Square of geometric shape - Numerical characteristics of the geometric shape showing the size of this figure (parts of the surface limited by a closed loop of this figure). The magnitude of the area is expressed by the number of square units consisting in it.
Triangle square formulas
- The formula of the area of \u200b\u200bthe triangle on the side and height
Area of \u200b\u200ba triangle equal to half the work of the length of the side of the triangle for the length of the height spent - The formula of the triangle area in three sides and radius of the circle described
- The formula of the triangle area in three sides and radius of the inscribed circle
Area of \u200b\u200ba triangle It is equal to the product of the half-versioner of the triangle on the radius of the inscribed circle. where S is the triangle area,
- the length of the side of the triangle,
- the height of the triangle,
- angle between the parties and
- radius inscribed circle,
R is the radius of the described circle,
Formulas square square
- Formula Square square side
Square area equal to the square of the length of his side. - Formula Square square diagonal
Square area Equal to half the length of its length diagonal.S \u003d. 1 2 2 where S is the square of the square,
- the length of the side of the square,
- Square diagonal length.
The formula of the square of the rectangle
- Square rectangle equal to the product of the length of its two adjacent sides
where S is the area of \u200b\u200bthe rectangle,
- The length of the sides of the rectangle.
Paralylogram area formulas
- Formula Square Pollogram side and height
Square Pollogram - The formula of the parallelogram on two sides and the corner between them
Square Pollogram It is equal to the product of its lengths multiplied by the corner between them.a · b · sin α
where S is the area of \u200b\u200bthe parallelogram,
- the length of the sides of the parallelogram,
- the length of the height of the parallelogram,
- The angle between the sides of the parallelogram.
Formulas of Romba
- Formula Square Rhombus side and height
Romba Square It is equal to the product of the length of its side and the length of the height of the height. - Formula Square Roma side and corner
Romba Square It is equal to the product of the square of its side of its side and the corner sinus between the sides of the rhombus. - Formula Square Roma on the lengths of his diagonals
Romba Square Equal to half the length of its lengths of diagonals. where s is the Roma Square,
- the length of the side of the rhombus,
- Length of the height of rhombus,
- angle between the sides of the rhombus,
1, 2 - lengths of diagonals.
Formulas Square Trapezia
- GEONON formula for trapezium
Where S is the square of the trapez
- the length of the foundation,
- the length of the side of the trapeze,
This online calculator helps to make calculation, definition and calculating land area in online mode. The presented program is able to properly prompt how to calculate the area of \u200b\u200bland of the wrong form.
Important! The area must approximately fit into the circle. Otherwise, the calculations will be not entirely accurate.
Indicate all the data in meters
A B, D A, C D, B C- The size of each side of the defense.
According to the data introduced, our program in online mode is calculated and determined by land area in square meters, hundreds and acres and hectares.
Methods for determining the size of the site manual method
To properly perform the calculation of the plot area, 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 measure, we determine the width and length of the defense. As a rule, it suffices to measure one width and one length, for rectangular or equilateral sections. For example, we have the following data: width - 20 meters and length - 40 meters.
Next, go to the calculation of the area of \u200b\u200bthe defense. In the correct form of the site, you can use the geometric formula for determining the area (S) of the rectangle. According to this formula, you need to multiply the width (20) for the length (40), that is, the product of the length of both sides. In our case, S \u003d 800 m².
After we have identified our area, we can define the amount of acres on the land plot. According to generally accepted data, in one hundred and 100 m². Next, with the help of simple arithmetic, we split our parameter s per 100. The finished result and will be equal to the size of the defense in the weave. For our example, this result is 8. Thus, we obtain that the area area is eight acres.
In the event that the territory of the land is very big, it is best to perform all measurements in other units - in hectares. According to generally accepted units of measurement - 1 hectare \u003d 100 acres. For example, if our land plot according to the obtained measurements is 10,000 m², then in this case its area is equal to 1 hectare or 100 hectares.
If your site is irregular, then in this case the amount of acres directly depends on the area. It is for this reason that, with the help of an online calculator, you can correctly calculate the definition parameter, and then deliming the resulting result by 100. Thus, you will receive calculations in the weave. This method provides the ability to measure the plots of complex forms, which is very convenient.
common data
The calculation of land area is based on classical calculations, which are performed according to the generally accepted geodesic formulas.
In total, several methods are available for calculating land area - mechanical (calculated according to plan with the help of dimensional pallet), graphic (determined by the project) and analytical (with the help of the area of \u200b\u200bthe area measured lines).
To date, the most accurate way is deservedly considered - analytical. Using this method, errors in calculations, as a rule, appear due to errors on the terrain of the measured lines. This method is also quite complicated if the boundaries of curvilinear or the number of angle on the plot more than ten.
A little easier by calculations is a graphic method. It is best used in the case when the boundaries of the site are presented in the form of a broken line, with a small amount of turns.
And the most affordable and easy way, and the most popular, but also at the same time the biggest error is a mechanical way. Using this method, you can easily and quickly calculate the area of \u200b\u200bland simple or complex shape.
Among the serious disadvantages of a mechanical or graphic method, the following, except for errors in measuring the site, is added when calculating the error is added due to the deformation of the paper or the error in the preparation of plans.
If there is several segments on the plane to sequentially, so that each next starts in the place where the previous one has ended, then the broken line will be. These segments are called links, and their places of their intersection are vertices. When the end of the last segment intersects with the initial point of the first, then the closed broken line is obtained, which dividing the plane into two parts. One of them is the ultimate, and the second endless.
A simple closed line along with the part of the plane concluded in it (the one that is finite) is called a polygon. Segments are parties, and the corners formed by them are vertices. The number of sides of any polygon is equal to the number of its vertices. The figure that has three sides is called a triangle, and four is a quadrangle. The polygon is numerically characterized by such a magnitude as area that shows the size of the shape. How to find a quadrangle area? This teaches the section of mathematics - geometry.
To find the square of the quadrilateral, you need to know what type does it relate - to convex or unwarked? All lies relatively straight (and it necessarily contains any of its sides) one way. In addition, there are such types of quadrangles as parallelograms with pairwise equal and parallel opposite sides (its varieties: a rectangle with straight corners, a rhombus with equal sides, a square with all straight corners and four equal parties), a trapezium with two parallel opposite sides and Deltoid with two pairs of adjacent sides that are equal.
The square of any polygon is found by applying a general method that is to break it on triangles, for each, calculate the area of \u200b\u200ban arbitrary triangle and folded the results. Any convex quadrilateer is divided into two triangles, a non-depth - two or three of it can be folded from the sum and difference of results. The area of \u200b\u200bany triangle is calculated as half the product of the base (A) to the height (ħ) conducted to the base. The formula that is applied in this case to calculate, is written as: S \u003d ½. a. ħ.
How to find a quadrilateral area, for example, a parallelogram? You need to know the length of the base (A), the length of the side (ƀ) and find the sine of the angle α formed by the base and the side (SINα), the formula for calculation will look like: S \u003d a. ƀ. sinα. Since the sine of the angle α is the product of the base of the parallelogram at its height (ħ \u003d ƀ) - the line is perpendicular to the base, then its area is calculated, multiplying its base: S \u003d a. ħ. To calculate the area of \u200b\u200brhombus and rectangle, this formula is also suitable. Since the rectangle has a side side ƀ coincides with a height ħ, then its area is calculated by the formula s \u003d a. ƀ. Because a \u003d ƀ, it will be equal to the square of it: s \u003d a. a \u003d a². It is calculated as half the sum of its parties, multiplied by height (it is carried out to the base of the trapezoid perpendicular): S \u003d ½. (A + ƀ). ħ.
How to find a quadrilateral area, if the length of its parties is unknown, but is known for its diagonal (E) and (F), as well as the sine of the angle α? In this case, the area is calculated as half of the work of its diagonals (lines that connect the tops of the polygon) multiplied by the sine of the angle α. The formula can be recorded in this form: S \u003d ½. (e. F). sinα. In particular, in this case, half the product of diagonals will be equal to (lines connecting opposite angles of rhombus): S \u003d ½. (e. F).
How to find a quadrilateral area that is not a parallelogram or a trapezium, it is usually called an arbitrary quadrangle. The area of \u200b\u200bsuch a figure is expressed through its half-versioner (ρ - the sum of two sides with the total vertex), the parties a, ƀ, c, d and the sum of the two opposite angles (α + β): s \u003d √ [(ρ - a). (Ρ - ƀ). (Ρ - C). (Ρ - d) - a. ƀ. c. d. Cos² ½ (α + β)].
If a φ \u003d 180o, then to calculate its area, use the brahmagupet formula (Indian astronomer and mathematician, who lived in 6-7 centuries of our era): S \u003d √ [(ρ - a). (Ρ - ƀ). (Ρ - C). (Ρ - D)]. If the quadrilateral is described by a circle, then (a + c \u003d ƀ + d), and its area is calculated: S \u003d √ [a. ƀ. c. d]. SIN ½ (α + β). If the quadrilateral is simultaneously described by one circle and inscribed in another circumference, then the following formula is used to calculate the area: S \u003d √.
Quadrangle The figure consisting of four vertices, three of which do not lie on one straight line, and segments connecting them.
There are many quadrangles. These include parallelograms, squares, rhombus, trapezoids. You can find on the sides, is easily calculated on diagonals. In an arbitrary quadrilateral, you can also use all the elements for the output of the formula of the quadrangle area. To begin with, consider the formula of the quadrangle area through the diagonal. In order for it to use the lengths of diagonals and the size of the acute angle between them. Knowing the necessary data can be carried out an example of calculating the area of \u200b\u200bthe quadrangle according to such a formula:
Half the product of diagonals and the sinus of an acute angle between them is the rank of quadrangle. Consider an example of calculating the quadrangle area through a diagonal.
Let a fetragon with two diagonals D1 \u003d 5 cm; d2 \u003d 4cm. The acute angle between them is equal to α \u003d 30 °. The formula of the quadrilateral area through the diagonal is easily used for well-known conditions. Substitute data: 
On the example of the calculation of the quadrangle area through the diagonal, we understand that the formula is very similar to the calculation.
Quadricle area on the sides
When the length of the side of the figure is known, you can apply the formula of the quadrangle area on the sides. To apply these calculations, you will need to find a half-version figure. We remember that the perimeter is the sum of the lengths of all sides. Semitter is half a perimeter. In our rectangle with the sides of A, B, C, D formula half-version will look like this: 
Knowing the sides, we derive the formula. The area of \u200b\u200bthe quadrilateral is the root from the product of the half-version difference with the length of each side:
Consider an example of calculating the quadrangle area through the parties. Dan arbitrary quadrilateral with sides of a \u003d 5 cm, b \u003d 4 cm, C \u003d 3 cm, D \u003d 6 cm. To begin with, we will find a half-meter: 
We use the value found to calculate the area:
The area of \u200b\u200bthe quadrangle specified by the coordinates
The formula of the quadrilateral area of \u200b\u200bcoordinates is used to calculate the area of \u200b\u200bfigures that are located in the coordinate system. In this case, it takes the calculation of the lengths of the necessary parties. Depending on the type of quadrilateral, the formula itself may vary. Consider an example of calculating the quadrilateral area using a square that lies in the XY coordinate system.
ABCD is given in XY coordinate system. Find the figure of the figure if the coordinates of the vertices A (2; 10); B (10; 8); C (8; 0); D (0; 2). 
We know that all sides of the figure are equal, and the formula of the square of the square is by the formula:
We find one of the parties, for example, AB: 
We substitute the values \u200b\u200bin the formula: 
We know that all sides are the same. We substitute the value in the formula for calculating the area: 
I. Preface
After all, it is not enough: I drove two weeks, you came to school and learned that they missed a very important topic, the tasks of which will be in the exams in the 9th grade - "triangles, quadrangles and their area". Here it would beight to rush to the geometry teacher with questions: "How to find a quadrangle area?" But half of the students are afraid to approach teachers, so that they are not found lagging behind, and the second half meets from teachers "Help", similar to "Look at the textbook, everything is written there!" Or "did not need to miss the lessons!" But in the textbook there is no information at all about the rules of finding the area of \u200b\u200btriangles and quadrangles. And the lessons were missed for a good reason, there is a certificate from the doctor. But many teachers only wave for these arguments with hand. Of course, they can be understood: they do not pay for an additional riding the material of the lesson in the heads of nothing understanding students. Many students throw it a useless case and after a year they fall on the exam, do not pay a dozen points for the task of finding the area of \u200b\u200btriangles and quadrangles. And only some go to the library and to familiar with the question: "How to find a quadrangle area?" And different people and books give different answers, and it turns out a big confusion of the rules. Below I will name the main ways to find the areas of triangles and quadrangles.
II. Quadrangles
Let's start with quadrangles. In schools and on exams only convex quadrangles are considered, so let's talk about them. At the average level of formation, parallelogram and trapezoids are being studied. The parallelograms are of several types: a rectangle, square, rhombus and arbitrary parallelogram, in which only its main signs are observed: the parties are parallel and equal, the sum of the neighboring angles is 180 o. But the ways of finding the squares in all these figures are different. Consider each separately.
1. Rectangle
S rectangle is located by the formula: S \u003d a * b, wherebut - horizontal side, b. - Vertical side. *
2. Squares square
S Square is in the formula: S \u003d a * and wherea. - side of a square.
3. Rombov Square
S rhombus is by the formula: S \u003d 0.5 * (D 1 * d 2), whered 1.- Big Dianogonal, ** d 2. - Little diagonal.
4. Square of an arbitrary parallelogram
S an arbitrary parallelogram is in the formula: S \u003d a * h a, A. - side of the parallelogram, h A.
Not all?
With parallelograms, we finished. "We must learn just that?" - Make sure you ask. I answer: from the parallelograms - yes, just that. But there were still a trapeze and triangles. So continue.
III. Trape c.and I
Square trapezium
S trapezia can be found in one formula, whether it is normal or equilibrium: S \u003d ((a + b): 2) * h, wherea, B. - EE bases, h. - EE height. This is all concerning the trapezium. Now to the question: "How to find a quadrangle area?" - You can not only answer yourself, but also enlighten others. And now we go to triangles.
IV. Triangle
In geometry for finding their area, three formulas were isolated: for rectangular, equilateral and arbitrary triangles.
1. The area of \u200b\u200bthe triangle
S an arbitrary triangle is calculated by the formula: S \u003d 0,5A * H a a. - the side of the triangle, h A. - Height spent on this side.
2. Square of equilateral triangles
S equilateral triangle can be found by the formula: S \u003d 0,5a * h, wherea. - the base of the triangle, h. - The height of this triangle.
3. Square of rectangular triangles
The area of \u200b\u200brectangular triangles is located by the formula: S \u003d (a * b): 2, wherebut - 1st cathet, b. - 2nd catat.
Conclusion
Well, this is, in my opinion, everything. About triangles, too, it is necessary to learn a little, isn't it? Now, promote everything I wrote here. "Trees-stick to learn, will need a month!" - Probably you exclaim. And who said that everything learns fast? But when you learn all this, you will not be afraid of questions on "How to find a square of a quadrangle" or "an arbitrary triangle area" on certification in grade 9. So, if you want to go somewhere at all, learn, learn and be scholars!
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Note
* - a. and b. Do not necessarily have places on me. When solving tasks, you can call the vertical side a., and horizontal - b;
** - Diagonal can be changed in places and change their names as well as in the note. *
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