Triangle, square, hexagon - these figures are known to almost everyone. But not everyone knows what a regular polygon is. But these are all the same geometric shapes. 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 figure. In addition, a 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 the same theorems. Any sideof a regular n-gon is related to the radius R of the circle circumscribed around 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 certain number of segments equal to each other, which, when connected, form a closed line. In this case, all the corners 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 star-shaped figures. For complex regular polygons, the sides are found by inscribing them in a circle. Let's 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 an inscribed regular triangle
An equilateral triangle is a regular polygon. The same formulas apply to it as to the square and the n-gon. A triangle will be considered correct if it has the same length sides. In this case, the angles are 60⁰. Construct a triangle with given side length a. Knowing its median and height,you can find the value of its sides. To do this, we will use the method of finding through the formula a \u003d 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 the given height. At the same time, it should be projected strictly on the base of the figure. So, knowing the height x, we find the side a of an isosceles triangle using the formula a \u003d b \u003d x: cosα. After finding the value of a, you can calculate the length of the base c. Let's apply the Pythagorean theorem. We will look for the value of half the base c: 2=√(x: cosα)^2 - (x^2)=√x^2 (1 - cos^2α): cos^2α=x ∙ tgα. Then c=2xtanα. Here is a simple way to find the number of sides of any inscribed polygon.
Calculate 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. You can calculate the sides of a square using the value of the diagonal. Let's consider this method in more detail. It is known that the diagonal bisects the angle. Initially, its value was 90 degrees. Thus, after division, two right-angled triangles are formed. Their base angles will be 45 degrees. Accordingly, each side of the square will be equal, that is: a \u003d c \u003d c \u003d d \u003d e ∙ cosα \u003d e √ 2: 2, where e is the diagonal of the square, or the base of the right triangle formed after division. It's not the only wayfinding the sides of a square. Let's inscribe 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 (360o: 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. It is easy to calculate it. To do this, you need to know the values of all sides. For some 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 \u003d 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, we calculate as follows: P=5 ∙ 6=30 cm. And so for each polygon.
Finding the perimeter of a parallelogram, a square and a rhombus
Depending on how many sides a regular polygon has, its perimeter is calculated. This makes the task much easier. Indeed, unlike other figures, in this case it is not necessary to look for all its sides, just one is enough. By the same principle, we find the perimeter atquadrangles, that is, a square and a rhombus. Despite the fact that these are different figures, the formula for them is the same P=4a, where a is the side. Let's take an example. If the side of a rhombus or square is 6 cm, then we find the perimeter as follows: P \u003d 4 ∙ 6 \u003d 24 cm. A parallelogram has only opposite sides. Therefore, its perimeter is found using a different method. So, we need to know the length a and the width b of the figure. Then we apply the formula P=(a + c) ∙ 2. A parallelogram, in which all sides and angles between them are equal, is called a rhombus.
Finding the perimeter of an equilateral and right triangle
The perimeter of a regular equilateral triangle can be found using the formula P=3a, where a is the length of a side. If it is unknown, it can be found through the median. In a right triangle, only two sides are equal. The basis can be found through the Pythagorean theorem. After the values of all three sides become known, we calculate the perimeter. It can be found by applying the formula P \u003d a + b + c, where a and b are equal sides, and c is the base. Recall that in an isosceles triangle a \u003d b \u003d a, therefore, a + b \u003d 2a, then P \u003d 2a + c. For example, the side of an isosceles triangle is 4 cm, find its base and perimeter. We calculate the value of the hypotenuse using the Pythagorean theorem c=√a2 + v2=√16+16=√32=5.65 cm. Now we calculate perimeter Р=2 ∙ 4 + 5, 65=13.65 cm.
How to find the corners of a regular polygon
Regular polygonoccurs in our lives 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 at first glance. In order to construct any n-gon, you need to know the value of its angles. But how do you find them? Even scientists of antiquity tried to build regular polygons. They guessed to fit them into circles. And then the necessary points were marked on it, connected by straight lines. For simple figures, the construction problem has been solved. Formulas and theorems have been 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 them and find angles. Let's see 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, which means that the number n is 15. We substitute the data we know into the formula and get S=180⁰ (15 - 2)=180⁰ x 13=2340⁰. We have found the sum of all interior angles of a 15-gon. Now we need to get the value of each of them. There are 15 angles in total. We do the calculation 2340⁰: 15=156⁰. This means that each internal angle is 156⁰, now using a ruler and a compass, you can build a regular 15-gon. But what about more complex n-gons? For 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 a 65537-gon. Since then, the problem is officially considered completely solved.
Calculation of angles of n-gonsin radians
Of course, there are several ways to find the corners of polygons. Most often they are calculated in degrees. But you can also express them in radians. How to do it? It is necessary to proceed 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. Consider these calculations using the example of the same fifteen-sided. So, the number n is 15. Apply the formula S=p(n - 2): n=3, 14(15 - 2): 15=3, 14 ∙ 13: 15=2, 72. This, of course, is not the only way to calculate angle in radians. You can simply divide the size of the angle in degrees by the number 57, 3. After all, that many degrees are equivalent to one radian.
Calculate the value of angles in degrees
Besides degrees and radians, you can try to find the value of the angles of a regular polygon in grads. This is done in the following way. Subtract 2 from the total number of angles, divide the resulting difference by the number of sides of a regular polygon. We multiply the found result by 200. By the way, such a unit of measurement of angles as hailstones is practically not used.
Calculation of external angles of n-gons
For any regular polygon, except for the internal one, you can also calculate the external angle. Its value is found in the same way as for other figures. So, to find the exterior angle of a regular polygon, you needknow the meaning of the inner. 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, if the inner corner of a square is 90 degrees, then the outer angle will be 180⁰ - 90⁰=90⁰. As we can see, it is not difficult to find it. The external angle can take a value from +180⁰ to -180⁰, respectively.
