A hexagon, by definition, is a polygon with six sides and angles. Regular hexagons have six sides and equal angles and are composed of six equilateral triangles and there are several ways to calculate their area, whether you are working with a regular or an irregular hexagon. If you want to know more about how to calculate the area of a hexagon, just follow these Steps.
Method 1 of 4: Calculating from a regular hexagon with a given measurement
Step 1. Write the formula to find the area of a hexagon if you already know the size of its side
Since a regular hexagon is composed of six equilateral triangles, the formula for finding its total area is derived from the one used for finding the area of an equilateral triangle. Said formula can be represented by Area = (3√3 s2)/ 2, Where s is the size of one side of the regular hexagon.
Step 2. Identify the size of one side
If you already know the length of one side, you can simply write it down; in this case, the size of one side equals 9 cm. If you don't know the side dimension but you know the perimeter or apothema (height of one of the equilateral triangles that make up the hexagon, perpendicular to the side), you can still find the size of the hexagon's side. Here's how to do it:
If you know the perimeter, just divide it by 6 and get the dimension of one side. For example, if the perimeter is 54 cm, divide this number by 6 to get the side size of 9 cm.
- If you only know the apothema, you can find the dimension of one side by putting it into the formula a = x√3 and then multiplying the answer by two. This is because the apothema represents the x√3 side of the 30-60-90 triangle created. If the apothema is 10√3, for example, x equals 10 and the side size equals 10 * 2, or 20.
Step 3. Enter the side size values into the formula
Once you know the dimension of just one side, or 9, just put this value into the original formula, which will look something like: Area = (3√3 x 92)/2
Step 4. Simplify your answer
Find the value of the equation and write the numerical answer. When working with the area, you should represent the answer in square units. Here's how to do it:
- (3√3 x 92)/2 =
- (3√3 x 81)/2 =
- (243√3)/2 =
- 420, 80/2 =
- 210, 40 cm2
Method 2 of 4: Calculating from a regular hexagon with a known apothema
Step 1. Write the formula to find the area of the hexagon with a given apothema
The formula is simply represented by Area = 1/2 x perimeter x apothema.
Step 2. Replace the variable with the value of the apothema
Let's say it's worth 5√3 cm.
Step 3. Use the apothema to find the perimeter
Since the apothema is perpendicular to one side of the hexagon, it creates one side of a 30-60-90 triangle. The sides of a triangle like this have the ratio x-x√3-2x, where the dimension of the smallest leg, which passes over a 60-degree angle, is represented by x√3, and the hypotenuse is represented by 2x.
- The apothema is the side represented by x√3. Then put your dimension into the formula a = x√3 and solve for it. If the apothema is equivalent to 5√3, for example, put this value in the formula and get 5√3 cm = x√3, or x = 5 cm.
- By finding the value of x, you will have found the size of the smallest leg in the triangle, or 5. Since it represents half the dimension of one side of the hexagon, multiply it by 2 and get its full size. 5 cm x 2 = 10 cm.
- Now that you know the size of one side is 10, just multiply it by 6 to find the hexagon's perimeter. 10 cm x 6 = 60 cm.
Step 4. Put all known amounts into the formula
The hardest part was finding the perimeter. Now, all you need to do is add the apothema and the perimeter to the formula and solve it:
- Area = 1/2 x perimeter x apothema.
- Area = 1/2 x 60 cm x 5√3 cm.
Step 5. Simplify the expression until you have removed the radicals from the equation
Remember to elaborate the final answer in square units.
- 1/2 x 60 cm x 5√3 cm =
- 30 x 5√3 cm =
- 150√3 cm =
- 259, 80 cm2
Method 3 of 4: Calculating from an Irregular Hexagon with Given Vertices
Step 1. List the x and y coordinates of all vertices
If you know the vertices of the hexagon, the first thing to do is create a spreadsheet with two columns and seven rows. Each column will be named with the names of the six points (Point A, Point B, Point C, etc.) and each column with the x or y coordinates of those points. List the x and y coordinates of Point A to the right of A, those of Point B to the right of B, and so on. Remember to repeat the coordinates from the first to the end of the list. Let's say you are working with the following points, in (x, y) format:
- A: (4, 10).
- B: (9, 7).
- C: (11, 2).
- D: (2, 2).
- E: (1, 5).
- F: (4, 7).
- A (again): (4, 10).
Step 2. Multiply the x coordinate by each point into the y coordinate of the subsequent point
You can think of this step as drawing a diagonal to the right and down a line for each x coordinate. List the results to the right of the spreadsheet, 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
Step 3. Multiply the y coordinates of each point by the x coordinates of the subsequent point
Think of this Step as drawing the same diagonal, but now to the right and down, on a line for each x coordinate below the line in question. After multiplying all the coordinates, add the results.
- 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.
Step 4. Subtract the sum of the second coordinate group from the sum of the first coordinate group
In this case, subtract 221 from 125. 125 – 221 = -96. Now, take the absolute value of the answer: 96. Areas can only have positive values.
Step 5. Divide the difference found by two
In the present problem, divide 96 by 2 and you have the area of this irregular hexagon. 96/2 = 48. Don't forget to write the answer in square units. The final answer in this case is 48 square units.
Method 4 of 4: Other Methods of Calculating the Area of an Irregular Hexagon
Step 1. Find the area of a regular hexagon with a missing triangle
If you know you are working with a regular hex with one or more of its triangles missing, the first thing to do is find the area of the entire hex as if it were complete. Then simply find the area of the empty or “missing” triangle and subtract the found value from the total area. This will give the area of the remaining irregular hexagon.
- For example, if you found that the area of the regular hexagon equals 60 cm2 and found that the area of the missing triangle equals 10 cm2, simply subtract the area of the missing triangle from the total area: 60 cm2 - 10 cm2 = 50 cm2.
- If you know that the hex has exactly one missing triangle, you can find the area of the hex by multiplying the total area by 5/6, since the hex retains the area of 5 of its 6 triangles. If two triangles are missing, just multiply the total area by 4/6 (2/3), and so on.
Step 2. Break an irregular hexagon into other triangles
You may find that the irregular hexagon is actually made up of four irregularly shaped triangles. To find the area of the irregular hexagon, you'll need to find the area of each individual triangle and then add up the results. There are a wide variety of ways used to find the area of a triangle depending on the information you have.
Step 3. Try to find other shapes in the irregular hexagon
If you just can't pick a few triangles to extract, look at the jagged hexagon more closely to see if you can decipher other shapes - maybe a triangle, rectangle, or square. Once you've bypassed other shapes, just find their respective areas and add them to the total area of the hex.