Theorem on the sum of interior angles of a polygon. regular polygon
Triangle, square, hexagon - these figures are known to almost everyone. But not everyone knows what a regular polygon is. But this is all the same Regular polygon is called the 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 subject to the same theorems. Any side of a regular n-gon is associated with the radius R of the circumscribed circle around it. Therefore, it can be calculated using the following formula: a = 2R ∙ sin180°. Through 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 one 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 right 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 is 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 isosceles triangle according to 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. In such a simple way, you 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, 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 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 \u003d b \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. Is not the only way finding 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 \u003d 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 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, square and 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 of quadrangles, 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 \u003d (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 the correct one can be found by the formula P \u003d 3a, where a is the length of the side. If it is unknown, it can be found through the median. AT 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 according to the Pythagorean theorem c \u003d √a 2 + in 2 \u003d √16 + 16 \u003d √32 \u003d 5.65 cm. Now we calculate the perimeter P \u003d 2 ∙ 4 + 5.65 \u003d 13.65 cm.
How to find the angles of a regular polygon
A regular polygon occurs 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 build 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 of 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 has officially been considered completely solved.
Calculation of angles of n-gons in 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, find out the number of sides regular polygon, then subtract 2 from it. So, we get the value: n - 2. Multiply the found difference by the number n ("pi" \u003d 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. Let's apply the formula S = p(n - 2) : n = 3.14(15 - 2) : 15 = 3.14 ∙ 13: 15 = 2.72. This is of course not the only way to calculate an angle in radians. You can simply divide the size of the angle in degrees by the number 57.3. After all, that many degrees is equivalent to one radian.
Calculation of the value of angles in degrees
In addition to 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 result found by 200. By the way, such a unit of measurement of angles as degrees is practically not used.
Calculation of external corners of n-gons
For any regular polygon, in addition to 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 outer corner of a regular polygon, you need to know the value of the inner one. 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 a square is 90 degrees, so 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, respectively, -180⁰.
your polygon. For example, if you need to find the angles of a regular polygon with 15 sides, plug n=15 into the equation. You will get S=180⁰(15-2), S=180⁰x13, S=2340⁰.
Next, divide the resulting sum of interior angles by their number. For example, in a polygon, the number of corners is the number of sides, that is, 15. Thus, you will get that the angle is 2340⁰/15=156⁰. Each interior angle of a polygon is 156⁰.
If it is more convenient for you to calculate the angles of the polygon in radians, proceed as follows. Subtract the number 2 from the number of sides and multiply the resulting difference by the number P (Pi). Then divide the product by the number of corners in the polygon. For example, if you need to calculate the angles of a regular 15-gon, proceed like this: P * (15-2) / 15 \u003d 13 / 15P, or 0.87P, or 2.72 (but, like, the number P remains unchanged) . Or just divide the size of the angle in degrees by 57.3 - that's how much is contained in one radian.
You can also try to calculate the angles of a regular polygon in degrees. To do this, subtract the number 2 from the number of sides, divide the resulting number by the number of sides and multiply the result by 200. This unit of measurement of angles is almost never used today, but if you decide to calculate angles in hail, do not forget that hail is divided into metric seconds and minutes (100 seconds per minute).
Perhaps you need to calculate the exterior angle of a regular polygon, in which case do this. Subtract the inner angle from 180⁰ - as a result, you will get the value of the adjacent, that is, the outer angle. It can take a value from -180⁰ to +180⁰.
If you managed to find out the angles of a regular polygon, you can easily build it. Draw one side of a certain length and use a protractor to set aside the desired angle from it. Measure exactly the same distance (all sides of a regular polygon are equal) and set aside the desired angle again. Continue until the sides meet.
Sources:
- angle in a regular polygon
A polygon is said to be circumscribed if all its sides are tangent to the circle inscribed in it. You can only describe a regular polygon, that is, one in which all sides are equal. Even ancient architects faced the solution of such a problem when it was necessary to design, for example, a column. Modern technologies allow you to do this with minimal time costs, but the principle of operation remains the same as in classical geometry.
You will need
- - compass;
- - protractor;
- - ruler;
- - paper.
Instruction
Draw a circle with the given . Define its center as O and draw one of the radii so that it is possible to start building. In order to describe a polygon around it, you need its only parameter - the number of sides. Designate it as n.
Remember, the angle of any circle. It is 360°. Based on this, it is possible to calculate the angles of the sectors, the sides of which will connect the center of the circle with the points of contact with the sides of the polygon. The number of these sectors is equal to the number of sides of the polygon, that is, n. Find the angle α using the formula α = 360°/n.
Using a protractor, set aside the resulting angle value from the radius and draw another radius through it. To make calculations accurate, use a calculator and round values only in exceptional cases. From this new radius, again set aside the corner of the sector and draw another line between the center and the circle line. Build all the corners in the same way.
Note. This material contains the theorem and its proof, as well as a number of problems illustrating the application of the theorem on the sum of angles of a convex polygon on practical examples.
Convex polygon angle sum theorem
.Proof.
To prove the theorem on the sum of angles of a convex polygon, we use the already proven theorem that the sum of the angles of a triangle is 180 degrees.
Let A 1 A 2... A n be a given convex polygon, and n > 3. Draw all the diagonals of the polygon from the vertex A 1. They divide it into n – 2 triangles: Δ A 1 A 2 A 3, Δ A 1 A 3 A 4, ... , Δ A 1 A n – 1 A n . The sum of the angles of the polygon is the same as the sum of the angles of all these triangles. The sum of the angles of each triangle is 180°, and the number of triangles is (n - 2). Therefore, the sum of the angles of a convex n-gon A 1 A 2... A n is 180° (n – 2).
A task.
In a convex polygon, three angles are 80 degrees and the rest are 150 degrees. How many corners are in a convex polygon?
Solution.
The theorem says: For a convex n-gon, the sum of the angles is 180°(n-2) .
So for our case:
180(n-2)=3*80+x*150, where
3 angles of 80 degrees are given to us according to the condition of the problem, and the number of other angles is still unknown to us, so we denote their number as x.
However, from the entry on the left side, we determined the number of corners of the polygon as n, since we know the values of three of them from the condition of the problem, it is obvious that x=n-3.
So the equation will look like this:
180(n-2)=240+150(n-3)
We solve the resulting equation
180n - 360 = 240 + 150n - 450
180n - 150n = 240 + 360 - 450
Answer: 5 peaks
A task.
How many vertices can a polygon have if each angle is less than 120 degrees?
Solution.
To solve this problem, we use the theorem on the sum of angles of a convex polygon.
The theorem says: For a convex n-gon, the sum of all angles is 180°(n-2) .
Hence, for our case, it is necessary to first estimate the boundary conditions of the problem. That is, make the assumption that each of the angles is equal to 120 degrees. We get:
180n - 360 = 120n
180n - 120n = 360 (we will consider this expression separately below)
Based on the equation obtained, we conclude: when the angles are less than 120 degrees, the number of corners of the polygon is less than six.
Explanation:
Based on the expression 180n - 120n = 360 , provided that the subtracted right side is less than 120n, the difference should be more than 60n. Thus, the quotient of division will always be less than six.
Answer: the number of polygon vertices will be less than six.
A task
A polygon has three angles of 113 degrees, and the rest are equal to each other and their degree measure is an integer. Find the number of vertices of the polygon.
Solution.
To solve this problem, we use the theorem on the sum of the external angles of a convex polygon.
The theorem says: For a convex n-gon, the sum of all exterior angles is 360° .
In this way,
3*(180-113)+(n-3)x=360
the right side of the expression is the sum of the external angles, on the left side the sum of the three angles is known by condition, and the degree measure of the rest (their number, respectively, n-3, since three angles are known) is denoted as x.
159 is decomposed only into two factors 53 and 3, and 53 is a prime number. That is, there are no other pairs of factors.
Thus, n-3 = 3, n=6, that is, the number of corners of the polygon is six.
Answer: six corners
A task
Prove that a convex polygon can have at most three acute angles.
Solution
As you know, the sum of the external angles of a convex polygon is 360 0 . Let us prove by contradiction. If a convex polygon has at least four acute internal angles, then among its external angles there are at least four obtuse ones, which means that the sum of all external angles of the polygon is greater than 4 * 90 0 = 360 0 . We have a contradiction. The assertion has been proven.
Video lesson 2: Polygons. Problem solving
Lecture: Polygon. Sum of angles of a convex polygon
Polygons- these are the figures that surround us everywhere - this is also the form of honeycombs in which bees store their honey, architectural structures, and much more.
As mentioned earlier, polygons are shapes that have more than two corners. They consist of a closed broken line.
Moreover, the corners of the polygons can be external and internal. For example, a star is a figure that has 10 corners, some of which are convex and others concave:
Examples of convex polygons:
Please note that the figure shows regular polygons - these are the ones studied in detail in school course mathematics.
Any polygon has the same number of vertices as the number of sides. Also note that neighboring vertices are those that have one common side. For example, a triangle has all adjacent vertices.
The more angles a regular polygon has, the greater their degree measure. However, the degree measure of an angle of a convex polygon cannot be greater than or equal to 180 degrees.
To determine the general degree measure of a polygon, you must use the formula.
Polygons. Types of polygons. Inner and outer corners of a convex polygon. The sum of the interior angles of a convex n-gon (theorem). The sum of the external angles of a convex n-gon (theorem). Regular polygons. Circle circumscribed about a regular polygon (theorem, corollary 1.2)
The interior angle of a convex polygon at a given vertex is the angle formed by its sides converging at that vertex. The exterior angle of a convex polygon at a given vertex is the angle adjacent to the interior angle at that vertex. inner corner outer corner
Theorem. The sum of the interior angles of a convex polygon is (n - 2) · 180 o, where n is the number of sides of the polygon. Given: a convex n-gon. Prove: α = (n – 2) ·180 o Proof Inside the n-gon, take an arbitrary point O and connect it to all vertices. The polygon will be divided into n triangles with a common vertex O. The sum of the angles of each triangle is 180 o, therefore, the sum of the angles of all triangles is 180 o n. This sum, in addition to the sum of all the internal angles of the polygon, includes the sum of the angles of the triangles at the vertex O, equal to 360 o. Thus, the sum of all internal angles of the polygon is 180 o n - 360 o \u003d (n - 2) 180 o. So, n \u003d (n - 2) 180 o. Ch.t.d. about
Theorem. The sum of the external angles of a convex polygon, taken one at each vertex, does not depend on n and is equal to 360, where n is the number of sides of the n-gon. Proof. Since the external corner of the polygon is adjacent to the corresponding internal angle, and the sum of the adjacent angles is 180, then the sum of the external angles of the polygon is: . External and internal internal So, the sum of the external angles of a convex polygon, taken one at each vertex, does not depend on n and is equal to 360 o, where n is the number of sides of the n-gon. Ch.t.d.
Theorem. Any regular polygon can be inscribed with a circle, and moreover, only one. Proof. Let А1,А2,…,А n be a regular polygon, О be the center of the circumscribed circle. ОА1А2 =ОА2А3= ОАnА1, therefore the heights of these triangles, drawn from the vertex О, are also equal to ОН1=ОН2=…=ОНn. Therefore, the circle with therefore the circle with center O and radius OH1 passes through the points H1, H2, ..., Hn and touches the sides of the polygon at these points, i.e. the circle is inscribed in the given polygon. Hn H1 H2 H3 A1 A2 A3 An
Let us prove that there is only one inscribed circle. Suppose there is another inscribed circle with center O and radius OA. Then its center is equidistant from the sides of the polygon, i.e. the point O1 lies on each of the bisectors of the angles of the polygon, and therefore coincides with the point O of the intersection of these bisectors. The radius of this circle is equal to the distance from the point O to the sides of the polygon, i.e. is equal to OH1. The theorem is proved. Corollary 1 A circle inscribed in a regular polygon touches the sides of the polygon at their midpoints. Corollary 2 The center of a circle circumscribed about a regular polygon coincides with the center of a circle inscribed in the same polygon.