Calculation of the hexagon formula. How to find the area of a hexagon

The topic of polygons is covered in the school curriculum, but they do not pay enough attention to it. Meanwhile, it is interesting, and this is especially true of a regular hexagon or hexagon - after all, many natural objects have this shape. These include honeycombs and more. This form is very well applied in practice.

Definition and construction

A regular hexagon is a plane figure that has six sides equal in length and the same number of equal angles.

If we recall the formula for the sum of the angles of a polygon

it turns out that in this figure it is equal to 720 °. Well, since all the angles of the figure are equal, it is easy to calculate that each of them is equal to 120 °.

Drawing a hexagon is very simple, all you need is a compass and a ruler.

The step by step instructions will look like this:

If desired, you can do without a line by drawing five circles of equal radius.

The figure thus obtained will be a regular hexagon, and this can be proved below.

Properties are simple and interesting

To understand the properties of a regular hexagon, it makes sense to break it into six triangles:

This will help in the future to more clearly display its properties, the main of which are:

circumscribed circle diameter;
diameter of the inscribed circle;
area;
perimeter.

The circumscribed circle and the possibility of construction

It is possible to describe a circle around a hexagon, and moreover, only one. Since this figure is correct, you can do it quite simply: draw a bisector from two adjacent angles inside. They intersect at point O, and together with the side between them form a triangle.

The angles between the side of the hexagon and the bisectors will be 60° each, so we can definitely say that a triangle, for example, AOB, is isosceles. And since the third angle will also be equal to 60 °, it is also equilateral. It follows that the segments OA and OB are equal, which means that they can serve as the radius of the circle.

After that, you can go to the next side, and also draw a bisector from the angle at point C. It will turn out another equilateral triangle, and side AB will be common to two at once, and OS will be the next radius through which the same circle goes. There will be six such triangles in total, and they will have a common vertex at point O. It turns out that it will be possible to describe the circle, and it is only one, and its radius is equal to the side of the hexagon:

That is why it is possible to construct this figure with the help of a compass and a ruler.

Well, the area of \u200b\u200bthis circle will be standard:

Inscribed circle

The center of the circumscribed circle coincides with the center of the inscribed one. To verify this, we can draw perpendiculars from the point O to the sides of the hexagon. They will be the heights of those triangles that make up the hexagon. And in an isosceles triangle, the height is the median with respect to the side on which it rests. Thus, this height is nothing but the perpendicular bisector, which is the radius of the inscribed circle.

The height of an equilateral triangle is calculated simply:

h²=a²-(a/2)²= a²3/4, h=a(√3)/2

And since R=a and r=h, it turns out that

r=R(√3)/2.

Thus, the inscribed circle passes through the centers of the sides of a regular hexagon.

Its area will be:

S=3πa²/4,

that is, three-quarters of that described.

Perimeter and area

Everything is clear with the perimeter, this is the sum of the lengths of the sides:

P=6a, or P=6R

But the area will be equal to the sum of all six triangles into which the hexagon can be divided. Since the area of a triangle is calculated as half the product of the base and the height, then:

S \u003d 6 (a / 2) (a (√3) / 2) \u003d 6a² (√3) / 4 \u003d 3a² (√3) / 2 or

S=3R²(√3)/2

Those who wish to calculate this area through the radius of the inscribed circle can be done like this:

S=3(2r/√3)²(√3)/2=r²(2√3)

Entertaining constructions

A triangle can be inscribed in a hexagon, the sides of which will connect the vertices through one:

There will be two of them in total, and their imposition on each other will give the Star of David. Each of these triangles is equilateral. This is easy to verify. If you look at the AC side, then it belongs to two triangles at once - BAC and AEC. If in the first of them AB \u003d BC, and the angle between them is 120 °, then each of the remaining ones will be 30 °. From this we can draw logical conclusions:

The height of ABC from vertex B will be equal to half the side of the hexagon, since sin30°=1/2. Those who wish to verify this can be advised to recalculate according to the Pythagorean theorem, it fits here perfectly.
The AC side will be equal to two radii of the inscribed circle, which is again calculated using the same theorem. That is, AC=2(a(√3)/2)=а(√3).
Triangles ABC, CDE and AEF are equal in two sides and the angle between them, and hence the equality of sides AC, CE and EA follows.

Intersecting with each other, the triangles form a new hexagon, and it is also regular. It's easy to prove:

Thus, the figure meets the signs of a regular hexagon - it has six equal sides and angles. From the equality of triangles at the vertices, it is easy to deduce the length of the side of the new hexagon:

d=а(√3)/3

It will also be the radius of the circle described around it. The radius of the inscribed will be half the side of the large hexagon, which was proved when considering the triangle ABC. Its height is exactly half of the side, therefore, the second half is the radius of the circle inscribed in the small hexagon:

r₂=а/2

S=(3(√3)/2)(а(√3)/3)²=а(√3)/2

It turns out that the area of the hexagon inside the star of David is three times smaller than that of the large one in which the star is inscribed.

From theory to practice

The properties of the hexagon are very actively used both in nature and in various fields of human activity. First of all, this applies to bolts and nuts - the hats of the first and second are nothing more than a regular hexagon, if you do not take into account the chamfers. The size of wrenches corresponds to the diameter of the inscribed circle - that is, the distance between opposite faces.

Has found its application and hexagonal tiles. It is much less common than a quadrangular one, but it is more convenient to lay it: three tiles meet at one point, not four. Compositions can be very interesting:

Concrete paving slabs are also produced.

The prevalence of the hexagon in nature is explained simply. Thus, it is easiest to fit circles and balls tightly on a plane if they have the same diameter. Because of this, honeycombs have such a shape.

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Theory. Area of a quadrilateral A quadrilateral is a geometric figure consisting of four points (vertices), no three of which lie on the same straight line, and four segments (sides) connecting these points in pairs. A quadrilateral is called convex if the segment connecting any two points of this quadrilateral will be inside it.

How to find the area of a polygon?

The formula for determining the area is determined by taking each edge of the polygon AB, and calculating the area of the triangle ABO with a vertex at the origin O, through the coordinates of the vertices. When walking around a polygon, triangles are formed, including the inside of the polygon and located outside it. The difference between the sum of these areas is the area of the polygon itself.

Therefore, the formula is called the surveyor's formula, since the "cartographer" is at the origin; if it walks the area counterclockwise, the area is added if it's on the left and subtracted if it's on the right in terms of the origin. The area formula is valid for any non-intersecting (simple) polygon, which may be convex or concave. Content

1 Definition
2 Examples
3 More complex example
4 Name explanation
5 See

Polygon area

Attention

It could be:

triangle;
quadrilateral;
five- or hexagon and so on.

Such a figure will certainly be characterized by two positions:

Adjacent sides do not belong to the same line.
Non-adjacent ones have no common points, that is, they do not intersect.

To understand which vertices are adjacent, you need to see if they belong to the same side. If yes, then neighboring. Otherwise, they can be connected by a segment, which must be called a diagonal. They can only be drawn in polygons that have more than three vertices.

What kinds of them exist? A polygon with more than four corners can be convex or concave. The difference of the latter is that some of its vertices may lie on different sides of a straight line drawn through an arbitrary side of the polygon.

How to find the area of a regular and irregular hexagon?

Knowing the length of the side, multiply it by 6 and get the perimeter of the hexagon: 10 cm x 6 \u003d 60 cm
Substitute the results in our formula:

trapezoid method.
A method for calculating the area of irregular polygons using the coordinate axis.
A method for splitting a hexagon into other shapes.

Depending on the initial data that you will know, the appropriate method is selected.

Important

Some irregular hexagons consist of two parallelograms. To determine the area of a parallelogram, multiply its length by its width and then add the two already known areas. Video on how to find the area of a polygon An equilateral hexagon has six equal sides and is a regular hexagon.

The area of an equilateral hexagon is equal to 6 areas of the triangles into which a regular hexagonal figure is divided. All triangles in a regular hexagon are equal, so to find the area of such a hexagon, it will be enough to know the area of at least one triangle. To find the area of an equilateral hexagon, of course, the formula for the area of a regular hexagon, described above, is used.

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Decorating a home, clothing, drawing pictures contributed to the process of formation and accumulation of information in the field of geometry, which people of those times obtained empirically, bit by bit and passed on from generation to generation. Today, knowledge of geometry is necessary for a cutter, a builder, an architect, and every ordinary person in everyday life. Therefore, you need to learn how to calculate the area of \u200b\u200bdifferent figures, and remember that each of the formulas can be useful later in practice, including the formula for a regular hexagon.
A hexagon is such a polygonal figure, the total number of angles of which is six. A regular hexagon is a hexagonal figure that has equal sides. The angles of a regular hexagon are also equal to each other.
In everyday life, we can often find objects that have the shape of a regular hexagon.

Irregular polygon area calculator by sides

You will need

- roulette;
— electronic rangefinder;
- a sheet of paper and a pencil;
- calculator.

Instruction 1 If you need the total area of an apartment or a separate room, just read the technical passport for the apartment or house, it shows the footage of each room and the total footage of the apartment. 2 To measure the area of a rectangular or square room, take a tape measure or an electronic rangefinder and measure the length of the walls. When measuring distances with a rangefinder, be sure to keep the beam direction perpendicular, otherwise the measurement results may be distorted. 3 Then multiply the resulting length (in meters) of the room by the width (in meters). The resulting value will be the floor area, it is measured in square meters.

Gauss area formula

If you need to calculate the floor area of a more complex structure, such as a pentagonal room or a room with a round arch, sketch a schematic sketch on a piece of paper. Then divide the complex shape into several simple ones, such as a square and a triangle, or a rectangle and a semicircle. Use a tape measure or rangefinder to measure the size of all sides of the resulting figures (for a circle, you need to know the diameter) and enter the results on your drawing.

5 Now calculate the area of each shape separately. The area of rectangles and squares is calculated by multiplying the sides. To calculate the area of a circle, divide the diameter in half and square (multiply it by itself), then multiply the result by 3.14.
If you only want half of the circle, divide the resulting area in half. To calculate the area of a triangle, find P by dividing the sum of all sides by 2.

Formula for calculating the area of an irregular polygon

If the points are numbered sequentially in a counterclockwise direction, then the determinants in the formula above are positive and the modulus in it can be omitted; if they are numbered in a clockwise direction, the determinants will be negative. This is because the formula can be viewed as a special case of Green's theorem. To apply the formula, you need to know the coordinates of the polygon vertices in the Cartesian plane.

For example, let's take a triangle with coordinates ((2, 1), (4, 5), (7, 8)). Take the first x-coordinate of the first vertex and multiply it by the y-coordinate of the second vertex, and then multiply the x-coordinate of the second vertex by the y-coordinate of the third. We repeat this procedure for all vertices. The result can be determined by the following formula: A tri.

The formula for calculating the area of an irregular quadrilateral

A) _(\text(tri.))=(1 \over 2)|x_(1)y_(2)+x_(2)y_(3)+x_(3)y_(1)-x_(2) y_(1)-x_(3)y_(2)-x_(1)y_(3)|) where xi and yi denote the corresponding coordinate. This formula can be obtained by opening the brackets in the general formula for the case n = 3. Using this formula, you can find that the area of a triangle is equal to half the sum of 10 + 32 + 7 - 4 - 35 - 16, which gives 3. The number of variables in the formula depends on the number of sides of the polygon. For example, the formula for the area of a pentagon will use variables up to x5 and y5: A pent. = 1 2 | x 1 y 2 + x 2 y 3 + x 3 y 4 + x 4 y 5 + x 5 y 1 − x 2 y 1 − x 3 y 2 − x 4 y 3 − x 5 y 4 − x 1 y 5 | (\displaystyle \mathbf (A) _(\text(pent.))=(1 \over 2)|x_(1)y_(2)+x_(2)y_(3)+x_(3)y_(4 )+x_(4)y_(5)+x_(5)y_(1)-x_(2)y_(1)-x_(3)y_(2)-x_(4)y_(3)-x_(5 )y_(4)-x_(1)y_(5)|) A for a quad - variables up to x4 and y4: A quad.

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, we substitute the length of one of the sides instead of it - 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, continue this way 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.

Mathematical properties

A feature of a regular hexagon is the equality of its side and the radius of the circumscribed circle, since

All angles are 120°.

The radius of the inscribed circle is:

The perimeter of a regular hexagon is:

The area of a regular hexagon is calculated by the formulas:

Hexagons tiling the plane, that is, they can fill the plane without gaps and overlaps, forming the so-called parquet.

Hexagonal parquet (hexagonal parquet)- tessellation of the plane with equal regular hexagons located side to side.

Hexagonal parquet is dual to triangular parquet: if you connect the centers of adjacent hexagons, then the segments drawn will give a triangular parquet. The Schläfli symbol of a hexagonal parquet is (6,3), which means that three hexagons converge at each vertex of the parquet.

Hexagonal parquet is the most dense packing of circles on the plane. In two-dimensional Euclidean space, the best filling is to place the centers of the circles at the vertices of a parquet formed by regular hexagons, in which each circle is surrounded by six others. The density of this packing is . In 1940, it was proved that this packing is the densest.

A regular hexagon with a side is a universal cover, that is, any set of diameter can be covered by a regular hexagon with a side (Pal's lemma).

A regular hexagon can be constructed using a compass and straightedge. Below is the construction method proposed by Euclid in the Elements, Book IV, Theorem 15.

Regular hexagon in nature, technology and culture

show the partition of the plane into regular hexagons. The hexagonal shape more than the others allows you to save on the walls, that is, less wax will be spent on honeycombs with such cells.

Some complex crystals and molecules, such as graphite, have a hexagonal crystal lattice.

Formed when microscopic water droplets in clouds are attracted to dust particles and freeze. The ice crystals that appear in this case, which at first do not exceed 0.1 mm in diameter, fall down and grow as a result of condensation of moisture from the air on them. In this case, six-pointed crystalline forms are formed. Due to the structure of water molecules, only 60° and 120° angles are possible between the rays of the crystal. The main water crystal has the shape of a regular hexagon in the plane. New crystals are then deposited on the tops of such a hexagon, new ones are deposited on them, and thus various forms of snowflake stars are obtained.

Scientists from Oxford University were able to simulate the emergence of such a hexagon in the laboratory. To find out how such a formation occurs, the researchers placed a 30-liter bottle of water on a turntable. She modeled the atmosphere of Saturn and its usual rotation. Inside, scientists placed small rings that rotate faster than the container. This generated miniature eddies and jets, which the experimenters visualized with green paint. The faster the ring rotated, the larger the eddies became, causing the nearby stream to deviate from a circular shape. Thus, the authors of the experiment managed to obtain various shapes - ovals, triangles, squares and, of course, the desired hexagon.

A natural monument of about 40,000 interconnected basalt (rarely andesitic) columns, formed as a result of an ancient volcanic eruption. Located in the north-east of Northern Ireland, 3 km north of the city of Bushmills.

The tops of the columns form a kind of springboard, which starts at the foot of the cliff and disappears under the surface of the sea. Most of the columns are hexagonal, although some have four, five, seven or eight corners. The tallest column is about 12 meters high.

About 50-60 million years ago, during the Paleogene period, the Antrim site was subject to intense volcanic activity when molten basalt permeated through the deposits, forming extensive lava plateaus. With rapid cooling, the volume of the substance decreased (this is observed when the mud dries). Horizontal compression resulted in the characteristic structure of hexagonal pillars.

The cross section of the nut has the form of a 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.