How To Find The Area Of A Hexagon Using Trigonometry

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A hexagon is a polygon that has vi sides and angles. Regular hexagons accept half dozen equal sides and angles and are composed of six equilateral triangles. At that place are a variety of ways to calculate the expanse of a hexagon, whether you're working with an irregular hexagon or a regular hexagon. If you lot want to know how to calculate the area of a hexagon, merely follow these steps.

i

Write downward the formula for finding the area of a hexagon if you know the side length. Since a regular hexagon is comprised of six equilateral triangles, the formula for finding the area of a hexagon is derived from the formula of finding the area of an equilateral triangle. The formula for finding the area of a hexagon is Area = (3√3 due southⁱⁱ)/ 2 where due south is the length of a side of the regular hexagon.^[1]
2
Identify the length of one one side. If you lot already know the length of a side, then you can simply write it downwards; in this case, the length of a side is nine cm. If you don't know the length of a side only know the length of the perimeter or apothem (the meridian of i of the equilateral triangles formed past the hexagon, which is perpendicular to the side), you tin can however find the length of the side of the hexagon. Here'southward how you lot exercise information technology:
- If you know the perimeter, then just dissever it by six to become the length of one side. For instance, if the length of the perimeter is 54 cm, then divide it by vi to get 9 cm, the length of the side.^[2]
- If y'all only know the apothem, you tin can observe the length of a side past plugging the apothem into the formula a = x√3 and then multiplying the answer by ii. This is considering the apothem represents the x√3 side of the 30-sixty-90 triangle that it creates. If the apothem is 10√3, for example, then x is ten and the length of a side is 10 * ii, or 20.
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iii

Plug the value of the side length into the formula. Since you know that the length of one side of the triangle is 9, just plug 9 into the original formula. It will look like this: Area = (3√iii 10 9²)/2
four
Simplify your answer. Find the value of equation and write the numerical respond. Since you're working with area, you should state your reply in square units. Hither's how you practice it:
- (iii√3 x ix²)/2 =
- (three√3 10 81)/2 =
- (243√3)/2 =
- 420.viii/ii =
- 210.4 cm²
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1

Write downwardly the formula for finding the area of a hexagon with a given apothem. The formula is simply Expanse = i/2 ten perimeter x apothem.^[iii]
2

Write downwardly the apothem. Permit's say the apothem is 5√3 cm.
3
Use the apothem to find the perimeter. Since the apothem is perpendicular to the side of the hexagon, it creates one side of a xxx-60-xc triangle. The sides of a 30-sixty-xc triangle are in the proportion of x-10√3-2x, where the length of the short leg, which is across from the thirty degree angle, is represented by x, the length of the long leg, which is across from the threescore degree angle, is represented by ten√3, and the hypotenuse is represented by 2x.^[4]
- The apothem is the side that is represented by x√iii. Therefore, plug the length of the apothem into the formula a = x√3 and solve. If the apothem's length is v√three, for instance, plug it into the formula and get 5√iii cm = x√three, or x = 5 cm.
- Past solving for 10, y'all have found the length of the brusk leg of the triangle, v. Since it represents half the length of one side of the hexagon, multiply it by 2 to get the full length of the side. 5 cm x 2 = 10 cm.
- Now that you know that the length of 1 side is x, just multiply it by 6 to detect the perimeter of the hexagon. 10 cm x six = threescore cm
4
Plug all of the known quantities into the formula. The hardest role was finding the perimeter. Now, all yous have to do is plug the apothem and perimeter into the formula and solve:
- Area = 1/two x perimeter x apothem
- Area = 1/2 x sixty cm x 5√3 cm
five
Simplify your respond. Simplify the expression until you've removed the radicals from the equation. State your concluding respond in foursquare units.
- 1/2 x 60 cm x 5√3 cm =
- 30 10 5√iii cm =
- 150√iii cm =
- 259. 8 cm^two
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1
List the 10 and y coordinates of all the vertices. If you know the vertices of the hexagon, the kickoff thing you should do is create a chart with 2 columns and 7 rows. Each row will be labeled by the names of the six points (Point A, Point B, Point C, etc), and each column will be labeled as the ten or y coordinates of those points. List the x and y coordinates of Point A to the right of Point A, the ten and y coordinates of Indicate B to the right of Signal B, and then on. Repeat the coordinates of the first point at the lesser of the listing. Allow'due south say y'all're working with the following points, in (x, y) format:^[5]
- A: (4, 10)
- B: (ix, 7)
- C: (11, ii)
- D: (two, 2)
- E: (1, 5)
- F: (iv, 7)
- A (once again): (4, 10)
ii
Multiply the x coordinate of each point by the y coordinate of the side by side point. You tin can think of this as cartoon a diagonal line to the right and downward one row from each x coordinate. List the results to the right of the chart. So, add the results.
- 4 ten 7 = 28
- nine x two = eighteen
- 11 10 2 = 22
- two x five = x
- 1 x 7 = seven
- 4 x ten = forty
  - 28 + eighteen + 22 + x + 7 + 40 = 125
3
Multiply the y coordinates of each betoken past the x coordinates of the next point. Think of this as drawing a diagonal line from each y coordinate downwards and to the left, to the x coordinate below information technology. In one case yous multiply all of these coordinates, add the results.
- 10 x 9 = xc
- vii 10 xi = 77
- 2 x 2 = four
- 2 x 1 = 2
- 5 x 4 = twenty
- 7 x four = 28
- 90 + 77 + 4 + 2 + xx + 28 = 221
four

Subtract the sum of the 2nd group of coordinates from the sum of the first grouping of coordinates. Only decrease 221 from 125. 125 - 221 = -96. Now, take the absolute value of this reply: 96. Area tin only be positive.
5

Divide this difference by two. Just divide 96 past 2 and you'll accept the expanse of the irregular hexagon. 96/2 = 48. Don't forget to write your respond in square units. The final answer is 48 square units.

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1
Notice the area of a regular hexagon with a missing triangle. If you lot know you're working with a regular hexagon that is missing one or more of its triangles, and then the start affair yous need to practise is observe the expanse of the unabridged regular hexagon equally if it were whole. Then, simply find the area of the empty or "missing" triangle, and that subtract that from the overall area. This will give you the area of the remaining irregular hexagon.^{[half dozen]}
- For case, if you've found that the area of the regular hexagon is 60 cmⁱⁱ and you've found that the area of the missing triangle is x cmⁱⁱ simply subtract the expanse of the missing triangle from the entire area: threescore cm² - 10 cm² = 50 cm^two.
- If you know that the hexagon is missing exactly one triangle, y'all can as well just find the area of the hexagon by multiplying the total area by 5/6, since the hexagon is retaining the area of 5 of its 6 triangles. If it'south missing two triangles, you can multiply the total area by 4/vi (2/iii), and so on.
2

Break up an irregular hexagon into other triangles. You may find that the irregular hexagon is really composed of 4 triangles that are irregularly shaped. To discover the surface area of the whole irregular hexagon, you need to find the expanse of each individual triangle and and then add them up. There are a variety of ways to discover the expanse of a triangle depending on the information that you have.^[7]
3
Await for other shapes in the irregular hexagon. If y'all can't simply pick apart a few triangles, expect through the irregular hexagon to run into if you tin locate other shapes -- possibly a triangle, a rectangle, and/or a square. Once yous've outlined the other shapes, just find their areas and add them up to get the surface area of the entire hexagon.^[eight]
- 1 blazon of irregular hexagon is comprised of two parallelograms. To get the areas of the parallelograms, just multiply their bases times their heights, just every bit you would practice to detect the area of a rectangle, and then add upward their areas.
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Add New Question

Question

I was just given the length of the diagonal. What do I do?

ane/tertiary of the length of the diagonal is the side of the hexagon. Using this, yous tin can calculate the area.
Question

I know the expanse, nothing else. I demand a regular hexagon. What practice I practice?

Assuming you're looking for the length of a side, solve for southward using the area formula in Method 1 to a higher place. Then plug in the known area.
Question

What is the expanse of a regular hexagon where the length of each side is 7m?

The side is i/2 of 7 = iii.5. The area of a regular hexagon is (3√3 *3.5^2)/ii =31.82 or 32 cm, approximately.
Question

How do I summate the area of a hexagon that has a perimeter of 48cm?

Assuming a regular hexagon, utilise Method 1 above.
Question

What is the formula for finding the side with a given apothem and the surface area?

It's super easy to find the area of a hexagon. A hexagon is fabricated up of six triangles and then only observe the area of each triangle, add them upward, and you take your answer.
Question

How do I calculate the surface area of a hexagonal prism?

A hexagonal prism has six rectangles and two hexagons, so for a curved expanse, it will exist "6 x (side of hexagon x height of prism)" and the total surface area would be "[6 x (side of hexagon x height of prism)] + two x (2.6 x side of hexagon)."
Question

What is the area of a regular hexagon where the length of each side is 10 cm?

The area of a regular hexagon where the length of each side is 10 cm is 259.81 cm^2.
Question

What is the human relationship betwixt area and side of an irregular hexagon?

There is no specific human relationship between them.
Question

How do I find an surface area of an equilateral triangle?

The formula is 0.433 multiplied by the square of any side of the triangle.
Question

How do I draw a regular hexagon with an expanse of 100 sq cm?

Find the perimeter and carve up information technology by the number of sides; the respond is the length of each side.

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Article Summary X

To summate the area of a hexagon, use the formula a = 3 × foursquare root of three × due south^two divided by ii, where a is the area and s is the length of a side of the hexagon. Merely plug in the length of one of the sides and then solve the formula to detect the expanse. If you lot don't take one of the side lengths but you do have the apothem, you lot tin can use the formula a = ane/ii × perimeter × apothem, where a is the expanse. To learn more than, similar how to summate the expanse of an irregular hexagon, keep reading the article!

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