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A regular polygon. The number of sides of a regular polygon

Triangle, square, hexagon - these figures are known to almost everyone. But not everyone knows about what a regular polygon is. But these are all the same geometric figures. A regular polygon is one that has equal angles and sides. There are a lot of such figures, but they all have the same properties, and the same formulas apply to them.

Properties of regular polygons

Any regular polygon, be it a square or an octagon, can be inscribed in a circle. This basic property is often used when constructing a shape. In addition, the circle can also be inscribed in a polygon. In this case, the number of points of contact will be equal to the number of its sides. It is important that a circle inscribed in a regular polygon will have a common center with it. These geometric figures are subject to one theorem. Any side of a regular n-gon is connected with the radius of the circumcircle R circumscribed about it. Therefore, it can be calculated using the following formula: a = 2R ∙ sin180 °. Through the radius of the circle, you can find not only the sides, but also the perimeter of the polygon.

How to find the number of sides of a regular polygon

Any regular n-gon consists of a number of equal segments that join together form a closed line. In this case, all the angles of the formed figure have the same value. Polygons are divided into simple and complex. The first group includes a triangle and a square. Complex polygons have more sides. They also include stellate figures. For complex regular polygons, the sides are found by inscribing them into a circle. We give a proof. Draw a regular polygon with an arbitrary number of sides n. Describe a circle around it. Specify the radius R. Now imagine that some n-gon is given. If the points of its angles lie on a circle and are equal to each other, then the sides can be found by the formula: a = 2R ∙ sinα: 2.

Finding the number of sides of the inscribed right triangle

An equilateral triangle is a regular polygon. Formulas to it apply the same as to the square, and n-gon. The triangle will be considered correct if it has the same length along the side. The angles are equal to 60 При. We construct a triangle with a given length of sides a. Knowing its median and height, one can find the significance of its sides. To do this, we use the method of finding through the formula a = x: cosα, where x is the median or height. Since all sides of the triangle are equal, we obtain a = b = c. Then the following assertion will hold: a = c = c = x: cosα. Similarly, one can find the value of the sides in an isosceles triangle, but x will be a given height. In this case, it should be projected strictly on the base of the figure. Thus, knowing the height x, we find the side a of an isosceles triangle by the formula a = b = x: cosa. After finding the value of a, we can calculate the length of the base c. We apply the theorem of Pythagoras. We will search for the value of the half of the base c: 2 = √ (x: cosα) ^ 2 - (x ^ 2) = √x ^ 2 (1 - cos ^ 2α): cos ^ 2α = x ∙ tgα. Then c = 2xtgα. In this simple way one can find the number of sides of any inscribed polygon.

Calculating the sides of a square inscribed in a circle

Like any other inscribed regular polygon, the square has equal sides and angles. The same formulas apply to it as to the triangle. Calculate the sides of the square can be through the value of the diagonal. Let's consider this method in more detail. It is known that the diagonal divides the angle in half. Initially, its value was 90 degrees. Thus, after division, two rectangular triangles are formed. Their corners at the base will be equal to 45 degrees. Accordingly, each side of the square will be equal, that is: a = c = c = q = e ∙ cosα = e√2: 2, where e is the diagonal of the square, or the base of the right triangle formed after division. This is not the only way to find the sides of a square. We will write this figure in a circle. Knowing the radius of this circle R, we find the side of the square. We will calculate it as follows: a4 = R√2. The radii of regular polygons are calculated by the formula R = a: 2tg (360 o: 2n), where a is the side length.

How to calculate the perimeter of an n-gon

The perimeter of an n-gon is the sum of all its sides. Calculate it is not difficult. To do this, you need to know the meaning of all parties. For certain types of polygons, there are special formulas. They allow you to find the perimeter much faster. It is known that any regular polygon has equal sides. Therefore, in order to calculate its perimeter, it is enough to know at least one of them. The formula will depend on the number of sides of the figure. In general, it looks like this: P = an, where a is the side value, and n is the number of angles. For example, to find the perimeter of a regular octagon with a side of 3 cm, multiply it by 8, that is, P = 3 ∙ 8 = 24 cm. For a hexagon with a side of 5 cm, calculate: P = 5 ∙ 6 = 30 cm. Of each polygon.

Finding the perimeter of a parallelogram, a square, and a rhombus

Depending on how many sides a regular polygon has, calculate its perimeter. This greatly simplifies the task. After all, unlike other figures, in this case you do not need to look for all its sides, just one. By the same principle, we find the perimeter of the quadrangles, that is, the square and the rhombus. Despite the fact that these are different figures, the formula for them is P = 4a, where a is the side. Let us give an example. If the side of the diamond or square is 6 cm, then we find the perimeter in the following way: P = 4 ∙ 6 = 24 cm. In the parallelogram, only the opposite sides are equal. Therefore, its perimeter is found using a different method. So, we need to know the length of a and the width of the figure. Then we apply the formula P = (a + b) ∙ 2. A parallelogram, in which all sides and angles are equal, is called a rhombus.

Finding the perimeter of an equilateral triangle and a right triangle

The perimeter of a regular equilateral triangle can be found by the formula P = 3a, where a is the side length. If it is unknown, it can be found through the median. In a rectangular triangle, only two sides have an equal value. The basis can be found through the Pythagorean theorem. After the values of all three sides become known, calculate the perimeter. It can be found by applying the formula P = a + b + c, where a and b are equal sides, and c is the base. Recall that in an isosceles triangle a = b = a, then a + b = 2a, then P = 2a + c. For example, the side of an isosceles triangle is 4 cm, we find its base and perimeter. We calculate the value of the hypotenuse according to the Pythagorean theorem with c = √a 2 + in 2 = √16 + 16 = √32 = 5.65 cm. Now calculate the perimeter P = 2 ∙ 4 + 5.65 = 13.65 cm.

How to find the corners of a regular polygon

A regular polygon occurs in our lives every day, for example, an ordinary square, a triangle, an octagon. It would seem that nothing is easier than building this figure yourself. But it's just just at first glance. In order to construct any n-gon, it is necessary to know the value of its angles. But how to find them? Even ancient scientists tried to build regular polygons. They guessed to fit them in a circle. And then they marked the necessary points on it, connected them by straight lines. For simple figures, the construction problem was solved. Formulas and theorems were obtained. For example, Euclid in his famous work "The Beginning" was engaged in solving problems for 3-, 4-, 5-, 6- and 15-gons. He found ways to construct and find angles. Consider how to do this for a 15-gon. First you need to calculate the sum of its internal angles. It is necessary to use the formula S = 180⁰ (n-2). So, we are given a 15-gon, so the number n is 15. We substitute the data known to us in the formula and get S = 180⁰ (15 - 2) = 180⁰ х 13 = 2340⁰. We found the sum of all the interior angles of a 15-gon. Now you need to get the value of each of them. Total angles 15. Do the calculation of 2340⁰: 15 = 156⁰. Hence, each internal angle is 156⁰, now with the help of ruler and compass you can construct the correct 15-gon. But what about the more complex n-gons? For many centuries scientists have struggled to solve this problem. It was only found in the 18th century by Carl Friedrich Gauss. He was able to build 65537-gon. Since then, the problem is officially considered fully resolved.

Calculation of angles of n-gons in radians

Of course, there are several ways to find the angles of polygons. Most often they are calculated in degrees. But you can express them in radians. How to do it? It is necessary to proceed as follows. First, we figure out the number of sides of a regular polygon, then subtract it from it 2. So, we get the value: n - 2. Multiply the difference by n ("pi" = 3.14). Now it remains only to divide the obtained product by the number of angles in the n-gon. Consider these calculations on an example of the same fifteen-cornered triangle. So, the number n is 15. Let's apply the formula S = n (n - 2): n = 3,14 (15 - 2): 15 = 3,14 ∙ 13: 15 = 2,72. This, of course, is not the only way to calculate the angle in radians. You can simply divide the size of the angle in degrees by the number 57.3. After all, so many degrees are equivalent to one radon.

Calculation of the angles in grads

In addition to degrees and radians, you can try to find the angles of a regular polygon in hail. This is done as follows. From the total number of angles, subtract 2, divide the resulting difference by the number of sides of the regular polygon. The result is multiplied by 200. By the way, such a unit of measurement of angles, as hail, is practically not used.

Calculation of external angles of n-gons

For any regular polygon, besides the inner one, it is possible to calculate also the outer angle. Its meaning is found in the same way as for the rest of the figures. So, to find the outer corner of a regular polygon, you need to know the meaning of the inner polygon. Further, we know that the sum of these two angles is always 180 degrees. Therefore, we do the calculations as follows: 180⁰ minus the value of the internal angle. We find the difference. It will be equal to the value of the angle adjacent to it. For example, the inner corner of the square is 90 degrees, then the outer corner will be 180⁰-90⁰ = 90⁰. As we see, it is not difficult to find it. The external angle can take a value from + 180⁰ to, respectively, -180.

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