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 tasks, 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.
With a question: How to find the area of a hexagon?, you can encounter not only in the geometry exam, etc., this knowledge will be useful in everyday life, for example, for the correct and accurate calculation of the area of \u200b\u200bthe room during the repair process. By substituting the required values into the formula, it will be possible to determine the required number of wallpaper rolls, tiles for the bathroom or kitchen, etc.
Some facts from history
Geometry was used as far back as ancient Babylon and other states that existed at the same time with him. Calculations helped in the construction of significant structures, because thanks to it, the architects knew how to maintain the vertical, correctly draw up a plan, and determine the height.
Aesthetics also had great importance, and here again geometry came into play. Today, this science is needed by a builder, cutter, architect, and not a specialist either.
Therefore, it is better to be able to calculate S figures, to understand that 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 meet objects of the correct hexagonal shape.
For example:
- screw;
- honeycombs;
- snowflake.
The hexagonal figure most economically fills the space on the plane. Take a look at the paving slabs, one fitted to the other so that there are no gaps.
Each angle is 120˚. The side of the figure is equal to the radius of the circumscribed circle.
Calculation
The required value can be calculated by dividing the figure into six triangles with equal sides.
Having calculated S of one of the triangles, it is easy to determine the general one. A simple formula, since a regular hexagon is, in fact, six equal triangles. Thus, to calculate it, the found area of \u200b\u200bone triangle is multiplied by 6.
If we draw a perpendicular from the center of the hexagon to any of its sides, we get a segment - apothem.
Let's see how to find the S of a hexagon if the apothem is known:
- S =1/2×perimeter×apothem.
- Let's take apothem equal to 5√3 cm.
- We find the perimeter using the apothem: since the apothem is perpendicular to the side of the 6-gon, the angles of the triangle formed with the apothem are 30˚-60˚-90˚. Each side of the triangle corresponds to: x-x√3-2x, where the short one, against the angle of 30˚, is x; the long side against the 60˚ angle 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 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 6-gon. Multiplying 5 by 2, we get 10 cm, which is the value of the length of the side.
- We multiply the resulting value by 6 and get the value of the perimeter - 60cm.
We substitute the results obtained into the formula: S=1/2×perimeter×apothem
S=½×60cm×5√3
We believe:
We simplify the answer 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:
- Breakdown of a 6-gon into other figures.
- trapezoid method.
- Calculation of S irregular polygons using coordinate axes.
The choice of method is dictated by the initial data.
Trapeze method
The hexagon is divided into separate trapezoids, after which the area of \u200b\u200beach resulting figure is calculated.
Using coordinate axes
We use the coordinates of the vertices of the polygon:
- We write down the coordinates of the vertices x and y in the table. We sequentially select vertices, "moving" counterclockwise, completing the list by re-recording the coordinates of the first vertex.
- Multiply the x value of the 1st vertex by the y value of the 2nd vertex, and keep multiplying. We summarize the results.
- We multiply the values of the y1-th vertex coordinates by the values of the x-coordinates of the 2nd vertex. We add up the results.
- Subtract the amount obtained in the 4th stage from the amount obtained in the third stage.
- We divide the result obtained at the previous stage and find what we were looking for.
Splitting a hexagon into other shapes
Polygons are broken down into other shapes: trapezoids, triangles, rectangles. Using the formulas for calculating the areas of the listed figures, the required values \u200b\u200bare calculated and added up.
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, therefore, to calculate the area of such a figure, it is sufficient to know the area of at least one triangle.
To find the desired value, use the area formula correct figure described above.
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 figure
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. Hence, 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.
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.
To find the area of a regular hexagon online using the formula you need, enter the numbers in the fields and click the "Calculate Online" button.
Attention! Dotted numbers (2.5) must be written with a dot(.), not a comma!
1. All angles of a regular hexagon are 120°
2. All sides of a regular hexagon are identical to each other
Regular hexagonal perimeter
4. The shape of the surface of a regular hexagon
5. Radius of the remote circle of a regular hexagon
6. Diameter of a round circle of a normal hexagon
7. Radius of the entered regular hexagonal circle
8. Relations between the radii of introduced and limited circles
like , and , and , from which a triangle follows - a right-angled one with a hypotenuse - is the same as . In this way,
10. The length of AB is
11. Sector Formula
Computing segment segments of a regular hexagon
Rice. 1. Regular hexagonal segments broken down into the same diamonds
1. The side of a regular hexagon is equal to the radius of the marked circle
2. Connecting dots with a hexagon, we get a series of equal rhombuses (Fig.
with squares
Rice. Segments of a regular hexagon broken down into the same triangles
3. Add a diagonal , , in rhombuses we get six identical triangles with surfaces
3. Segments of a normal hexagon divided into triangles
4. Since the normal hexagon is 120°, the area and they will be the same
5. Areas and we use the quadratic formula of a real triangle 
.
Considering that in our case the height is , but the basis is , we get it
Area of a normal hexagon This is the number that is characteristic of a regular hexagon in units of area.
Real hexagon (hexagon) This is a hexagon in which all pages and corners are the same.
[edit] Legend
Enter an entry:
— page length;
N- number of clients, n=6;
R Is the radius of the entered circle;
R This is the radius of the circle;
α - half of the central corner, α = π / 6;
P6- the size of a regular hexagon;
SΔ- the surface of an equal triangle with a base, equal to the side, and the sides are equal to the radius of the circle;
S6 This is the area of a normal hexagon.
[edit] Formulas
The formula is used for the area of a regular n-gon in n=6:
S_6=\frac(3a^2)(2)CTG\frac(\pi)(6)\Leftrightarrow\Leftrightarrow S_6=6S_(\triangle)\S_(\triangle)=\frac(e^2)( 4) CTG\frac(\pi)(6)\Leftrightarrow\Leftrightarrow S_6=\frac(1)(2)P_6r\P_6=\right(\math)(Math)\Leftrightarrow S_6=6R^2\sin\frac (\pi)(6)\cos\frac((pi)Frac(\pi)(6)\R=\frac(a)(2\sin\frac(\pi)(6))\Leftrightarrow\Leftrightarrow S_6 = 6r^2tg \frac(pi)(6),\r=R\cos\frac(\pi)(6)
Using trigonometric angle angles for corners α = π / 6:
S_6=\FRAC(3\sqrt(3))(2)^2\Leftrightarrow\Leftrightarrow S_6=6S_(\triangle)\S_(\triangle)=\FRAC(\sqrt(3))(4)^ 2\Leftrightarrow \Leftrightarrow S_6=\frac(1)(2)P_6r\P_6=6a,\r=\FRAC(\sqrt(3))(2)A\Leftrightarrow\Leftrightarrow S_6=\FRAC(3\sqrt( 3)) (2) R^2, \R=A\Leftrightarrow\\r=\frac(\sqrt(3))(2)R leftrightarrow S_6=2\sqrt(3)r^2
where (Math)\(pi\)sin\frac(6)=\frac(1)(2)\cos\frac(\pi)(6)=\FRAC(\sqrt(3))(2) , tg \frac(\pi)(6)=\frac(\sqrt(3))(3)pi)(6)=\sqrt(3)
[edit] Other polygons
Total Hexagon Area // KhanAcademyNussian
Bee bees become hexagonal without the help of bees
A typical mesh pattern can be made if the cells are triangular, square, or hexagonal.
The hexagonal shape is larger than the rest, allowing you to store on the walls, leaving less juice on the combs with such cages. For the first time this "economy" of bees was noted in IV. century. E. and at the same time it was suggested that the bees in the construction of clocks "should be controlled by a mathematical plan."
However, with researchers from Cardiff University, the bees' technical fame is greatly exaggerated: the correct geometric shape of the hexagonal honeycomb cell arises from the appearance of their physical strength and only helpers of insects.
Why is it transparent?
Mark Medovnik
Born from crystals?
Nikolai Yushkin
In their structure, the simplest simplest biosystems and hydrocarbon crystals are the simplest.
If such a mineral is supplemented with protein components, then we get a real proto-organism. Thus begins the beginning of the concept of crystallization of the origin of life.
Controversy about the structure of water
Malenkov G.G.
Controversies about the structure of water have been a matter of concern for decades in the scientific community as well as non-science people. This interest is not accidental: the structure of water is sometimes attributed to healing properties, and many believe that this structure can be controlled in some way. physical method or just the power of the mind.
And what is the opinion of scientists who have studied the mysteries of water in liquid and solid state for decades?
Honey and medical treatment
Stoymir Mladenov
Using the experience of other researchers and the results of experimental and clinical experimental studies, the author draws attention to the healing properties of bees and the method of its use in medicine as part of their capabilities.
To make this work more enduring in appearance and to enable the reader to gain a more holistic view of the economic and medical significance bees in the book will briefly discuss other bee products that are inextricably linked to the life of bees, namely bee venom, royal jelly, pollen, wax and propolis, as well as the relationship between science and these products.
Caustics in the plane and in the universe
Caustics are all-encompassing optical surfaces and curves that occur when light is reflected and destroyed.
Caustics can be described as lines or surfaces with a concentrated beam of light.
How does a transistor work?
They are everywhere: in every electrical appliance, from the TV to the old Tamagotchi.
We don't know anything about them because we perceive them as reality. But without them, the world would have completely changed. Semiconductors. About what it is and how it works.
Let the cockroach turn out to be turbulent
An international team of scientists has determined how easy it is for flies to fly in very windy conditions. It turned out that even under conditions of significant impacts, a special mechanism for creating lift forces allows insects to stay on the move with minimal additional energy costs.
The mechanism of self-organization of nanocrystals of carbonates and silicates in the biomorphic structure has been established
Elena Naimark
Spanish scientists have discovered a mechanism that can cause the spontaneous formation of carbonate and silicate crystals of a very complex and unusual shape.
These crystalline neoplasms are similar to biomorphs - inorganic structures obtained with the participation of living organisms. And the mechanism leading to such mimicry is surprisingly simple - it is only a spontaneous fluctuation of the pH of a solution of carbonates and silicates at the boundary between a solid crystal and a liquid medium that is formed.
False High Pressure Samples
Komarov S.M.
with what formula to find the area of a regular hexagon from page 2?
- these are six one-sided triangles with a side of 2
the surface of an equilateral triangle is a and the square root is 3 divided by 4, where a = 2 - The area of the tower is 12 * the base of the height. A hexagon is a hexagonal polygon divided into six equal triangles.
all equilateral triangles with 60 degree angle and side 2 cm find height of pythagorean theorem 2 in squares = 1 height of square per square root so height = 3S = 12 * 2 * 3 + square root square root of 3 hours TP 6 means 6 roots of 3
- A feature of a regular hexagon is the equality of its side t and the radius of the remote circle (R = t).
The normal area of a hexagon is calculated using the equation:
Real hexagon
- The normal area of a hexagon is 3x for the square root. 3 x R2 / 2, where R is the radius of the circle around it. In a regular hexagon, there is the same side of the hexagon = 2, then the area will be equal to the square of the root 6x. from 3.
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