Triangle, square, hexagon - these figures are known to almost everyone. But far from everyone knows what a regular polygon is. But these are all the same geometric figures. A regular polygon is called one that has equal angles and sides to each other. 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 building a shape. In addition, the circle can also be inscribed in the 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 circle R described near 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 segments equal to each other, which, when connected, form a closed line. Moreover, all the angles of the resulting figure have the same value. Polygons are divided into simple and complex. The first group includes a triangle and a square. Complex polygons have a larger number of sides. They also include star shapes. For complex regular polygons, sides are found by inscribing them in a circle. We give a proof. Draw a regular polygon with an arbitrary number of sides n. Describe the circle around it. Define 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 an inscribed regular triangle
An equilateral triangle is a regular polygon. The formulas are applied to it the same as to the square and the n-gon. A triangle will be considered correct if it has the same sides in length. The angles are 60⁰. We construct a triangle with a given length of the sides a. Knowing its median and height, you can find the meaning of its sides. To do this, we will 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 get a = b = c. Then the following statement will be true: a = b = c = x: cosα. Similarly, you 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 onto the base of the figure. So, knowing the height x, we find the side a of an isosceles triangle by the formula a = b = x: cosα. After finding the value of a, we can calculate the length of the base c. We apply the Pythagorean theorem. We look for the value of the half base c: 2 = √ (x: cosα) ^ 2 - (x ^ 2) = √x ^ 2 (1 - cos ^ 2α): cos ^ 2α = x ∙ tgα. Then c = 2xtgα. In such a simple way, you can find the number of sides of any inscribed polygon.
Calculation of the sides of a square inscribed in a circle
Like any other inscribed regular polygon, a square has equal sides and angles. The same formulas apply to it as to the triangle. The sides of the square can be calculated using the diagonal value. 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 angles at the base will be equal to 45 degrees. Accordingly, each side of the square will be equal, that is: a = b = c = d = e ∙ cosα = e√2: 2, where e is the diagonal of the square, or the base formed after the division of the right triangle. This is not the only way to find the sides of the square. Fill this shape into 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 length of the side.
How to calculate the perimeter of an n-gon
The perimeter of an n-gon is the sum of all its sides. It is not difficult to calculate it. To do this, you need to know the values of all parties. For some types of polygons, special formulas exist. 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 value of the side, and n is the number of angles. For example, to find the perimeter of a regular octagon with a side of 3 cm, you need to multiply it by 8, that is, P = 3 ∙ 8 = 24 cm. For a hexagon with a side of 5 cm, calculate this: P = 5 ∙ 6 = 30 cm. And so for each polygon.
Finding the perimeter of a parallelogram, square and rhombus
Depending on how many sides a regular polygon has, its perimeter is calculated. This greatly facilitates the task. Indeed, unlike other figures, in this case it is not necessary to look for all its sides, just one. By the same principle, we find the perimeter of the quadrangles, that is, the square and rhombus. Despite the fact that these are different figures, the formula for them is one P = 4a, where a is the side. We give an example. If the side of the rhombus or square is 6 cm, then we find the perimeter as follows: P = 4 ∙ 6 = 24 cm. Only the opposite sides are equal for a parallelogram. Therefore, its perimeter is found using another method. So, we need to know the length of a and the width of the figures. Then we apply the formula P = (a + c) ∙ 2. A parallelogram in which all sides are equal and the angles between them is called a rhombus.
Finding the perimeter of an equilateral and rectangular triangle
The perimeter of a regular equilateral triangle can be found by the formula P = 3a, where a is the length of the side. If she is unknown, she can be found through the median. In a right triangle, only two sides are of equal importance. The base can be found through the Pythagorean theorem. After the values of all three sides become known, we calculate the perimeter. It can be found using 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, which means 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 hypotenuse value by the Pythagorean theorem with = √ 2 + 2 = √16 + 16 = √32 = 5.65 cm. Now we calculate the perimeter P = 2 ∙ 4 + 5.65 = 13.65 cm.
How to find the corners of a regular polygon
A regular polygon is found in our life every day, for example, an ordinary square, triangle, octagon. It would seem that there is nothing easier than building this figure yourself. But this is just only 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 scientists of antiquity tried to build regular polygons. They guessed to fit them in a circle. And then the necessary points were marked on it, connected by straight lines. For simple figures, the construction problem was solved. Formulas and theorems were obtained. For example, Euclid in his famous work "Beginning" was engaged in solving problems for 3-, 4-, 5-, 6- and 15-gons. He found ways to build them 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. Substitute the data we know into the formula and get S = 180⁰ (15 - 2) = 180⁰ x 13 = 2340⁰. We found the sum of all the internal corners of the 15-gon. Now you need to get the value of each of them. There are 15 angles in total. We do the calculation 2340⁰: 15 = 156⁰. So, each internal angle is 156⁰, now with the help of a ruler and a compass you can build a regular 15-gon. But what about more complex n-gons? For many centuries, scientists have struggled to solve this problem. It was found only in the 18th century by Karl Friedrich Gauss. He was able to build a 65537-gon. Since then, the problem has been officially considered completely 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 act as follows. First we find out the number of sides of a regular polygon, then subtract 2 from it. So, we get the value: n - 2. Multiply the found difference by the number n ("pi" = 3.14). Now it remains only to divide the resulting product by the number of angles in the n-gon. Let's consider these calculations using the example of the same pentagon. So, the number n is 15. We 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 angle in degrees by 57.3. After all, just so many degrees is equivalent to one radian.
Calculation of angles in degrees
In addition to degrees and radians, the values of the angles of a regular polygon can be tried to find in grad. This is done as follows. Subtract 2 from the total number of angles, divide the resulting difference by the number of sides of a regular polygon. We multiply the result by 200. By the way, such a unit of measurement of angles as hail is practically not used.
Calculation of the external angles of n-gons
For any regular polygon, except for the internal, you can also calculate the external angle. Its value is found in the same way as for other figures. So, to find the outer corner of a regular polygon, you need to know the meaning of the inner one. Further, we know that the sum of these two angles is always 180 degrees. Therefore, the calculations are done as follows: 180⁰ minus the value of the internal angle. Find the difference. It will be equal to the value of the angle adjacent to it. For example, the internal angle of the square is 90 degrees, which means that the external 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, -180⁰.