What is the area of the 6th square? Regular hexagon: why it is interesting and how to build it

Is there a pencil near you? Take a look at its cross-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 giant 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 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 the regular hexagon has this property.
The angles are equal to each other, and each measure is 120°.
The perimeter of a hexagon can be found using the formula Р=6*R, if the radius of the circle described 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 circle.
The area occupied by a regular hexagon is determined as follows: S=(3*√(3)*R 2)/2. If the radius is unknown, substitute the length of one of the sides - as is known, it corresponds to the length of the radius of the circumscribed circle.

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

Now let's look at constructing a regular hexagon. There are several methods, the simplest of which involves using 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 angle of the compass, we place the tip at this point, mark the next notch on the circle, and continue this until we get all 6 points. Now all that remains is to connect them together with straight segments, and you will get the desired figure.

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

The most famous figure, which has more than four corners, is a regular hexagon. In geometry it is often used in problems. And in life, this is exactly what honeycombs look like when cut.

How is it different from the wrong one?

Firstly, a hexagon is a figure with 6 vertices. Secondly, it can be convex or concave. The first is different 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 meaning. 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. That is, each angle is equal to 120 degrees.

In everyday activities, the regular hexagon is found in the snowflake and 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 circle circumscribed around it;
Using such a figure, it is possible to fill the plane, and there will be no gaps between them and no overlaps.

Introduced designations

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

Some formulas include internal corner, semiperimeter and apothem (which is perpendicular to the middle of any side from the center of the polygon). The letters used for them are: α, р, m.

Formulas that describe a figure

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

Since the perimeter of a hexagon is the sum of all sides, it will be determined as follows: P = 6 * a. Taking into account the fact that the side is equal to the radius of the inscribed circle, for the perimeter there is the following formula for a regular hexagon: P = 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: P = 4 r * √3.

For the area of a regular hexagon, the following may be useful: S = p * r = (a 2 * 3 √3) / 2.

Tasks

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

Solution. Cylinder volume 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 will need to calculate the radius of the circle inscribed in the hexagon. The formula for this is given above. This means r = 2√3 (cm). Then the area of the circle: S = π * r 2 = 3.14 * (2√3) 2 = 37.68 (cm 2).

Answer. V = 150.72 cm 3.

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

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

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

Answer. r = 1.5 cm, P = 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 you need to calculate the side. It is clear that the radius is multiplied by two and divided by the root of three. It is necessary to get rid of irrationality in the denominator. Therefore, the result of the actions takes the following form: (12 √3) / (√3 * √3), that is, 4√3.

Answer. a = 4√3 cm.

A hexagon is a polygon with 6 sides and 6 corners. Depending on whether a hexagon is regular or not, there are several methods for finding its area. We'll look at everything.

How to find the area of a regular hexagon

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

Given side length:

Area formula: S = (3√3*a²)/2
If the length of 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.
An 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.

The apothem is given:

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

Trapezoid method:

Divide the hexagon into free 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 middle line.

The coordinates of the hexagon vertices are known:

First, let's write down the coordinates of the points, 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
We 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
We add up the results:
2 + 24 + 3 – 40 – 45 – 18 = -74
From the first result we subtract the second:
60 -(-74) = 60 + 74 = 134
Divide the resulting number by two:
134/2 = 67
Answer: 67 square units.

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

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

With a question: “How to find the area of a hexagon?”, you can encounter not only in a geometry exam, etc., this knowledge is also useful in everyday life, for example, for correctly and accurately calculating the area of a room during the renovation process. By substituting the required values into the formula, you will be able to determine the required number of rolls of wallpaper, tiles for the bathroom or kitchen, etc.

Some facts from history

Geometry has been used since ancient Babylon and other states that existed at the same time as him. Calculations helped in the construction of significant structures, since thanks to it the architects knew how to maintain the vertical, draw up a plan correctly, and determine the height.

Aesthetics also had great value, and here geometry came into play again. Today this science is needed by the builder, cutter, architect, and non-specialist too.

Therefore, it is better to be able to calculate S figures, to understand that the formulas can be useful in practice.

Area of a regular 6-gon

So we have hexagonal figure with equal sides and angles. In everyday life, we often have the opportunity to encounter objects of regular hexagonal shape.

For example:

screw;
honeycomb;
snowflake.

A hexagonal figure most economically fills space on a plane. Look at the paving slabs, one fits the other so that there are no gaps left.

Each angle is 120˚. The side of the figure is equal to the radius of the circumcircle.

Calculation

The required value can be calculated by dividing the figure into six triangles with equal sides.

Having calculated the S of one of the triangles, it is not difficult to determine the general one. A simple formula, since a regular hexagon is essentially six equal triangles. Thus, to calculate it, the found area of one triangle is multiplied by 6.

If you draw a perpendicular from the center of the hexagon to any of its sides, you get a segment - apothem.

Let's see how to find S of a hexagon if the apothem is known:

S =1/2×perimeter×apothem.
Let's take an apothem equal to 5√3 cm.

Find the perimeter using the apothem: since the apothem is perpendicular to the side of the hexagon, the angles of the triangle formed using the apothem are 30˚-60˚-90˚. Each side of the triangle corresponds to: x-x√3-2x, where the short side, opposite the 30˚ angle, is x; the long side against the angle 60˚ is x√3, and the hypotenuse is 2x.
The apothem x√3 can be substituted into the formula a=x√3. If the apothem is equal to 5√3, substituting this value, we get: 5√3cm=x√3, or x=5cm.
The short side of the triangle is 5 cm, since this value is half the length of the side of the hexagon. Multiplying 5 by 2, we get 10cm, which is the length of the side.
Let's multiply the resulting value by 6 and get the perimeter value - 60 cm.

We substitute the obtained results into the formula: S=1/2×perimeter×apothem

S=½×60 cm×5√3

We count:

We simplify the answer received to get rid of the roots. The result will be expressed in square centimeters: ½×60cm×5√3cm=30×5√3cm=150√3cm=259.8s m².

How to find the area of an irregular hexagon

There are several options:

Breaking down a 6-gon into other shapes.
Trapezoid method.
Calculation of S irregular polygons using coordinate axes.

The choice of method is dictated by the initial data.

Trapezoid method

The hexagon is divided into individual trapezoids, after which the area of each resulting figure is calculated.

Using Coordinate Axes

We use the coordinates of the polygon vertices:

We record the coordinates of the vertices x and y in the table. We select vertices sequentially, “moving” counterclockwise, completing the list by re-recording the coordinates of the first vertex.
We multiply the x coordinate values of the 1st vertex by the y value of the 2nd vertex, and continue to multiply in this way. Let's add up the results.
We multiply the coordinate values of the y1st vertex by the x coordinate values of the 2nd vertex. Let's add up the results.
Subtract the amount received in the 4th stage from the amount received in the third stage.
We divide the result obtained at the previous stage and find what we were looking for.

Breaking a hexagon into other shapes

Polygons are divided into other shapes: trapezoids, triangles, rectangles. Using formulas for calculating the areas of the listed figures, the required values are calculated and added.

An irregular hexagon can consist of two parallelograms. To calculate the area of a parallelogram, its length is multiplied by its width, and then the already known two areas are added.

Area of an equilateral hexagon

A regular hexagon has six equal sides. The area of an equilateral figure is equal to 6S triangles into which a regular hexagon is divided. Each triangle in a regular hexagon is equal, so to calculate the area of such a figure it is enough to know the area of at least one triangle.