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How Many Sides Does A Hexagon Have? Definition, Types, And Properties Of Hexagon

What is a hexagon? How many sides does a hexagon have? A hexagon is a closed two-dimensional shape composed of six straight lines. It has 6 sides, 6 vertices, and 6 interior angles. The name is made up of the words ‘hex’, which means six, and ‘gonia,’ which means corners. In this article, we will go over the hexagon shape in depth. Let’s get started!

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What is a hexagon?

Hexagons are 2-dimensional geometrical shapes with 6 sides that have the same or different length dimensions. A hexagonal floor tile, pencil cross-section, snowflakes, honeycomb, and other real-life examples of the hexagon shape can be observed. It may be a regular hexagon (with 6 equal side lengths and angles) or an irregular hexagon (with 6 unequal side lengths and angles).

Types of Hexagon

Hexagon shapes are classified into the following types:

• Regular hexagon
• Irregular hexagon
• Convex hexagons
• Concave hexagons

Hexagon classification is based on their sides.

What is a regular hexagon shape?

A regular hexagon 2D geometric polygon with 6 sides of equal length and 6 angles of equal size. It doesn’t have curved sides and all lines are straight. A regular hexagon’s internal angles total 720 degrees. There are 6 rotational symmetries and 6 reflectional symmetries in these shapes.

The image below depicts a regular hexagon:

A regular hexagon must include three characteristics:

• All hexagonal sides must be the same length.
• Each of the interior angles must be 120 degrees.
• When all of the interior angles are added together, they must equal 720 degrees.

Use these Hexagon Shape Worksheets to support your students develop a sixth sense for these 6-sided shapes.

Here is some interesting fact about regular hexagon:

A hexagon can be divided into 6 equilateral triangles. These will all be of the same size and will be able to be reassembled like a honeycomb. Tessellation occurs when we do this with a large number of hexagons.

What is an irregular hexagon shape?

An irregular hexagon also has 6 sides and 6 angles of varying lengths and sizes. This can result in some odd-looking shapes that you might not immediately recognize as hexagons. There are tons of ways to arrange six sides to form a hexagon.

Here are a few examples of irregular hexagons.

Hexagonal classification based on angles

Convex Hexagon

A convex hexagon is characterized by:

• Interior angles are smaller than 180 degrees.
• Angles with unequal lengths and sides
• There are no symmetrical lines.
• There are no inward pointed angles.
• The vertices are pointing outwards.
• It looks like a regular hexagon

Hexagon Concave

A concave hexagon always has:

• Angles greater than 180° are found in a concave hexagon.
• There is only one line of symmetry across its center
• At least an angle points inward.

Sides of a Hexagon

A hexagon has 6 sides, as shown in the diagram above. All of the edges are straight and create a closed shape. A regular hexagon has six equal sides, whereas an irregular hexagon has at least two sides that are different in length. The perimeter of the hexagon can be calculated by adding the sums of all six sides.

If the perimeter of a regular hexagon is given, we can determine the length of each side as «Perimeter 6». For example, if the perimeter of a regular hexagon is 60 units, then each hexagon side is  60 ÷ 6 = 10 units long.

Read more >> What Is A Triangular Prism? Definition, And Properties Of Triangular Prism

Angles of Hexagon

The properties of hexagonal angles are as follows:

• Each hexagon includes 6 exterior and 6 interior angles.
• The total value of all hexagon angles is 720°.
• The total value of the exterior angles is 360°.
• All Irregular hexagon 2 angles are different in measurements.
• The measurement of each interior angle in a regular hexagon is 120°
• The measurement of each exterior angle in a regular hexagon is 60°

Diagonals of a Hexagon

A diagonal is a line segment that links 2 non-adjacent vertices of a polygon. The total of polygon diagonals is calculated by n(n-3)/2, where ‘n’ represents the number of polygon sides. The number of diagonals in a hexagon is calculated as 6(6 — 3)/2 = 6(3)/2, which equals 9. Three of the nine diagonals pass through the hexagon’s center.

In a hexagon, there are 2 kinds of diagonals: long diagonals (3 diagonals that pass through the center) and short diagonals. The formula ‘2s’ can be used to calculate the length of a long diagonal in a regular hexagon. And the length of a short diagonal can be calculated using the formula ‘3s,’ where s represents the length of each hexagonal side.

Properties of Hexagon

• The hexagon’s significant properties are as follows:
• It has a two-dimensional flat shape.
• It has 6 sides, 6 edges, and 6 vertices.
• Each side length is either equal or unequal in length. Not all hexagons include equal angles or sides.
• The total value of its internal angles is 720°.
• The total value of its external angles is 360°.
• A regular hexagon’s internal and external angles are all equal to 120° and 60°, respectively.
• Hexagons have 2 types of diagonals: one long and one short.
• A convex hexagon is similar to a regular hexagon. They have internal angles that are less than 180°.
• The opposite sides of a regular hexagon are always parallel to each other.
• To calculate the perimeter of a hexagon, sum up the lengths of all its sides.
• A hexagon can be easily divided into equilateral triangles (six).
• There are nine diagonals in total.
• Arbitrary hexagons are hexagons with unequal-length sides.

Where can I find hexagons in real life?

Do you believe that a specific shape, such as a hexagon, could appear in the world? Most shapes can be seen in everyday life if you take a closer look. Here are a few places you might come across hexagons.

• Honeycombs: Regular hexagons are 1 of 3 polygons that tesselate a plane, which means they can be duplicated indefinitely to fill a space with no gaps. When bees create honeycombs, they prefer hexagons. Always!
• Pencils: You may not have noticed, but regular pencils are hexagonal in shape. This is because it will save space when storing them, simplifies the process of to grip, and makes gluing 2 halves together easier when manufacturing them.
• Snowflakes: Did you realize that all snowflakes are hexagonal? When ice crystals form, the molecules align in a hexagonal pattern. Mother Nature has defined that this type of structure is the most efficient way for snowflakes to form.
• Saturn: You may be wondering what a planet has to do with hexagons. Isn’t a planet circular? You’re correct, but if you look at Saturn’s equator, you’ll notice a massive hexagon shape. It is even larger than the Earth! Scientists believe this is a massive storm with numerous pressure points. The gasses form a hexagon shape as a result of these points.

Hexagon FAQs

What are the various types of hexagons?

Hexagons are classified into four types. There are four types of hexagons: regular hexagons, irregular hexagons, concave hexagons, and concave hexagons.

How many sides does a hexagon have?

It is made up of six sides, six edges, and six vertices.

How many degrees are there in a hexagon?

The total value of a hexagon’s interior angles is 720°. All sides of a regular hexagon have the same length, and all interior angles are equal. As a result, each interior angle equals 720/6 = 120°. Thus, the interior angle in a regular hexagon is 120°.

Hope that with the above information, you can find the best way to help your kids grasp the answer to the question “How many sides does a hexagon have?” If you are planning to teach your kids about this essential topic, you can make your own collections of hexagon worksheets using our worksheet maker. 

Hexagon – How Many Sides Does A Hexagon Have?

March 23, 2023 by Nguyen Thanh Son

how many sides does a hexagon have

how many sides does a hexagon have

The two-dimensional shaped Hexagon is used in geometry. We find Hexagon shaped things everywhere, around us. Its name explains its sides quite clearly, such as:

The root word ‘Hex’ means six, a Greek prefix.

‘Gonia’ means corners.

Collectively it means six corners. Isn’t it easy to remember this 2D shape with six angles, six sides, six straight lines, six angles? To learn more about this closed 2D shape and discover what differentiates it from other polygons, give this blog post a read. We have shared the most important information about Hexagons here.

A closed shape with six sides, six equal lengths, six interior angles, and six vertices is known as Hexagon. The sum of its total internal angles is 720°.

According to math tutors and professors, the most asked question during geometry class is how many sides a hexagon has or do all hexagons have six equal sides. Let us delve into the depth of hexagons and figure out how many sides all types have.

Hexagon shape are for following types:

The properties of a regular hexagon are

The properties of an irregular hexagon are

A convex hexagon has

A concave hexagon has

The shape of hexagon comes in two forms:

Most of the things around us are in a hexagon shape; the following are a few real-life examples:

All hexagons have six sides. The sum of all sides is the perimeter of the hexagon. The sides of the hexagon have straight edges and form a closed shape.

Each side of a regular hexagon is equal in measure; however, the two sides of an irregular hexagon are different. That’s why calculating the length of regular hexagon sides is quite easier than of irregular hexagons. For example, to calculate the length of each side, divide its perimeter by 6. i.e., Perimeter ÷ 6

Following are the properties related to angles of the hexagon;

The important properties of the hexagon are given below:

A line segment that joins two non-adjacent vertices of a polygon is called the diagonal of a hexagon. Each vertex of the hexagon has three diagonals, so, collectively, a hexagon has 18 diagonals.
Basically, the hexagon has two types of diagonals:

The length of each short and long diagonal can be measured using the formula √3s and 2s, respectively. Here, s represents the length of each side.

Solution

the perimeter of the hexagon = 108 units.
Divide the perimeter by 6.
Perimeter ÷ 6 ⇒ 108 ÷ 6
= 18 units

Hence, 18 units is the length of each hexagon’s sides.

Hexagon: a six-sided regular polygon or 6-gon

Number of sides	6
Number of vertices	6
Interior angle	120° (Sums up to 720°)
Exterior angle	60° (Sums up to 360°)
Perimeter	P= 6* a (a = side)
Area	A = 3√3/2 R2

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what is interesting about it and how to build it. Formulas that describe the figure

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 get lost when they see problems with a regular hexagon and think 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.

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.
It is equal to .
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 topic of polygons is covered in the school curriculum, but not enough attention is paid 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 planar 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

, then 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 easy, all you need is a compass and a ruler.

The step-by-step instruction will look like this:

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

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

The properties are simple and interesting

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

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

1. diameter of the circumscribed circle;
2. inscribed circle diameter;
3. area;
4. perimeter.

Circumscribed circle and the possibility of construction

It is possible to circumscribe 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°, 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 corner 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 a circle, and it is only one, and its radius is equal to the side of the hexagon:

That is why it is possible to build this figure using a compass and ruler .

Well, the area of ​​this circle will be standard:

Inscribed circle

The center of the circumscribed circle will coincide 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 what is 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=6(a/2)(a(√3)/2)= 6a²(√3)/4=3a²(√3)/2
or

S=3R²(√3)/2

Those who wish to calculate this area through the radius of the inscribed circle can also do this:

S=3(2r/√3)²(√3)/2=r²(2√3)

Interesting 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 superposition 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 here we can draw logical conclusions:

1. 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.
2. Side AC will be equal to two radii of the inscribed circle, which is again calculated by the same theorem. That is, AC=2(a(√3)/2)=a(√3).
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 each other, the triangles form a new hexagon, and it is also regular. The proof is simple:

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 circumscribed circle 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=a/2

S=(3(√3)/2)(a(√3)/3)²=a(√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 hexagon in which the star is inscribed.

From theory to practice

The properties of the hexagon are widely used both in nature and in various areas 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.

Hexagonal tiles have also found their way. 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 tiles for paving 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.

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 til the plane, that is, they can fill the plane without gaps or 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 meet 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 .
At 1940, it was proved that this package is the most dense.

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 method of construction 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 the University of Oxford 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.

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 that 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 m 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.

Nut section
looks like a regular hexagon.

Construction of a regular hexagon inscribed in a circle.
The construction of a hexagon is based on the fact that its side is equal to the radius of the circumscribed circle. Therefore, to build, it is enough to divide the circle into six equal parts and connect the found points to each other (Fig. 60, a).

A regular hexagon can be built using a T-square and a 30X60° square. To perform this construction, we take the horizontal diameter of the circle as the bisector of angles 1 and 4 (Fig. 60, b), build sides 1-6, 4-3, 4-5 and 7-2, after which we draw sides 5-6 and 3- 2.

Construction of an equilateral triangle inscribed in a circle
. The vertices of such a triangle can be constructed using a compass and a square with angles of 30 and 60 °, or only one compass.

Consider two ways of constructing an equilateral triangle inscribed in a circle.

First way
(Fig. 61,a) is based on the fact that all three angles of the triangle 7, 2, 3 contain 60° each, and the vertical line drawn through point 7 is both the height and the bisector of angle 1. Since the angle 0-1 -2 is equal to 30°, then to find the side

1-2 it is enough to construct an angle of 30° from point 1 and side 0-1. To do this, set the T-square and square as shown in the figure, draw a line 1-2, which will be one of the sides of the desired triangle. To build side 2-3, set the T-square to the position shown by the dashed lines, and draw a straight line through point 2, which will define the third vertex of the triangle.

Second way
is based on the fact that if you build a regular hexagon inscribed in a circle, and then connect its vertices through one, you get an equilateral triangle.

To construct a triangle (Fig. 61, b), mark a vertex-point 1 on the diameter and draw a diametrical line 1-4. Further, from point 4 with a radius equal to D / 2, we describe the arc until it intersects with the circle at points 3 and 2. The resulting points will be two other vertices of the desired triangle.

Construction of a square inscribed in a circle
. This construction can be done using a square and a compass.

The first method is based on the fact that the diagonals of the square intersect in the center of the circumscribed circle and are inclined to its axes at an angle of 45°. Based on this, we install a T-square and a square with angles of 45 ° as shown in Fig. 62, a, and mark points 1 and 3. Further, through these points, we draw the horizontal sides of the square 4-1 and 3-2 with the help of a T-square. Then, using a T-square along the leg of the square, we draw the vertical sides of the square 1-2 and 4-3.

The second method is based on the fact that the vertices of the square bisect the arcs of the circle enclosed between the ends of the diameter (Fig. 62, b). We mark points A, B and C at the ends of two mutually perpendicular diameters, and from them with a radius y we describe the arcs until they intersect.

Next, through the intersection points of the arcs, we draw auxiliary straight lines, marked on the figure with solid lines. Their points of intersection with the circle will define vertices 1 and 3; 4 and 2. The vertices of the desired square obtained in this way are connected in series with each other.

Construction of a regular pentagon inscribed in a circle.

To fit a regular pentagon into a circle (Fig. 63), we make the following constructions.

Mark point 1 on the circle and take it as one of the vertices of the pentagon. Divide segment AO in half. To do this, with the radius AO from point A, we describe the arc to the intersection with the circle at points M and B. Connecting these points with a straight line, we get the point K, which we then connect to point 1. With a radius equal to the segment A7, we describe the arc from point K to the intersection with the diametrical line AO ​​at point H. Connecting point 1 with point H, we get the side of the pentagon. Then, with a compass opening equal to the segment 1H, describing the arc from vertex 1 to the intersection with the circle, we find vertices 2 and 5. Having made notches from vertices 2 and 5 with the same compass opening, we obtain the remaining vertices 3 and 4. We connect the found points sequentially with each other.

Construction of a regular pentagon on its given side.

To construct a regular pentagon along its given side (Fig. 64), we divide the segment AB into six equal parts. From points A and B with radius AB we describe arcs, the intersection of which will give point K. Through this point and division 3 on the line AB we draw a vertical line.

Get the point 1-vertex of the pentagon. Then, with a radius equal to AB, from point 1 we describe the arc to the intersection with the arcs previously drawn from points A and B. The intersection points of the arcs determine the vertices of the pentagon 2 and 5. We connect the found vertices in series with each other.

Construction of a regular heptagon inscribed in a circle.

Let a circle of diameter D be given; you need to inscribe a regular heptagon into it (Fig. 65). Divide the vertical diameter of the circle into seven equal parts. From point 7 with a radius equal to the diameter of the circle D, we describe the arc until it intersects with the continuation of the horizontal diameter at point F. Point F is called the pole of the polygon. Taking point VII as one of the vertices of the heptagon, we draw rays from the pole F through even divisions of the vertical diameter, the intersection of which with the circle will determine the vertices VI, V and IV of the heptagon. To obtain vertices / — // — /// from points IV, V and VI, we draw horizontal lines until they intersect with the circle. We connect the found vertices in series with each other. The heptagon can be constructed by drawing rays from the F pole and through odd divisions of the vertical diameter.

The above method is suitable for constructing regular polygons with any number of sides.

The division of a circle into any number of equal parts can also be done using the data in Table. 2, which shows the coefficients that make it possible to determine the dimensions of the sides of regular inscribed polygons.

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 incorrect?

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 daily 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 indicated above, it 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 designations

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.

Some formulas involve an interior angle, a semi-perimeter, and an apothem (which is a perpendicular to the midpoint of any side from the center of the polygon). 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, the following may be useful: S = p * r = (a 2 * 3 √3) / 2.

Problems

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 is to 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).

Reply
. 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?

Decision.
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 is 1.5 cm. The correct value for the perimeter is 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 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 cm0205 The real color of the hurricane. Photo taken June 26, 2013 (Source: NASA/JPL-Caltech/Space Science Institute/Val Klavans)

Saturn is one of the most unusual planets in the solar system. Of course, each planet is unique, but Saturn has several differences from other planets at once. First, it is a gas giant. Second, Saturn has rings. Thirdly, there has been a storm here for many hundreds of years, and this object has a hexagonal shape.

Now it turned out that this atmospheric vortex also changes its color over time. Over four years of observation, the color of the hexagon has changed from blue to golden. Moreover, experts still cannot answer the question “why?”. There are only a few guesses. One of them is that the color changes due to seasonal changes in the atmosphere of the gas giant.

Saturn’s hexagonal storm was discovered about 30 years ago. Its diameter is 32,000 km, depth — at least 100 kilometers. The speed of movement of atmospheric masses here is 150 m/s. The «eye» of a hurricane is about 50 times the size of the average «eye» of a hurricane on Earth. Last year, astronomers posted a video of the rotation of this object, compiled from many photographs taken by the Cassini spacecraft. We are talking about using approximately 30,000 photographs of Saturn.

Scientists say that each point of the hurricane rotates around the center at a speed approximately equal to the speed of the planet’s rotation around its axis. According to experts, the hurricane is constantly creating more and more clouds. The hurricane itself is located in the region of the north pole of the planet. Perhaps this atmospheric object has existed not for tens, but for hundreds of years.

Over the years of studying the hurricane, astronomers have received a large amount of information about this atmospheric phenomenon, and they understand the processes occurring inside the hurricane quite well. But why a hurricane changes color is still a mystery. In addition, the force that makes a section of the atmosphere of the giant planet rotate is also a mystery. Making a vortex is not that difficult, scientists say, but keeping it spinning for hundreds of years is much more difficult. “Water can easily be made to spin in a bucket. Rotating, it will form a funnel in the center. But there are no buckets on Saturn, and besides, the hurricane here is hexagonal, not a circle.”

According to NASA, the change in color of the hurricane may be caused by seasonal changes. A Saturnian year is 29 Earth years long. The change of seasons on Saturn occurs about once every seven years. The planet has been receiving more sun rays lately than in the previous season — it was «winter» here. Perhaps the increase in the amount of energy received by the planet is the reason for the «gilding» of the hurricane.

“The change in color is probably only a consequence of the change of seasons on Saturn. In particular, the color transition from bluish to golden may be due to the passage of photochemical reactions in the atmosphere, ”said NASA in a statement. Interestingly, in 2017 it will be possible to observe the Saturnian solstice.

How does it all work? Perhaps, the scientists say, the hexagon functions as a barrier that prevents aerosol and particles formed from the outside from getting in. A closer look at the structure of the hurricane shows that two different types of particles are collected within the hurricane and outside its walls.

“There are many small clouds inside the hexagon, and more large clouds outside it. The hexagon acts as a barrier, in some ways it looks like a hole in the Earth’s ozone layer,” said Kunio Sayanagi, a member of the Cassini project team.

During the polar winter of Saturn, which lasted from November 1995 to August 2009, aerosols, which are formed due to photochemical reactions, were not observed in the atmosphere at the North Pole of the gas giant. Basically, we are talking about the interaction of the rays of the sun and the atmosphere of the planet. In this case, the hexagon has a bluish color.

Since August 2009, Saturn has been getting more and more sunlight. It is possible that over the course of a few years, the sun’s rays contributed to the formation of a large number of aerosols in the hexagon and in the atmosphere at the entire north pole. As a result, more «fog» appeared here and Saturn became golden.

“Other phenomena may also play a role here,” the NASA website says. «Scientists believe that seasonal changes, including an increase in the amount of solar energy absorbed by Saturn, are likely to affect the winds at the poles.»

Now the interplanetary station «Cassini» is in orbit of Saturn, studying this planet and its satellites. Previously, with its help, scientists learned that precipitation often falls on Saturn in the form of rains from liquid helium. According to astronomers, helium moves from the upper atmosphere to the lower layers, resulting in rain.