triangle, square, hexagon - these figures known to almost everyone.But that is a regular polygon, knows not everyone.But it's all the same geometric shapes.Regular polygon called the one that has equal angles to each other and the sides.Such figures are many, but they all have the same properties, and apply to them the same formula.
properties of regular polygons
Any regular polygon, whether square or octagon, can be inscribed in a circle.This basic property is often used in the construction of the figure.In addition, a circle and can enter into a polygon.The number of points of contact will be equal to the number of its sides.It is important that the circle inscribed in a regular polygon will have with him a common center.These geometric figures are subject to one theorems.Any party regular n-gon is related to the radius of the circle around him R. Therefore, it can be calculated using the following formula: a = 2R ∙ sin180 °.After the circle radius can be found not only the parties but also the perimeter of a polygon.
Find the number of sides of a regular polygon
Any regular n-gon is composed of a number equal to each other segments which combine to form a closed line.In this figure all the angles formed have the same value.The polygons are divided into simple and complex.The first group includes a triangle and a square.Complex polygons have a greater number of sides.They also include a star-shaped figure.In the part of the complex of regular polygons are by inscribing them in a circle.Here is the proof.Draw a regular polygon with an arbitrary number of sides n.Describe a circle around him.Ask a radius R. Now imagine that some given n-gon.If the point of its corners lie on a circle and equal to each other, the parties can be found by the formula: a = 2R ∙ sinα: 2.
Finding the number of sides of the inscribed equilateral triangle
equilateral triangle - a regular polygon.Formula applied to it the same as that of the square, and the n-gon.Triangle will be considered valid if it has the same length on the side.The angles are equal 60⁰.We construct a triangle with sides a predetermined length.Knowing him and the median height, you can find the value of its sides.For this we use a method of finding the formula through a = x: cosα, where x - the median or height.Since all sides of the triangle are equal, we get a = b = c.Then the following statement is true and = B = C = x: cosα.Similarly, we can find the value of the parties within an equilateral triangle, but will be given x height.At the same time it must be projected strictly on the basis of the figures.So, knowing the height of x, find a side of an isosceles triangle using the formula A = B = x: cosα.After finding the value and can calculate the length of the base with.We apply the Pythagorean theorem.We seek the value of half the base c: 2 = √ (x: cosα) ^ 2 - (x ^ 2) = √x ^ 2 (1 - cos ^ 2α): cos ^ 2α = x ∙ tgα.Then c = 2xtgα.That's the simple way you can find any number of sides of the inscribed polygon.
Calculating the sides of the square inscribed in a circle
Like any other regular polygon inscribed square has equal sides and angles.Shall apply the same formula as the triangle.Calculate the side of the square, you can diagonally through the value.Consider this method in more detail.It is known that a diagonal divides in half the angle.Initially, the value was 90 degrees.Thus, after division gives rise to two right-angled triangles.Their base angles are equal to 45 degrees.Accordingly each side of the square is equal, that is: a = b = c = d = e ∙ cosα = e√2 2, where e - is the diagonal of a square, or a base formed after the division of a right triangle.This is not the only way of finding the sides of the square.Inscribe this figure into a circle.Knowing the radius of the circle R, find the side of the square.We calculate it as follows a4 = R√2.The radii of regular polygons is calculated from the formula R = a: 2tg (360o: 2n), where a - side length.
How to calculate the perimeter of the n-gon
perimeter of the n-gon is the sum of all its sides.Calculate it easy.You need to know the values of all parties.For some types of polygons exist special formulas.They allow you to find the perimeter of a lot faster.It is known that any regular polygon has equal sides.Therefore, in order to calculate the 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: R = an, where a - value side, and n - number of angles.For example, to find the perimeter of a regular octagon with a side of 3 cm, you must multiply it by 8, that is, P = 3 ∙ 8 = 24 cm. The hexagon with a side of 5 cm are calculated as follows: P = 5 ∙ 6 = 30 cm. And so foreach polygon.
Finding the perimeter of a parallelogram, square and diamond
Depending on how many sides a regular polygon has calculated its perimeter.It's much easier task.Indeed, in contrast to the other pieces, in this case, no need to search all its aspects, one is enough.On the same principle is at the perimeter of rectangles, that is, square and diamond.Despite the fact that they are different shapes, the formula for which R a = 4a where a - side.Here is an example.If a party or diamond-shaped square is 6 cm, find the perimeter of the following: P = 4 ∙ 6 = 24 cm. Parallelogram are just opposite sides.Therefore its perimeter is found using another method.Thus, we need to know the length and width and shape.Then apply the formula P = (a + b) ∙ 2. parallelogram whose sides all equal and the angles between them, called diamond.
Finding the perimeter of an equilateral triangle and rectangular
Perimeter proper equilateral triangle can be found from the formula P = 3a, where a - length of the side.If it is unknown, it can be found through the median.In a right triangle is equal to the value are just two sides.The base can be found through the Pythagorean theorem.After become known values of all three parties calculate perimeter.It can be found using the formula R = a + b + c, where a and b - equal sides, and with - a base.Recall that in an equilateral triangle is A = B = A, then A + B = 2a, then P = 2a + c.For example, the side of an isosceles triangle is equal to 4 cm, find its base and perimeter.Calculate the value of the hypotenuse of a Pythagorean theorem = √a2 + e2 = √16 + 16 = √32 = 5,65 cm. We now calculate the perimeter P = 2 ∙ 4 + 5.65 = 13.65 cm.
How to find the right angles
polygon regular polygon is found in our lives every day, for example, the usual square, triangle, octagon.It would seem that there is nothing easier than to build it on their own figure.But that's just at first glance.In order to build any n-gon, it is necessary to know the value of its angles.But how to find them?Even ancient scientists were trying to build regular polygons.They guessed fit them into the circle.And then on it notes the need to point, connecting them with straight lines.For simple shapes was solved the problem of constructing.The formulas and theorems have been obtained.For example, Euclid in his famous work "The Beginning" was engaged in solving the problem for 3-, 4-, 5-, 6- and 15-gon.He found ways to build and find the angles.Here's how to do it for the 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, then n is the number 15. Substitute the known data in the formula and get S = 180⁰ (15 - 2) = 180⁰ x 13 = 2340⁰.We found the sum of all interior angles of a 15-sided polygon.Now you need to get the value of each.Total corners 15. Do the calculation 2340⁰: 15 = 156⁰.Therefore, each internal angle is 156⁰, is now using a ruler and compass, you can construct a regular 15-gon.But what about more complex n-gon?For many centuries, scientists struggled to solve this problem.It was found only in the 18th century, Carl Friedrich Gauss.He was able to build a 65,537-square.Since then the problem is officially considered to be fully solved.
calculation of the n-gon angle in radians
Of course, there are several ways of finding the angles of polygons.Most often they are calculated in degrees.But we can express them in radians.How to do it?It is necessary to proceed as follows.First, find out the number of sides of a regular polygon, and then subtract from it 2. So, we get the value: n - 2. Multiply the difference found in the number n ("pi" = 3.14).Now you just divide that product by the number of corners in the n-gon.Consider these calculations on the example of the same pyatnadtsatiugolnika.Thus, the number n is equal to 15. We apply the formula S = n (n - 2): n = 3,14 (15 - 2): 15 = 3,14 ∙ 13: 15 = 2.72.This, of course, 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 equivalent to one radian.
Calculation of angles in grads
addition to degrees and radians, the value of the angles of a regular polygon, you can try to find in the Castle.This is done as follows.Of the total number of angles we subtract 2, dividing the resulting difference by the number of sides of a regular polygon.Found the result is multiplied by 200. By the way, this unit of measurement of angles as grads, hardly used.
Calculation outer corners n-gon
Any regular polygon, except for internal, you can also calculate the outer corner.Its value is the same as for the other figures.So, to find an external angle of a regular polygon, you must know the value of internal.Further, we know that the sum of these two angles is always 180 degrees.Therefore, calculations are as follows: 180⁰ minus the inner corner.We find the difference.It will be the value of the angle adjacent to it.For example, the inner corner of the square is 90 degrees, so the appearance will be 180⁰ - 90⁰ = 90⁰.As we can see, it is easy to find.External angle can be set between + 180⁰ to, respectively, -180⁰.
