Calculation of the side of a polygon. Construction of a regular n-gon

Programming environment:

Visual Studio 2013

IN in this example constructing a polygon based on the number of sides n, coordinates of the polygon center and distance R from the center of the polygon to its side. All this data is entered by the user and begins to be processed by clicking the "Build" button. The program allows you to draw polygons with different parameters on one shape.

Function button1_Click receives input parameters and processes them for correctness. In case of incorrect data: a negative number of sides or a negative distance, the program reports that the data is incorrect (if negative coordinates are entered, the polygon is shifted relative to the visibility area and, at certain values, may be completely outside the visibility area (outside the form), as in the case of entering sufficient of great importance distances). If the data entered by the user is correct, then control passes to the function lineAngle, which directly constructs a polygon.

Program code:

using System ; using System.Collections.Generic ; using System.ComponentModel ; using System.Data ; using System.Drawing ; using System.Linq ; using System.Text ; using System.Threading.Tasks ; using System.Windows.Forms ; namespace pravilnyy_mnogougolnik ( public partial class Form1 : Form ( public Form1() ( InitializeComponent() ; ) int n; //number of sides int R; //distance from center to side Point Cntr; //center Point p; //array of points of the future polygon //create an array of points of our polygon private void lineAngle(double angle) ( double z = 0 ; int i= 0 ; while (i< n+ 1 ) { p[ i] . X = Cntr. X + (int ) ( Math. Round (Math. Cos (z/ 180 * Math. PI ) * R) ) ; p[ i] . Y = Cntr. Y - (int ) ( Math. Round (Math. Sin (z/ 180 * Math. PI ) * R) ) ; z= z+ angle; i++; } } private void button1_Click(object sender, EventArgs e) { label10. Text = "" ; //receive input data and check it for correctness n = Convert. ToInt32(textBox4.Text); R = Convert. ToInt32(textBox5.Text); Cntr. X = Convert. ToInt32(textBox6.Text); Cntr. Y = Convert. ToInt32(textBox7.Text); if(n< 0 || R < 0 ) label10. Text = "Invalid input data!"; else //input data is correct, draw a polygon( p = new Point[ n + 1 ] ; lineAngle((double ) (360.0 / (double ) n) ) ; int i = n; Graphics g = pictureBox2. CreateGraphics(); while (i > 0 ) ( g. DrawLine ( new Pen(Color. Black, 2) , p[ i] , p[ i - 1 ] ) ; i = i - 1 ; ) ) ) //leave the drawn polygon, reset the input values for the new input private void button2_Click(object sender, EventArgs e) ( textBox4. Text = "0" ; textBox5. Text = "0" ; textBox6. Text = "0" ; textBox7. Text = "0" ; label10. Text = "" ; ) // erase everything drawn without resetting the last input data private void button3_Click(object sender, EventArgs e) ( pictureBox2. Image = null ; label10. Text = "" ; ) ) )

Converter of distance and length units Converter of area units Join us © 2011-2017 Dovzhik Mikhail Copying of materials is prohibited. In the online calculator you can use values in the same units of measurement! If you have difficulty converting units of measurement, use the distance and length unit converter and the area unit converter. Additional features of the quadrilateral area calculator

You can move between input fields by pressing the “right” and “left” keys on the keyboard.

Theory. Area of a quadrilateral Quadrilateral - geometric figure, consisting of four points (vertices), no three of which lie on the same line, and four segments (sides) connecting these points in pairs. A quadrilateral is called convex if the segment connecting any two points of this quadrilateral is located inside it.

How to find out the area of a polygon?

The formula for determining the area is determined by taking each edge of the polygon AB, and calculating the area of the triangle ABO with its vertex at the origin O, through the coordinates of the vertices. When walking around a polygon, triangles are formed that include the inside of the polygon and those located outside it. The difference between the sum of these areas is the area of the polygon itself.

Therefore, the formula is called the surveyor's formula, since the "cartographer" is located at the origin; if he walks around the area counterclockwise, the area is added if it is on the left and subtracted if it is on the right from the point of view of the origin. The area formula is valid for any self-disjoint (simple) polygon, which can be convex or concave. Content

1 Definition
2 Examples
3 More complex example
4 Explanation of name
5 See

Area of a polygon

Attention

It could be:

triangle;
quadrilateral;
pentagon or hexagon and so on.

Such a figure will certainly be characterized by two positions:

Adjacent sides do not belong to the same straight line.
Non-adjacent ones have no common points, that is, they do not intersect.

To understand which vertices are neighboring, you will need to see if they belong to the same side. If yes, then neighboring ones. Otherwise, they can be connected by a segment, which must be called a diagonal. They can only be carried out in polygons that have more than three vertices.

What types of them exist? A polygon with more than four corners can be convex or concave. The difference between the latter is that some of its vertices may lie on opposite sides of a straight line drawn through an arbitrary side of the polygon.

How to find the area of a regular and irregular hexagon?

Knowing the length of the side, multiply it by 6 and get the perimeter of the hexagon: 10 cm x 6 = 60 cm
Let's substitute the results obtained into our formula:

square roots

regular hexagon

Trapezoid method.
A method for calculating the area of irregular polygons using the coordinate axis.
A method for breaking a hexagon into other shapes.

Depending on the initial data that you know, a suitable method is selected.

Important

Some irregular hexagons consist of two parallelograms. To determine the area of a parallelogram, multiply its length by its width and then add the two famous squares. Video on how to find the area of a polygon An equilateral hexagon has six equal sides and is a regular hexagon.

The area of an equilateral hexagon is equal to 6 areas of the triangles into which a regular hexagonal figure is divided. All triangles in a hexagon of regular shape are equal, so to find the area of such a hexagon it will be enough to know the area of at least one triangle. To find the area of an equilateral hexagon, we use, of course, the formula for the area of a regular hexagon described above.

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Decorating a home, clothing, and drawing pictures contributed to the process of forming and accumulating information in the field of geometry, which the people of those times obtained empirically, bit by bit, and passed on from generation to generation. Today, knowledge of geometry is necessary for the cutter, the builder, the architect, and everyone to the common man at home. Therefore, you need to learn to calculate the area of various figures, and remember that each of the formulas can be useful later in practice, including the formula for a regular hexagon.
A hexagon is a polygonal figure whose total number of angles is six. A regular hexagon is a hexagonal figure that has equal sides. The angles of a regular hexagon are also equal to each other.
IN Everyday life we can often find objects that have the shape of a regular hexagon.

Area calculator of an irregular polygon by sides

You will need

- roulette;
— electronic rangefinder;
- a sheet of paper and a pencil;
- calculator.

Instruction 1 If you need total area apartment or a separate room, just read the technical passport for the apartment or house, it shows the footage of each room and the total footage of the apartment. 2 To measure the area of a rectangular or square room, take a tape measure or electronic rangefinder and measure the length of the walls. When measuring distances with a rangefinder, be sure to ensure that the direction of the beam is perpendicular, otherwise the measurement results may be distorted. 3 Then multiply the resulting length (in meters) of the room by the width (in meters). The resulting value will be the floor area, it is measured in square meters.

Gaussian area formula

If you need to calculate the floor area of a more complex structure, such as a pentagonal room or a room with a round arch, draw a sketch on a piece of paper. Then divide the complex shape into several simple ones, such as a square and a triangle or a rectangle and a semicircle. Using a tape measure or rangefinder, measure the size of all sides of the resulting figures (for a circle you need to know the diameter) and record the results on your drawing.

5 Now calculate the area of each figure separately. Calculate the area of rectangles and squares by multiplying the sides. To calculate the area of a circle, divide the diameter in half and square it (multiply it by itself), then multiply the resulting value by 3.14.
If you only need half a circle, divide the resulting area in half. To calculate the area of a triangle, find P by dividing the sum of all sides by 2.

Formula for calculating the area of an irregular polygon

If the points are numbered sequentially in a counterclockwise direction, then the determinants in the formula above are positive and the modulus in it can be omitted; if they are numbered in a clockwise direction, the determinants will be negative. This is because the formula can be considered as a special case of Green's theorem. To apply the formula, you need to know the coordinates of the vertices of the polygon in the Cartesian plane.

For example, let's take a triangle with coordinates ((2, 1), (4, 5), (7, 8)). Let's take the first x-coordinate of the first vertex and multiply it by the y-coordinate of the second vertex, and then multiply the x-coordinate of the second vertex by the y-coordinate of the third. Let's repeat this procedure for all vertices. The result can be determined by the following formula: A tri.

Formula for calculating the area of an irregular quadrilateral

A) _(\text(tri.))=(1 \over 2)|x_(1)y_(2)+x_(2)y_(3)+x_(3)y_(1)-x_(2) y_(1)-x_(3)y_(2)-x_(1)y_(3)|) where xi and yi denote the corresponding coordinate. This formula can be obtained by opening the parentheses in general formula for the case n = 3. Using this formula, you can find that the area of the triangle is equal to half the sum of 10 + 32 + 7 − 4 − 35 − 16, which gives 3. The number of variables in the formula depends on the number of sides of the polygon. For example, the formula for the area of a pentagon would use variables up to x5 and y5: A pent. = 1 2 | x 1 y 2 + x 2 y 3 + x 3 y 4 + x 4 y 5 + x 5 y 1 − x 2 y 1 − x 3 y 2 − x 4 y 3 − x 5 y 4 − x 1 y 5 | (\displaystyle \mathbf (A) _(\text(pent.))=(1 \over 2)|x_(1)y_(2)+x_(2)y_(3)+x_(3)y_(4 )+x_(4)y_(5)+x_(5)y_(1)-x_(2)y_(1)-x_(3)y_(2)-x_(4)y_(3)-x_(5 )y_(4)-x_(1)y_(5)|) A for a quadrilateral - variables up to x4 and y4: A quad.

This online calculator helps to calculate, determine and calculate the area of a land plot online. The presented program can correctly suggest how to calculate the area of land plots of irregular shape.

Important! The important area should fit approximately into the circle. Otherwise, the calculations will not be entirely accurate.

We indicate all data in meters

A B, D A, C D, B C— The size of each side of the plot.

According to the entered data, our program performs online calculations and determines the area of land in square meters, acres, acres and hectares.

Method for determining the size of a plot manually

To correctly calculate the area of plots, you do not need to use complex tools. We take wooden pegs or metal rods and install them in the corners of our site. Next, using a measuring tape, determine the width and length of the plot. As a rule, it is enough to measure one width and one length, for rectangular or equilateral areas. For example, we have the following data: width – 20 meters and length – 40 meters.

Next we move on to calculating the area of the plot. If the shape of the area is correct, you can use geometric formula determining the area (S) of a rectangle. According to this formula, you need to multiply the width (20) by the length (40), that is, the product of the lengths of the two sides. In our case S=800 m².

After we have determined our area, we can determine the number of acres per plot of land. According to generally accepted data, one hundred square meters is 100 m². Next, using simple arithmetic, we will divide our parameter S by 100. The finished result will be equal to the size of the plot in acres. For our example, this result is 8. Thus, we find that the area of the plot is eight acres.

In the case where the land area is very large, it is best to carry out all measurements in other units - in hectares. According to generally accepted units of measurement - 1 Ha = 100 acres. For example, if our land plot, according to the measurements obtained, is 10,000 m², then in this case its area is equal to 1 hectare or 100 acres.

If your plot is of irregular shape, then the number of acres directly depends on the area. It is for this reason that using online calculator You will be able to correctly calculate the parameter S of the plot, and then divide the result by 100. Thus, you will receive calculations in acres. This method makes it possible to measure plots complex shapes, which is very convenient.

Total information

Calculation of the area of land plots is based on classical calculations, which are performed according to generally accepted geodetic formulas.

There are several methods available for calculating the area of land - mechanical (calculated according to the plan using measuring palettes), graphic (determined by the project) and analytical (using the area formula based on measured boundary lines).

Today, the most accurate method is deservedly considered to be analytical. Using this method, errors in calculations, as a rule, appear due to errors in the terrain of the measured lines. This method It is also quite complex if the boundaries are curved or the number of angles on the plot is more than ten.

The graphical method is a little easier to calculate. It is best used when the boundaries of the site are presented in the form of a broken line, with a small number of turns.

And the most accessible and simple method, and the most popular, but at the same time the biggest error is the mechanical method. Using this method, you can easily and quickly calculate the area of land of simple or complex shape.

Among the serious disadvantages of the mechanical or graphical method, the following are distinguished: in addition to errors in measuring the area, during calculations an error is added due to the deformation of the paper or an error in drawing up plans.

Calculation of the side of a polygon. Construction of a regular n-gon

How to find out the area of ​​a polygon?

Area of ​​a polygon

How to find the area of ​​a regular and irregular hexagon?