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Area of a Hexagon

There are infinitely many shapes around us, out of which few are predefined geometrical shapes and others are non geometrical. The branch of mathematics that studies about the geometrical figures is known as geometry. In geometry, we come across with a type of shapes termed as a polygon. A polygon is defined as a family of two-dimensional objects which are bounded by three or more line segments. A polygon is made up of many angles as well as sides; as the name itself suggests, poly + gonia, i.e. many terms.

Hexagon is a type of polygon having six sides and six angles. There are two types of hexagons - one is regular, while another is irregular. A regular hexagon has all the sides and angles equal. On the other hand, an irregular hexagon does not contain equal sides and angles. The total space inside the boundary of the hexagon is called as the area of the hexagon.


More elaborately, a h
exagon is a six sided polygon. Hexagon has six edges and six vertices. The measurement of each internal angle in a regular hexagon is given by $102^{\circ}$.  

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Types of Hexagons

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There are two types of hexagons. They are,

  • Regular hexagon
  • Irregular hexagon

In a regular hexagon, the sides and angles are equal in measurement. So, we can easily find out the area of the hexagon with the help of the formula.
regular hexagon figure

In the case of irregular hexagon, the sides and angles are not equal. And so, we cant easily find the area of a irregular hexagon.
irregular hexagon figure

Formula Area Hexagon

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To find the area of a irregular hexagon, we need to divide the hexagon into triangles and then find the sum of areas of all the triangles.

The formula for the area of a regular hexagon is as follows,

Area of a Regular Hexagon = $\frac{3\sqrt{3}}{ 2}$ * x2

Where, x is the side of the hexagon.


Surface Area

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Surface area can be found only for 3D figures not for 2D figures. But, the length of the boundary can be found by adding all the three sides.

Perimeter of a hexagon = Sum of all the sides.

For regular hexagon, Perimeter = 6s, s is the length of each side.

Find the Area of a Hexagon

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Let us see with the help of an example how to find the area of an irregular hexagon:

Area of a Hexagon

Step 1: Divide above diagram into triangles (regular polygons).

Step 2: Find the area of the triangles A, B, C, D, E and F. Finally add the area of all triangles to get the result.

Step 3: From the triangle A, It has equal sides 5

Area of the equal triangle $\frac{\sqrt{3}}{2}$ × a2

Where, a is the side of the triangle

Area of triangle A = $\frac{\sqrt{3}}{2}$ × 52

= 0.866 × 25

= 21.65

Step 5: Area of triangle B = $\frac{1}{2}$ * base × height [Right triangle]

Base = 3, Hypotenuse = 5.

To find the height, we use Pythagorean Theorem c2 = a2 + b2

52 = 32 + b2

b = 4

So, height = 4.

=> Area = $\frac{1}{2}$ × 3 × 4

Area of triangle B = 6

Step 6: Area of  triangle C = $\frac{1}{2}$ × base × height [Right triangle]

Base = 3 Height = 4.

Area = $\frac{1}{2}$ × 3 × 4

Area of triangle C = 6

Step 7: Area of triangle D = $\frac{1}{2}$ × base × height [Right triangle]

Base = 3 and Hypotenuse = 5.

To find height we use Pythagorean Theorem c2 = a2 + b2

52 = 32 + b2

B = 4

So the height = 4.

=> Area = $\frac{1}{2}$ * 3 * 4

Area of triangle D = 6

Step 8: Similarly, area of  triangles E and F are 6 [From the above method]

Area of the hexagon = Area of all triangles

= 21.65 + 6 + 6 + 6 + 6 + 6

= 51.65  square unit

Examples

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Solved Examples

Question 1: Find the area of regular hexagon whose side is 7 cm.
Solution:

Step 1: Area of a hexagon equation is A = $\frac{3\sqrt{3}}{ 2}$ × x2

Here, side x = 7 cm

Step 2: Substitute the value of side in area formula,

Area = $\frac{3\sqrt{3}}{ 2}$ × 72

= 127. 31 cm2 .



Question 2: If the base length is 4 cm and apothem height is 16 cm, then find the area of the hexagon.
Solution:

Step 1: Area of a hexagon = 3 × b × h

Step 2: Substituting the value in the above formula, we get

= 3 × 4 × 16

= 192 cm2.



Practice Problems

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Given below are some of the practice problems on area of a hexagon.

Practice Problems

Question 1: Find the area of regular hexagon whose side is 9 cm.
Question 2: If the base length is 6 cm and apothem height is 18 cm, then find the area of hexagon.
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