A hexagon is a polygon that has six sides and angles. A regular hexagon has six equal sides and angles and consists of six equilateral triangles. There are various ways to calculate the area of a hexagon, whether it's a regular hexagon or an irregular hexagon. If you want to know how to calculate the area of a hexagon, just follow these steps.
Step
Method 1 of 4: Calculating the Area of a Regular Hexagon If You Know The Lengths Of The Sides
Step 1. Write a formula to find the area of a hexagon if you know the side lengths
Since a regular hexagon consists of six equilateral triangles, the formula for calculating the area of a hexagon can be obtained from the formula for calculating the area of an equilateral triangle. The formula for calculating the area of a hexagon is Area = (3√3 s2)/ 2 with description s is the side length of a regular hexagon.
Step 2. Find the length of the side
If you already know the length of the side, then you can write it right away; in this case, the length of the side is 9 cm. If you don't know the side lengths but know the perimeter or apothem (height of the triangle that makes up the hexagon, which is perpendicular to the side of the hexagon), then you can still find the side lengths of the hexagon. Here's how:
- If you know the perimeter, then just divide by 6 to get the length of the side. For example, if the perimeter is 54 cm, then divide by 6 to get 9, which is the length of the side.
- If you only know the apothem, you can calculate the side length by plugging the apothem into the formula a = x√3 and then multiplying the result by two. This is because the apothem represents the x√3 part of the 30-60-90 triangle it makes. For example, if the apothem is 10√3, then x is 10 and the side length is 10*2, which is 20.
Step 3. Enter the side length values into the formula
Since you know that the side length of the triangle is 9, plug 9 into the original formula. This will look like this: Area = (3√3 x 92)/2
Step 4. Simplify your answer
Find the value of the equation and write down the number of the answer. Since you want to calculate the area, you must state the answer in square units. Here's how:
- (3√3 x 92)/2 =
- (3√3 x 81)/2 =
- (243√3)/2 =
- 420.8/2 =
- 210.4cm2
Method 2 of 4: Calculating the Area of a Regular Hexagon If You Know The Apothem
Step 1. Write a formula to calculate the area of a hexagon if you know the apothem
The formula is only Area = 1/2 x perimeter x apothem.
Step 2. Write down the apothem
Let's say the apothem is 5√3 cm.
Step 3. Use the apothem to calculate the perimeter
Since the apothem is perpendicular to the side of the hexagon, it makes a 30-60-90 angle triangle. The side of a triangle with an angle of 30-60-90 will be proportional to xx√3-2x, with the length of the short side, which is opposite the 30 degree angle represented by x, the length of the long side, which is opposite the 60 degree angle, represented by x 3, and the hypotenuse is represented by 2x.
- The apothem is the side represented by x√3. Therefore, plug the length of the apothem into the formula a = x√3 and solve. For example, if the length of the apothem is 5√3, plug it into the formula and get 5√3 cm = x√3, or x = 5 cm.
- Now that you've got the x value, you've found the length of the short side of the triangle, which is 5. Since this value is half the length of the side of the hexagon, multiply by 2 to get the actual side length. 5cm x 2 = 10cm.
- Now that you know the length of the side is 10, just multiply it by 6 to get the perimeter of the hexagon. 10 cm x 6 = 60 cm
Step 4. Plug all the known values into the formula
The hardest part is finding the circumference. Now all you have to do is plug the apothem and perimeter into the formula and solve:
- Area = 1/2 x perimeter x apothem
- Area = 1/2 x 60 cm x 5√3 cm
Step 5. Simplify your answer
Simplify the equation until you remove the square root from the equation. Express your final answer in square units.
- 1/2 x 60 cm x 5√3 cm =
- 30 x 5√3 cm =
- 150√3 cm =
- 259. 8 cm2
Method 3 of 4: Calculating the Area of an Irregular Hexagon If You Know The Points
Step 1. Find the list of x and y coordinates of all points
If you know the points of the hexagon, the first thing you should do is create a graph with two columns and seven rows. Each row will be named with the names of the six points (Point A, Point B, Point C, etc.), and each column will be populated with the x or 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. Rewrite the coordinates of the first point on the bottom line of the list. Assume that you use the following dots, 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 of each point by the y-coordinate of the next point
Think of it like drawing a diagonal line to the right and down one line from each x-coordinate. Write the results to the right of the graph. Then add up the results.
- 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-coordinate of each point by the x-coordinate of the next point
Think of it like drawing a diagonal line going down from each y-coordinate and then to the left, towards the x-coordinate below it. After multiplying all the coordinates, add up 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 group of coordinates from the sum of the first group of coordinates
Subtract 221 from 125. 125 - 221 = -96. Then, take the absolute value of this result: 96. Area can only be positive..
Step 5. Divide the difference by two
Divide 96 by 2 and you get the area of the irregular hexagon. 96/2 = 48. Don't forget to write your answer in square units. The final answer is 48 square units.
Method 4 of 4: Another Way to Calculate the Area of an Irregular Hexagon
Step 1. Find the area of a regular hexagon with the missing triangle
If you know that the regular hexagon that you want to calculate does not have a complete triangular section, then the first thing you should do is find the area of the entire regular hexagon as if it were a whole. Then, find the area of the "missing" triangle, and subtract it from the total area. Thus, you will get the area of the irregular hexagon
- For example, if you already know that the area of a regular hexagon is 60 cm2 and you also know that the area of the missing triangle is 10 cm2, just subtract the area of the missing triangle from the total area: 60 cm2 - 10 cm2 = 50 cm2.
- If you know that the hexagon is missing exactly one triangle, you can immediately calculate the area of the hexagon by multiplying the total area by 5/6, since the hexagon has the area of 5 of the 6 triangles. If the hexagon is missing two triangles, you can multiply the total area by 4/6 (2/3), and so on.
Step 2. Break the irregular hexagon into several triangles
You may notice that an irregular hexagon is actually made up of four irregularly shaped triangles. To find the total area of an irregular hexagon, you must calculate the area of each triangle and add them all together. There are various ways to calculate the area of a triangle depending on the information you have.
Step 3. Find another shape of the irregular hexagon
If you can't break it down into triangles, take a look at the irregular hexagon to see if you can find another shape – maybe a triangle, rectangle, and/or square. When you find other shapes, find their areas and add them to get the total area of the hexagon.
