Lesson from the series “ Geometric algorithms»

Hello dear reader.

The solution to many problems in computational geometry is based on finding polygon area. In this lesson, we will derive a formula for calculating the area of a polygon through the coordinates of its vertices, and write a function to calculate this area.

Task. Calculate area of polygon, specified by the coordinates of its vertices, in the order of their traversal clockwise.

Insights from Computational Geometry

To derive the formula for the area of a polygon, we need information from computational geometry, namely, the concept of the oriented area of a triangle.

The oriented area of a triangle is an ordinary area provided with a sign. Sign of the oriented area of a triangle ABC the same as the oriented angle between the vectors and . That is, its sign depends on the order in which the vertices are listed.

On rice. 1 triangle ABC– rectangular. Its oriented area is equal to (it is greater than zero, since the pair is oriented positively). The same value can be calculated in another way.

Let ABOUT– arbitrary point of the plane. In our figure the area triangle ABC obtained by subtracting the areas OAB and OCA from the area of triangle OBC. So you just need add oriented areas triangles OAB, OBC and OCA. This rule works for any choice of point ABOUT.

Similarly, to calculate the area of any polygon, you need to add up the oriented areas of the triangles

The total will be the area of the polygon, taken with a plus sign if, when traversing the polygon, the polygon is on the left (counterclockwise traversal of the boundary), and with a minus sign if it is on the right (clockwise traversal).

So, calculating the area of a polygon has been reduced to finding the area of a triangle. Let's see how to express it in coordinates.

Vector artwork two vectors on a plane is the area of a parallelogram constructed on these vectors.

Cross product expressed in terms of vector coordinates:

The area of the triangle will be equal to half this area:

It is convenient to take the origin of coordinates as point O, then the coordinates of the vectors on the basis of which the oriented areas are calculated will coincide with the coordinates of the points.

Let (x 1, y 1), (x 2, y 2), ..., (x N, y N) - coordinates of the vertices of a given polygon in clockwise or counterclockwise traversal order. Then its oriented area S will be equal to:

This is ours working formula, it is used in our program.

If the coordinates of the vertices were specified in counterclockwise order, then the number S, calculated using this formula will be positive. Otherwise it will be negative, and to obtain the usual geometric area we need to take its absolute value.

So, let's consider a program for finding the area of a polygon given by the coordinates of the vertices.

Program geom6; Const n_max=200; (maximum number of points+1) type b=record x,y:real; end; myArray= array of b; var input:text; A:myArray; s:real; i,n:integer; procedure ZapMas(var n:integer; var A:myArray); (Filling the array) begin assign(input,"input.pas"); reset(input); readln(input, n); for i:=1 to n do read(input, a[i].x,a[i].y); close(input); end; function Square (A:myarray): real; (Calculating the area of a polygon) var i:integer; S: real; begin a.x:=a.x; a.y:=a.y; s:=0; for i:=1 to n do s:= s + (a[i].x*a.y - a[i].y*a.x); s:=abs(s/2); Square:= S end; (Square) begin (main) Zapmas(n, a); PrintMas(a); S:= Square(a); writeln("S= ",s:6:2); end.

The coordinates of the vertices are read from the file input.pas., stored in an array A as records with two fields. For the convenience of traversing the polygon, n+1 elements are introduced into the array, the value of which is equal to the value of the first element of the array.

Input data:
5
0.6 2.1 1.8 3.6 2.2 2.3 3.6 2.4 3.1 0.5

Output:
S= 3.91

We solved the problem of finding the area of a polygon from the coordinates of its vertices. The tasks are becoming more difficult. If you have comments on this article or suggestions, write in the comments. I will be very grateful for your cooperation.

See you in the next lesson.

In this article we will talk about how to express the area of a polygon into which a circle can be inscribed, through the radius of this circle. It’s worth noting right away that not every polygon can fit a circle. However, if this is possible, then the formula by which the area of such a polygon is calculated becomes very simple. Read this article to the end or watch the attached video tutorial, and you will learn how to express the area of a polygon in terms of the radius of the circle inscribed in it.

Formula for the area of a polygon in terms of the radius of the inscribed circle

Let's draw a polygon A 1 A 2 A 3 A 4 A 5, not necessarily correct, but one into which a circle can be inscribed. Let me remind you that an inscribed circle is a circle that touches all sides of the polygon. In the picture it is a green circle with a center at the point O:

We took the 5-gon as an example here. But in fact, this is not of significant importance, since the further proof is valid for both a 6-gon and an 8-gon, and in general for any arbitrary “gon”.

If you connect the center of the inscribed circle with all the vertices of the polygon, then it will be divided into as many triangles as there are vertices in the given polygon. In our case: for 5 triangles. If we connect the dot O with all points of tangency of the inscribed circle with the sides of the polygon, then you get 5 segments (in the figure below these are segments OH 1 , OH 2 , OH 3 , OH 4 and OH 5), which are equal to the radius of the circle and perpendicular to the sides of the polygon to which they are drawn. The latter is true, since the radius drawn to the point of contact is perpendicular to the tangent:

How to find the area of our circumscribed polygon? The answer is simple. You need to add up the areas of all the resulting triangles:

Let's consider what the area of a triangle is. In the picture below it is highlighted in yellow:

It is equal to half the product of the base A 1 A 2 to height OH 1, drawn to this base. But, as we have already found out, this height is equal to the radius of the inscribed circle. That is, the formula for the area of a triangle takes the form: , Where r— radius of the inscribed circle. The areas of all remaining triangles are found similarly. As a result, the required area of the polygon is equal to:

It can be seen that in all terms of this sum there is a common factor that can be taken out of brackets. The result will be the following expression:

That is, what remains in brackets is simply the sum of all sides of the polygon, that is, its perimeter P. Most often in this formula the expression is simply replaced by p and they call this letter “semi-perimeter”. As a result, the final formula takes the form:

That is, the area of a polygon into which a circle of known radius is inscribed is equal to the product of this radius and the half-perimeter of the polygon. This is the result we were aiming for.

Finally, he will note that a circle can always be inscribed in a triangle, which is a special case of a polygon. Therefore, for a triangle this formula can always be applied. For other polygons with more than 3 sides, you first need to make sure that a circle can be inscribed in them. If so, you can safely use this simple formula and use it to find the area of this polygon.

Material prepared by Sergey Valerievich

1.1 Calculation of areas in ancient times

1.2 Different approaches to studying the concepts of “area”, “polygon”, “polygon area”

1.2.1 The concept of area. Area Properties

1.2.2 Concept of polygon

1.2.3 The concept of the area of a polygon. Descriptive definition

1.3 Various formulas for the areas of polygons

1.4 Derivation of formulas for the areas of polygons

1.4.1 Area of a triangle. Heron's formula

1.4.2 Area of a rectangle

1.4.3 Area of a trapezoid

1.4.4 Area of a quadrilateral

1.4.5 Universal formula

1.4.6 Area of n-gon

1.4.7 Calculating the area of a polygon from the coordinates of its vertices

1.4.8 Pick's formula

1.5 Pythagorean theorem on the sum of the areas of squares built on legs right triangle

1.6 Equal arrangement of triangles. Bolyay-Gerwin theorem

1.7 Ratio of areas of similar triangles

1.8 Figures with the largest area

1.8.1 Trapezoid or rectangle

1.8.2 Remarkable property of the square

1.8.3 Sections of other shapes

1.8.4 Triangle with the largest area

Chapter 2. Methodological features of studying the areas of polygons in mathematics classes

2.1 Thematic planning and features of teaching in classes with in-depth study of mathematics

2.2 Methodology for conducting lessons

2.3 Results of experimental work

Conclusion

Literature

Introduction

The topic “Areas of Polygons” is an integral part of school course mathematics, which is quite natural. After all, historically the very emergence of geometry is associated with the need to compare land plots of one shape or another. However, it should be noted that educational opportunities for covering this topic in high school are far from being fully used.

The main task of teaching mathematics at school is to ensure students’ strong and conscious mastery of the system of mathematical knowledge and skills required in everyday life And labor activity every member modern society sufficient for studying related disciplines and continuing education.

Along with solving the main problem, in-depth study of mathematics involves the formation in students of a sustainable interest in the subject, identifying and developing their mathematical abilities, orientation towards professions significantly related to mathematics, preparation for studying at a university.

The qualifying work includes the content of a mathematics course secondary school and a number of additional questions directly adjacent to this course and deepening it along the main ideological lines.

The inclusion of additional questions has two interrelated purposes. On the one hand, this is the creation, in conjunction with the main sections of the course, of a base for satisfying the interests and development of the abilities of students with a penchant for mathematics; on the other hand, it is the fulfillment of the content gaps of the main course, giving the content in-depth study necessary integrity.

The qualifying work consists of an introduction, two chapters, a conclusion and cited literature. The first chapter discusses the theoretical foundations of studying the areas of polygons, and the second chapter deals directly with methodological features studying areas.

Chapter 1. Theoretical foundations studying the areas of polygons

1.1Calculation of areas in ancient times

The beginnings of geometric knowledge related to the measurement of areas are lost in the depths of thousands of years.

Even 4 - 5 thousand years ago, the Babylonians were able to determine the area of a rectangle and trapezium in square units. The square has long served as a standard for measuring areas due to its many remarkable properties: equal sides, equal and right angles, symmetry and general perfection of form. Squares are easy to construct, or you can fill a plane without gaps.

IN ancient China The measure of area was a rectangle. When masons determined the area of a rectangular wall of a house, they multiplied the height and width of the wall. This is the definition accepted in geometry: the area of a rectangle is equal to the product of its adjacent sides. Both of these sides must be expressed in the same linear units. Their product will be the area of the rectangle, expressed in the corresponding square units. Say, if the height and width of a wall are measured in decimeters, then the product of both measurements will be expressed in square decimeters. And if the area of each facing raft is a square decimeter, then the resulting product will indicate the number of tiles needed for the cladding. This follows from the statement underlying the measurement of areas: the area of a figure composed of non-intersecting figures is equal to the sum of their areas.

The ancient Egyptians 4,000 years ago used almost the same techniques as we do to measure the area of a rectangle, triangle, and trapezoid: the base of the triangle was divided in half and multiplied by the height; for a trapezoid, the sum of the parallel sides was divided in half and multiplied by the height, etc. To calculate area

quadrilateral with sides (Fig. 1.1), formula (1.1) was used

those. The half sums of the opposite sides were multiplied.

This formula is clearly incorrect for any quadrilateral; it follows, in particular, that the areas of all rhombuses are the same. Meanwhile, it is obvious that the areas of such rhombuses depend on the size of the angles at the vertices. This formula is only true for a rectangle. With its help, you can approximately calculate the area of quadrilaterals whose angles are close to right angles.

To determine the area

isosceles triangle(Fig. 1.2), in which the Egyptians used an approximate formula:

(1.2) Fig. 1.2 The error made in this case is smaller, the smaller the difference between the side and the height of the triangle, in other words, the closer the vertex (and ) to the base of the height from . That is why the approximate formula (1.2) is applicable only for triangles with a relatively small angle at the apex.

But already the ancient Greeks knew how to correctly find the areas of polygons. In his Elements, Euclid does not use the word “area,” since by the word “figure” itself he understands a part of a plane bounded by one or another closed line. Euclid does not express the result of measuring area by number, but compares areas different figures among themselves.

Like other scientists of antiquity, Euclid deals with the issues of transforming some figures into others of equal size. The area of a composite figure will not change if its parts are arranged differently, but without intersecting. Therefore, for example, it is possible, based on the formulas for the area of a rectangle, to find formulas for the areas of other figures. Thus, a triangle is divided into parts from which a rectangle of equal size can then be formed. From this construction it follows that the area of a triangle is equal to half the product of its base and height. By resorting to such a recut, they find that the area of a parallelogram is equal to the product of the base and the height, and the area of a trapezoid is the product of half the sum of the bases and the height.

When masons have to tile a wall with a complex configuration, they can determine the area of the wall by counting the number of tiles used for cladding. Some tiles, of course, will have to be chipped so that the edges of the cladding coincide with the edge of the wall. The number of all tiles used in the work estimates the wall area with an excess, the number of unbroken tiles – with a deficiency. As the size of the cells decreases, the amount of waste decreases, and the wall area, determined through the number of tiles, is calculated more and more accurately.

One of the later Greek mathematicians and encyclopedists, whose works were mainly of an applied nature, was Heron of Alexandria, who lived in the 1st century. n. e. Being an outstanding engineer, he was also called "Heron the Mechanic". In his work "Dioptrics" Heron describes various machines and practical measuring instruments.

One of Heron’s books was called “Geometrics” and is a kind of collection of formulas and corresponding problems. It contains examples on calculating the areas of squares, rectangles and triangles. About finding the area of a triangle based on its sides, Heron writes: “Let, for example, one side of the triangle has a length of 13 measuring cords, the second 14 and the third 15. To find the area, proceed as follows. Add 13, 14 and 15; it will be 42. Half of this will be 21. Subtract from this the three sides one by one; first subtract 13 - you are left with 8, then 14 - you are left with 7, and finally 15 - you are left with 6. Now multiply them: 21 times 8 gives 168, take this 7 times - you get 1176, and take this 6 more times - you get 7056. From here square root will be 84. That’s how many measuring cords there will be in the area of the triangle.”

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 A quadrilateral is a geometric figure consisting of four points (vertices), no three of which lie on the same straight 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

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 correct form are equal, therefore, 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.

404 not found

Decorating homes, clothing, and painting contributed to the process of forming and accumulating information in the field of geometry, which people of those times obtained experimentally, 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 in everyday life. Therefore, you need to learn to calculate the area 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 come across 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 more than complex design For example, a pentagonal room or a room with a round arch, draw a sketch on a piece of paper. Then divide complex shape into several simple ones, for example, into 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 thought of as special case 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 the 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 number of sides of a 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.

A polygon is a flat or convex figure that consists of intersecting straight lines (more than 3) and forms large number points of intersection of lines. Another polygon can be defined as a broken line that closes. In another way, the intersection points can be called the vertices of the figure. Depending on the number of vertices, the figure may be called a pentagon, hexagon, and so on. The angle of a polygon is the angle formed by the sides meeting at one vertex. The angle is inside the polygon. Moreover, the angles can be different, up to 180 degrees. There are also external corners, which are usually adjacent to the internal one.

The straight lines that subsequently intersect are called the sides of the polygon. They can be adjacent, adjacent or non-adjacent. A very important characteristic presented geometric figure is that its non-adjacent sides do not intersect, and therefore do not have common points. Adjacent sides of a figure cannot be on the same straight line.

Those vertices of a figure that belong to the same line can be called adjacent. If you draw a line between two vertices that are not adjacent, you get the diagonal of a polygon. As for the area of a figure, this is the internal part of the plane of a geometric figure with a large number of vertices, which is created by the polygon segments dividing it.

There is no single solution for determining the area of the presented geometric figure, since there can be an infinite number of variants of the figure and for each variant there is its own solution. However, some of the most common options for finding the area of a figure still need to be considered (they are most often used in practice and are even included in the school curriculum).

First of all, let's consider a regular polygon, that is, a figure in which all the angles formed by equal sides are also equal. So, how do you find the area of a polygon in a specific example? For this case, finding the area of a polygonal figure is possible if the radius of the circle inscribed in the figure or circumscribed around it is given. To do this, you can use the following formula:

S = ½∙P∙r, where r is the radius of a circle (inscribed or circumscribed), and P is the perimeter of a geometric polygonal figure, which can be found by multiplying the number of sides of the figure by their length.

How to find the area of a polygon

To answer the question of how to find the area of a polygon, it is enough to follow the following interesting property of a polygonal figure, which was once discovered by the famous Austrian mathematician Georg Pieck. For example, using the formula S = N + M/2 -1, you can find the area of a polygon whose vertices are located at the nodes of a square grid. In this case, S is, accordingly, the area; N – the number of square grid nodes that are located inside a figure with many corners; M is the number of those nodes of the square grid that are located on the vertices and sides of the polygon. However, despite its beauty, Pick's formula is practically not used in practical geometry.

The simplest and most famous method for determining area, which is studied in school, is to divide a polygonal geometric figure into simpler parts (trapezoids, rectangles, triangles). Finding the area of these figures is not difficult. In this case, the area of the polygon is determined simply: you need to find the areas of all those figures into which the polygon is divided.

Basically, the definition of the area of a polygon is determined in mechanics (dimensions of parts).

The area of ​​a polygon through the radius of the inscribed circle. How to find the area of ​​a polygon? Equal area of ​​the polygon

Formula for the area of ​​a polygon in terms of the radius of the inscribed circle