagon is symmetrical as each of its side lengths is equalThe opposite sides of a regular hexagon are always parallel to each other.
As with any polygon, a regular hexagon also has a different formula to calculate the area, perimeter, and a number of diagonals. Let us look into each one of them.
Diagonals of a Hexagon
A diagonal is a segment of a line, that connects any two non-adjacent vertices of a polygon. The number of diagonals of a polygon is given by n(n-3)/2, where 'n' is the number of sides of a polygon. The number of diagonals in a hexagon is given by, 6 (6 - 3) / 2 = 6(3)/2, which is 9. Out of the 9 diagonals, 6 of them pass through the center of the hexagon.
Sum of Interior Angles of Hexagon
The sum of internal angles formed by a regular hexagon is 720˚ (because each angle is 120˚ and there are 6 such angles adding up to 720˚). It is given by the formula for regular polygon, where n is a number of sides, which has a value of 6 for hexagonal shape. The formula is (n-2) × 180°. Therefore, (6-2) ×180° which gives us 720°.
The area of a regular hexagon is the space or the region occupied by the shape. It is measured in square units. Let us divide the hexagon into 6 equilateral triangles as shown below. Let us calculate the area of one triangle and multiply it by 6 to get the entire area of the hexagon.
Area of one equilateral triangle is √3a2/4 square units. Hence, the area of a regular hexagon formed by combining 6 such triangles is,
6 × √3a2/4= 3√3a2/2 square units
Therefore, the formula for the regular hexagon area is 3√3a2/2 square units.
Perimeter of a Hexagon
Perimeter is the total length of the boundary or the outline of a shape. Considering the side of a regular hexagon as 'a' units, the regular hexagon perimeter is given by summing up the length of all the sides which is equal to 6a units. Therefore, the perimeter of a regular hexagon = 6a units, and the perimeter of an irregular hexagon = (a + b + c + d + e + f) units, where, a, b, c, d, e, and f are the side-lengths of the hexagon.
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Example 1: What is the area of a regular hexagon with sides equal to 3 units?
Area of a regular hexagon = 3√3a2/2 square units.Given side 'a' = 3 unitsTherefore, area = 3(√3)32/2= (3 × √3 × 9) /2= (27× √3) / 2= 23.382 square units,
Example 2: Find the length of each side of a regular hexagon, if the hexagon's area is 150√3 square units. Use the length of the sides to find the perimeter of the hexagon.
Applying the formula of area of a regular hexagon,
Area of a regular hexagon = 3√3a2/2 square units.Therefore, 150√3 = 3√3a2/2300√3 = 3√3a2Canceling √3 on both sides,300/3 = a2100 = a2a = √100Therefore, the length of each side, a = 10 units.
Therefore, the length of the sides of the hexagon = 10 units.Perimeter of a regular hexagon = 6a units.a = 10 units. Therefore,Perimeter = 6 × 10Therefore, the given regular hexagon perimeter = 60 units.
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