How to find the area of a rhombus?To give an answer, you must first understand what we consider a diamond.
Firstly, it is a quadrangle.Secondly, it has four equal sides.Thirdly, the diagonals of the point of intersection perpendicular.Fourth, the point of intersection diagonally divided into equal parts.Fifthly, the same share the diagonal corners of the rhombus into two equal parts.Sixth, in the sum of two angles which are adjacent to one of the sides is the unwrapped angle that is 180 degrees.And if you say simply, the diamond - a sloping square.
If you take a square whose sides are fastened flexibly, and easily pull it in two opposite corner, the square will lose its squareness and turn into a diamond.Therefore, the diamond with right angles - this is a real square.
first introduced the concept of diamond and Papp Hero of Alexandria, mathematics of ancient Greece.The word "diamond" of the Greek can be translated as "drum".
To find the area of a rhombus is worth considering that the diamond - is a parallelogram.And the area of the parallelogram can be found by multiplying together the base, that is the direction and height.
To prove this provision should be deleted from the upper corners of the rhombus vertex normals.For example, given a diamond QWER.From the peaks of the upper corners of Q and W QT and perpendiculars WY.And QT perpendicular falls on the side of RE, and perpendicular WY is on the continuation of this side.
So, get new QWYT quadrilateral with parallel sides and right angles, which is the basis of the above, we can boldly call box.
area of this rectangle is the part of multiplication and height.Now we need to prove that the area of the resulting rectangle on the area corresponding to the present condition of the diamond.
Considering obtained by constructing additional triangles QYR and WET, we can say that they are on the leg and a hypotenuse.For legs of triangles are conducted perpendiculars, which at the same time are also parties to the resulting rectangle.A hypotenuse - is the part of the diamond.
Rhombus is the sum of the square of the triangle and trapezoid QYR QYEW.The resulting rectangle is the same triangle and trapezoid QYEW WET, the area of which is equal to the area of a triangle QYR.Hence the conclusion suggests itself: the importance of the area of the rhombus QWER corresponds to the area of a rectangle QWYT.
Now it becomes clear how to find the area of a rhombus of side and height: they need to multiply.
You can find the area of a rhombus, a rhombus knowing the angle and direction.It is only necessary to know what is the sine of the angle, and multiply it by twice the side.Find the sine can be using a calculator or table Bradis.
Sometimes, talking about how to find the area of a rhombus is used sine of the angle and the radius of the inscribed circle, which definitely is the maximum.
However, most often calculate the area of a rhombus through diagonally.From this formula it follows that the area is poluproizvedeniyu diagonals.
Prove it's pretty simple, having considered the two triangles and QWE ERQ, which received during the diamond in one diagonal.These triangles are equal on three sides or bottom and an adjacent two corners.
spending in the second diagonal of the diamond, we will get the height of these triangles, as the diagonals intersect at the point X at an angle of 90 degrees.The area of a triangle is equal to the product QWE QE, which is one inch at WX - the second half diagonally divided by the two.
Now, the question of how to find the area of a rhombus, the answer is clear: this expression should be doubled.For the convenience of the algebraic expression that you can bring one diagonal designated by the letter z, and the second - the letter u.Get:
2 (z X 1 / 2u: 2) = z x 1 / 2u, it just comes out - poluproizvedenie diagonals.
