The area of an equilateral hexagon formula. Regular hexagon

Do you know what a regular hexagon looks like?
This question was not asked by chance. Most students in grade 11 do not know the answer to it.

A regular hexagon is one in which all sides are equal and all angles are also equal..

Iron nut. Snowflake. A cell of honeycombs in which bees live. Benzene molecule. What do these objects have in common? - The fact that they all have a regular hexagonal shape.

Many schoolchildren are lost when they see tasks for a regular hexagon, and they believe that some special formulas are needed to solve them. Is it so?

Draw the diagonals of a regular hexagon. We got six equilateral triangles.

We know that the area of an equilateral triangle is .

Then the area of a regular hexagon is six times larger.

Where is the side of a regular hexagon.

Please note that in a regular hexagon, the distance from its center to any of the vertices is the same and equal to the side of the regular hexagon.

This means that the radius of a circle circumscribed around a regular hexagon is equal to its side.
The radius of a circle inscribed in a regular hexagon is easy to find.
He is equal.
Now you can easily solve any USE problems in which a regular hexagon appears.

Find the radius of a circle inscribed in a regular hexagon with side .

The radius of such a circle is .

Answer: .

What is the side of a regular hexagon inscribed in a circle with a radius of 6?

We know that the side of a regular hexagon is equal to the radius of the circle circumscribed around it.

The most famous figure with more than four corners is the regular hexagon. In geometry, it is often used in problems. And in life, this is exactly what honeycombs have on the cut.

How is it different from wrong?

First, a hexagon is a figure with 6 vertices. Secondly, it can be convex or concave. The first one differs in that four vertices lie on one side of a straight line drawn through the other two.

Thirdly, a regular hexagon is characterized by the fact that all its sides are equal. Moreover, each corner of the figure also has the same value. To determine the sum of all its angles, you will need to use the formula: 180º * (n - 2). Here n is the number of vertices of the figure, that is, 6. A simple calculation gives a value of 720º. So each angle is 120 degrees.

In everyday activities, a regular hexagon is found in a snowflake and a nut. Chemists see it even in the benzene molecule.

What properties do you need to know when solving problems?

To what is stated above should be added:

the diagonals of the figure, drawn through the center, divide it into six triangles, which are equilateral;
the side of a regular hexagon has a value that coincides with the radius of the circumscribed circle around it;
using such a figure, it is possible to fill the plane, and between them there will be no gaps and no overlaps.

Introduced notation

Traditionally, the side of a regular geometric figure is denoted by the Latin letter "a". To solve problems, area and perimeter are also required, these are S and P, respectively. A circle is inscribed in a regular hexagon or circumscribed about it. Then values for their radii are entered. They are denoted respectively by the letters r and R.

In some formulas, an internal angle, a semi-perimeter and an apothem (which is a perpendicular to the middle of any side from the center of the polygon) appear. Letters are used for them: α, p, m.

Formulas that describe a shape

To calculate the radius of an inscribed circle, you need this: r= (a * √3) / 2, and r = m. That is, the same formula will be for the apothem.

Since the perimeter of a hexagon is the sum of all sides, it will be determined as follows: P = 6 * a. Given that the side is equal to the radius of the circumscribed circle, for the perimeter there is such a formula for a regular hexagon: P \u003d 6 * R. From the one given for the radius of the inscribed circle, the relationship between a and r is derived. Then the formula takes the following form: Р = 4 r * √3.

For the area of a regular hexagon, this might come in handy: S = p * r = (a 2 * 3 √3) / 2.

Tasks

No. 1. Condition. There is a regular hexagonal prism, each edge of which is equal to 4 cm. A cylinder is inscribed in it, the volume of which must be determined.

Solution. The volume of a cylinder is defined as the product of the area of the base and the height. The latter coincides with the edge of the prism. And it is equal to the side of a regular hexagon. That is, the height of the cylinder is also 4 cm.

To find out the area of its base, you need to calculate the radius of the circle inscribed in the hexagon. The formula for this is shown above. So r = 2√3 (cm). Then the area of the circle: S \u003d π * r 2 \u003d 3.14 * (2√3) 2 \u003d 37.68 (cm 2).

Answer. V \u003d 150.72 cm 3.

No. 2. Condition. Calculate the radius of a circle that is inscribed in a regular hexagon. It is known that its side is √3 cm. What will be its perimeter?

Solution. This task requires the use of two of the above formulas. Moreover, they must be applied without even modifying, just substitute the value of the side and calculate.

Thus, the radius of the inscribed circle turns out to be 1.5 cm. For the perimeter, the following value turns out to be correct: 6√3 cm.

Answer. r = 1.5 cm, Р = 6√3 cm.

No. 3. Condition. The radius of the circumscribed circle is 6 cm. What value will the side of a regular hexagon have in this case?

Solution. From the formula for the radius of a circle inscribed in a hexagon, one easily obtains the one by which the side must be calculated. It is clear that the radius is multiplied by two and divided by the root of three. It is necessary to get rid of the irrationality in the denominator. Therefore, the result of actions takes the following form: (12 √3) / (√3 * √3), that is, 4√3.

Answer. a = 4√3 cm.

Is there a pencil near you? Take a look at its section - it is a regular hexagon or, as it is also called, a hexagon. The cross section of a nut, a field of hexagonal chess, some complex carbon molecules (for example, graphite), a snowflake, a honeycomb and other objects also have this shape. A gigantic regular hexagon was recently discovered in. Doesn't it seem strange that nature so often uses structures of this particular shape for its creations? Let's take a closer look.

A regular hexagon is a polygon with six equal sides and equal angles. From the school course, we know that it has the following properties:

The length of its sides corresponds to the radius of the circumscribed circle. Of all, only a regular hexagon has this property.
The angles are equal to each other, and the magnitude of each is 120 °.
The perimeter of a hexagon can be found using the formula Р=6*R if the radius of the circle circumscribed around it is known, or Р=4*√(3)*r if the circle is inscribed in it. R and r are the radii of the circumscribed and inscribed circles.
The area occupied by a regular hexagon is determined as follows: S=(3*√(3)*R 2)/2. If the radius is unknown, instead of it we substitute the length of one of the sides - as you know, it corresponds to the length of the radius of the circumscribed circle.

The regular hexagon has one interesting feature due to which it has become so widespread in nature - it is able to fill any surface of the plane without overlaps and gaps. There is even the so-called Pal lemma, according to which a regular hexagon whose side is equal to 1/√(3) is a universal tire, that is, it can cover any set with a diameter of one unit.

Now consider the construction of a regular hexagon. There are several ways, the easiest of which involves the use of a compass, pencil and ruler. First, we draw an arbitrary circle with a compass, then we make a point in an arbitrary place on this circle. Without changing the solution of the compass, we put the tip at this point, mark the next notch on the circle, and continue until we get all 6 points. Now it remains only to connect them with each other with straight segments, and the desired figure will turn out.

In practice, there are times when you need to draw a large hexagon. For example, on a two-level plasterboard ceiling, around the attachment point of the central chandelier, you need to install six small lamps at the lower level. It will be very, very difficult to find a compass of this size. How to proceed in this case? How do you draw a big circle? Very simple. You need to take a strong thread of the desired length and tie one of its ends opposite the pencil. Now it remains only to find an assistant who would press the second end of the thread to the ceiling at the right point. Of course, in this case, minor errors are possible, but they are unlikely to be noticeable to an outsider at all.

A hexagon is a polygon with 6 sides and 6 angles. Depending on whether a hexagon is regular or not, there are several methods for finding its area. We will review everything.

How to find the area of a regular hexagon

Formulas for calculating the area of a regular hexagon - a convex polygon with six identical sides.

Given side length:

Area formula: S = (3√3*a²)/2
If the length of the side a is known, then substituting it into the formula, we can easily find the area of the figure.
Otherwise, the length of the side can be found through the perimeter and apothem.
If the perimeter is given, then we simply divide it by 6 and get the length of one side. For example, if the perimeter is 24, then the side length will be 24/6 = 4.
Apothem is a perpendicular drawn from the center to one of the sides. To find the length of one side, we substitute the length of the apothem into the formula a = 2*m/√3. That is, if the apothem m = 2√3, then the length of the side a = 2*2√3/√3 = 4.

Given an apothem:

Area formula: S = 1/2*p*m, where p is the perimeter, m is the apothem.
Let us find the perimeter of the hexagon through the apothem. In the previous paragraph, we learned how to find the length of one side through an apothem: a \u003d 2 * m / √3. It remains only to multiply this result by 6. We get the perimeter formula: p \u003d 12 * m / √3.

Given the radius of the circumscribed circle:

The radius of a circle circumscribed around a regular hexagon is equal to the side of this hexagon.
Area formula: S = (3√3*a²)/2

Given the radius of the inscribed circle:

Area formula: S = 3√3*r², where r = √3*a/2 (a is one of the sides of the polygon).

How to find the area of an irregular hexagon

Formulas for calculating the area of an irregular hexagon - a polygon whose sides are not equal to each other.

Trapeze method:

We divide the hexagon into arbitrary trapezoids, calculate the area of each of them and add them up.
Basic formulas for the area of a trapezoid: S = 1/2*(a + b)*h, where a and b are the bases of the trapezoid, h is the height.
S = h*m, where h is the height, m is the midline.

The coordinates of the vertices of the hexagon are known:

To begin with, let's write down the coordinates of the points, moreover, placing them not in a chaotic order, but sequentially one after another. For example:
A: (-3, -2)
B: (-1, 4)
C: (6, 1)
D: (3, 10)
E: (-4, 9)
F: (-5, 6)
Next, carefully, multiply the x-coordinate of each point by the y-coordinate of the next point:
-3*4 = -12
-1*1 = -1
6*10 = 60
3*9 = 27
-4*6 = -24
-5*(-2) = 10
Add up the results:
-12 – 1 + 60 + 27 – 24 + 10 = 60
Next, multiply the y-coordinate of each point by the x-coordinate of the next point.
-2*(-1) = 2
4*6 = 24
1*3 = 3
10*(-4) = -40
9*(-5) = -45
6*(-3) = -18
Add up the results:
2 + 24 + 3 – 40 – 45 – 18 = -74
Subtract the second from the first result:
60 -(-74) = 60 + 74 = 134
The resulting number is divided by two:
134/2 = 67
Answer: 67 square units.

Also, to find the area of a hexagon, you can break it into triangles, squares, rectangles, parallelograms, and so on. Find the areas of its constituent figures and add them up.

So, the methods for finding the area of a hexagon for all occasions have been studied. Now go ahead and apply what you've learned! Good luck!

The area of ​​an equilateral hexagon formula. Regular hexagon