Despite the fact that mathematics is the Queen of Science, and the arithmetic - Queen of mathematics, the greatest difficulty in studying schoolchildren causes geometry. Planimetry is a section of geometry, which studies flat pieces. One of these figures is rhombus. Most tasks by solving quadrangles are reduced to finding their area. We systematize well-known formulas and various methods for calculating the area of \u200b\u200brhombus.
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Rhombus is a parallelogram, all four sides of which are equal. Recall that the parallelogram has four corners and four pairwise parallel equal side. Like any quadrangle, rhombus has a number of properties that are reduced to the following: when the diagonal is crossed, an angle of equal to 90 degrees (AC ⊥ BD) is formed, the intersection point divides each one into two equal segments. The diagonally of rhombus is also bisector of its angles (∠dca \u003d ∠bca, ∠abd \u003d ∠cbd, etc.). From here it follows that they share a rhombus on four equal rectangular triangles. The sum of the lengths of the diagonals erected into the second degree is equal to the length of the side to the second degree multiplied by 4, i.e. BD. 2 + AC 2 \u003d 4AB. 2.
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There are many methods used in planimetry to calculate the area of \u200b\u200brhombus, the use of which depends on the source data. If the side length and any angle are known, you can use the following formula: Roma Square is equal to the square side multiplied by the corner sinus. From the trigonometry course it is known that sin (π - α) \u003d sin α, which means that the sinus of any angle can be used in the calculations - both sharp and stupid. A special case is a rhombus, which has all the corners direct. This is a square. It is known that the sinus of the direct angle is equal to one, so the square of the square is equal to the length of its side, erected into the second degree.
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If the size of the parties is unknown, we use the length of the diagonals. In this case, the Roma Square is equal to half the work of large and small diagonals.
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With a known length of diagonals and the magnitude of any angle, the rhombus area is determined in two ways. The first: the area is half the square of the larger diagonal, multiplied by the tangent of half a degree of acute angle, i.e. S \u003d 1/2 * D 2* TG (α / 2), where D is a large diagonal, α is an acute angle. If you know the size of a smaller diagonal, we use the 1/2 * D formula 2* TG (β / 2), where D is a smaller diagonal, β - dull angle. Recall that the measure of an acute angle is less than 90 degrees (measures of direct angle), and a dull angle, respectively - more than 90 0.
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Roma Square can be found using the side length (remind, all sides of the rhombus are equal) and height. The height is a perpendicular, lowered on the opposite corner of the side or its continuation. So that the base of the height is located inside the rhombus, it should be lowered from a stupid angle.
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Sometimes in the task you need to find the area of \u200b\u200bROMM, based on the data related to the inscribed circle. In this case, it is necessary to know its radius. There are two formulas that can be used to calculate. So, to answer the question of the assigned question, you can double the work of the side of the rhombus and the radius of the inscribed circle. In other words, you must multiply the diameter of the inscribed circle to the side of the rhombus. If the value of the corner is presented in the problem, then the area is via the private between the square of the radius multiplied by four, and the sine corner.
As you can see, there are many ways to find the square of the rhombus. Of course, to remember each of them, you will need patience, attentiveness and, of course, time. But in the future you can easily select a method suitable for your task, and make sure that the geometry is easy.
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