Advice 1: How to find the diagonal right prism

Finding a diagonal right prism is often used as an intermediate step for more complex tasks. The General formula is easily derived by consideration of two right triangles.
How to find the diagonal right prism
To find the diagonal right prism you need to understand just a few definitions.

A prism is called a polyhedron, having as bases two equal polygon (triangle, quadrangle, etc.) lying in parallel planes and the lateral faces parallelograms.
Direct the prism is called prism, whose lateral faces rectangles.
A right prism is called a straight prism, the base of which are regular polygon (equilateral triangle, square, etc.)
АВСDА1В1С1D1 - regular quadrangular prism.
АА1В1В - the side faces of the regular quadrangular prism.
All four side faces of the prism are equal.
ABCD and А1В1С1D1 -base of the prism (squares lying in parallel planes).
A diagonal of a polyhedron is the line segment joining two not adjacent vertices, i.e. vertices that do not belong to the same face.
The figure shows that the point A and the point 1 does not belong to one face and therefore the cut of AC1 - diagonal of the prism.
To find the diagonal of prism necessary to consider the triangle АСС1. This triangle is rectangular. Diagonal prism AC1 in the triangle will be the hypotenuse, and the segments as SS1 and the other two sides. From the Pythagorean theorem (in a right triangle the hypotenuse squared is equal to the sum of the squares of the legs) it follows that:
АС12 = AC2 + СС12 (1);
Next, you should consider the triangle АСD. Triangle АСD is also rectangular (because the base of the prism is square). For convenience, you can designate the base of the letter. Therefore, by the Pythagorean theorem:
AC2 = A2 +A2 AC = √2A (2);
If we denote the height of the prism with the letter h, and substitute the expression (2) into the expression (1), we get:
АС12 = 2A2+h2, AC1 = √(2a^2+h^2 ), where a - side of the base, h is the height.
This formula is valid for any right prism.

Advice 2: How to find the diagonal

Each polyhedron, a rectangle, a parallelogram and a diagonal. It usually connects the corners of any of these geometric shapes. The value of the diagonal has to find the solution of problems in elementary and higher mathematics.
How to find the diagonal
A diagonal is any straight line connecting the corners of the polyhedra. Its location depends on the type of shapes (rhombus, square, parallelogram) and the data given in the problem. The easiest way to find the diagonal of a rectangle is the following.The two sides of the rectangle a and b. Knowing that all of its angles equal to 90°, and its diagonal is the hypotenuse of the two triangles, we can conclude that the diagonal of this fighti can be found by using the Pythagorean theorem. In this case, the sides of the rectangle are the legs of triangles. It follows that the diagonal of a rectangle is:d=√(a^2+b^2)a special case of applying this method to finding diagonals is a square. Its diagonal can also be found by the Pythagorean theorem, but given that all its sides are equal, the diagonal of a square is equal to a√2. The value a is the square side.
If given a parallelogram, its diagonal find, as a rule, cosine theorem. However, in exceptional cases, for a given value of the second diagonal can be found from the first equation:d1=√2(a^2+b^2)-d2^2Теорема of cosines applies when not given a second diagonaland are given only the sides and corners. It is a generalized Pythagorean theorem. For example, given a parallelogram whose sides are b and c. Through two opposite corners of a parallelogram is the diagonal of a. Since a, b and c form a triangle, we can apply the theorem of cosines, which can be computed diagonal:a^2=√b^2+c^2-2bc*cosα When given the area of a parallelogram and one of the diagonals, and the angle between the two diagonals, the diagonal can be calculated in the following way:d2=S/d1*cos
arambam is called a parallelogram whose all sides are equal. Let him have the two sides equal to a, and, the unknown diagonal. Then, knowing the theorem of cosines, the diagonal can be calculated by the formula:d=a^2+a^2-2a*a*cosα=2a^2(1-cosα)
The diagonal of a trapezoid is several ways. To calculate you need to know, as a rule, three values - top and bottom base, and at least one lateral side. This can be seen on the example of a rectangular trapezoid.For example, given a rectangular trapezoid. First you need to find a small segment that is a leg of a right triangle. It is equal to the difference between the upper and lower bases. As a rectangular trapezoid, the drawing shows that the height is equal to the side of the trapezoid. As a result, you can find another lateral side of the trapezoid. If you know the upper base and lateral side, the cosine theorem can be found first diagonal:c^2=a^2+b^2-2ab*soavtory diagonal is based on values of the first lateral side and the upper base on the Pythagorean theorem. In this case, this diagonal is the hypotenuse of a right triangle.
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