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A hexagon is a polygon with six sides and six angles. It is a geometric shape that is commonly encountered in various fields, such as mathematics, architecture, and design. The area of a hexagon is a fundamental measurement that determines the amount of space it occupies. Calculating the area of a hexagon may seem challenging, but with the right formula and understanding of its properties, it can be a straightforward process. In this guide, we will explore the step-by-step method of calculating the area of a hexagon, allowing you to confidently apply this knowledge in practical situations. Whether you are a student, professional, or simply someone with a curiosity for geometry, learning how to calculate the area of a hexagon will enhance your understanding of shapes and measurements.
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A hexagon is a polygon with six faces and six angles. A regular hexagon has six equal faces and six angles and consists of six equilateral triangles. There are many ways to calculate the area of a hexagon regardless of whether it is a regular hexagon or an irregular hexagon. If you want to know how to calculate the area of a hexagon, just follow these steps.
Steps
Find the area of a regular hexagon when the length of one side is known
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Write down the formula for calculating the area of a hexagon when the length of the side is known. Since an equilateral hexagon consists of six equilateral triangles, the formula for its area is derived from the formula for the area of an equilateral triangle. The formula for the area of a regular hexagon is Area = (3√3 s 2 )/2 where s is the length of one side. [1] X Research Source
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Determine the length of an edge. If you already know the length of an edge, simply write it down; in this case the side length is 9 cm. If you don’t know the length of the side but know the length of the circumference or the midline (the height of the perpendicular falling from the center of the hexagon to one side), you can still find out the length of the side of the hexagon. Here’s how to do it:
- If you know the perimeter, you can simply divide it by 6 to get the length of the side. For example, if the perimeter length is 54 cm, divide it by 6 to get 9 cm, which is the side length.
- If you only know the midline, you can find the length of the side by substituting the median value into the formula a = x√3 and then multiplying the answer by two. The reason is that the main median is the x√3 side of the 30-60-90 triangle it creates. For example, if the median is 10√3, then x is 10 and the side length is 10 * 2, or 20.
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Substitute the value of the side length into the formula. Since you know that the length of one side of the triangle is 9, just substitute 9 into the original formula. The result is as follows: Area = (3√3 x 9 2 )/2.
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Shorten the answer. Find the value of the equation and write the answer in numbers. Since you are talking about area, you must put your answer in square units. Here’s how to do it:
- (3√3 x 9 2 )/2 =
- (3√3 x 81)/2 =
- (243√3)/2 =
- 420.8/2 =
- 210.4 cm 2
Find the area of a regular hexagon when the median line is known
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Write down the formula for calculating the area of a regular hexagon when the midpoint is known. The simple formula is Area = 1/2 x perimeter x median . [2] X Research Source
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Write down the median length. Assume the median is 5√3 cm.
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Use the median to find the circumference. Since the median is perpendicular to the side of the hexagon, it forms a 30-60-90 triangle. Triangle faces 30-60-90 have a scale of xx√3-2x, where the length of the short side opposite the 30 degree angle is represented by x, the length of the long side opposite the 60 degree angle is x√3, and hypotenuse is 2x. [3] X Research Sources
- The midpoint is the edge represented by x√3. Therefore, substitute the median length into the formula a = x√3 and solve the equation. For example, if the median length is 5√3, substitute it in the formula and get 5√3 cm = x√3, or x = 5 cm.
- By solving the equation for x, you have calculated that the length of the short side of the triangle is 5. Since it is half the length of one side of the hexagon, multiply it by 2 to get the length of one side. 5 cm x 2 = 10 cm.
- Now that you know the length of a side is 10, just multiply it by 6 to find the perimeter of the hexagon. 10 cm x 6 = 60 cm
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Substitute all known indices into the formula. The hardest part is finding the perimeter. Now all you have to do is put the median and perimeter values into the formula and solve the equation:
- Area = 1/2 x perimeter x median
- Area = 1/2 x 60 cm x 5√3 cm
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Shorten the answer. Simplify the expression until you can remove the radical sign from the equation. Remember to use square units in the final result.
- 1/2 x 60 cm x 5√3 cm =
- 30 x 5√3 cm =
- 150√3 cm =
- 259.8 cm 2
Calculate the area of an irregular hexagon given the vertices
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List the x and y coordinates of all the vertices. If you know the vertices of the hexagon, the first thing you need to do is create a graph with two columns and seven rows. Each row will list the names of six points (Point A, Point B, Point C, etc.) and each column will record the x and y coordinates of those points. Write the x and y coordinates of Point A to the right of Point A, the x and y coordinates of Point B to the right of Point B, and so on. Record the coordinates of the first point at the bottom of the list. Suppose you have the following points, in the format (x, y): [4] X Research Source
- A: (4, 10)
- B: (9, 7)
- C: (11, 2)
- D: (2, 2)
- E: (1, 5)
- F: (4, 7)
- A (repeat): (4, 10)
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Multiply the x coordinate of each point by the y coordinate of the next point. Record the results on the right side of the chart. Then add the results together.
- 4 x 7 = 28
- 9 x 2 = 18
- 11 x 2 = 22
- 2 x 5 = 10
- 1 x 7 = 7
- 4 x 10 = 40
- 28 + 18 + 22 + 10 + 7 + 40 = 125
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Multiply the y coordinate of each point by the x coordinate of the next point. After multiplying all these coordinates, add the results together.
- 10 x 9 = 90
- 7 x 11 = 77
- 2 x 2 = 4
- 2 x 1 = 2
- 5 x 4 = 20
- 7 x 4 = 28
- 90 + 77 + 4 + 2 + 20 + 28 = 221
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Subtract the sum of the first group of coordinates from the sum of the second group of coordinates. Just subtract 125 for 221. 125-221 = -96. Now, take the absolute value of the above result: 96. Area can only be positive.
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Divide the above difference by two. Just divide 96 by 2 and you will have the area of the hexagon. 96/2 = 48. Don’t forget to write your answer in square units. The final answer is 48 square units.
Other methods to calculate the area of an irregular hexagon
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Find the area of a regular hexagon that is missing a triangle. If your regular hexagon is missing one or more triangles, the first thing you need to do is find the area of the entire regular hexagon as if it were whole. Then simply find the area of the empty or “missing” triangle, and subtract the total area of the figure from the area of the missing part. The result will be the area of the remaining irregular hexagon.
- For example, if you calculate that the area of a regular hexagon is 60 cm 2 and the area of the missing triangle is 10 cm 2 , simply subtract the total area of the hexagon by the area of the vacated triangle: 60 cm 2 – 10 cm 2 = 50 cm 2 .
- If you know the missing hexagon is exactly a triangle, you can also calculate the area of the hexagon by multiplying the total area by 5/6, since this hexagon occupies 5 of the 6 triangles of It. If it’s missing two triangles, you can multiply the total area by 4/6 (2/3), and so on.
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Divide the irregular hexagon into triangles. You can see that the irregular hexagon actually consists of four triangles of different shapes. To find the area of the entire hexagon, you need to find the area of each individual triangle then add them together. There are many ways to find the area of a triangle depending on the information you have.
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Find other shapes in the irregular hexagon. If you can’t divide the hexagon into several triangles, see if you can divide it into other shapes — maybe triangles, rectangles, and/or squares. Once you’ve identified the shapes, simply find their areas and add them together to get the area of the entire hexagon.
- There is a type of irregular hexagon consisting of two parallelograms. To calculate the area of a parallelogram, simply multiply the base by their height, just like calculating the area of a rectangle, and then add the results together.
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wikiHow is a “wiki” site, which means that many of the articles here are written by multiple authors. To create this article, 23 people, some of whom are anonymous, have edited and improved the article over time.
This article has been viewed 216,030 times.
A hexagon is a polygon with six faces and six angles. A regular hexagon has six equal faces and six angles and consists of six equilateral triangles. There are many ways to calculate the area of a hexagon regardless of whether it is a regular hexagon or an irregular hexagon. If you want to know how to calculate the area of a hexagon, just follow these steps.
In conclusion, calculating the area of a hexagon is a relatively straightforward process that requires only a few simple equations. By breaking the hexagon into smaller shapes, such as triangles or rectangles, and correctly applying the respective formulas, one can accurately determine the area. It is important to remember that the process involves finding the perimeter first, using that to calculate the apothem, and finally using the apothem and perimeter to determine the area. Understanding and applying these steps allows individuals to confidently and efficiently calculate the area of a hexagon without any confusion or errors.
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