Advice 1: How to find the area of the diagonal section

If both sides of a plane there are points belonging to three-dimensional figures (e.g., polyhedra), this plane can be called a secant. A two-dimensional figure formed by the General points in the plane and a polyhedron in this case is called a cross section. This section will be diagonal, if one of the diagonals of the base belongs to the section plane.
How to find the area of the diagonal section
The diagonal section of the cube has the shape of a rectangle, the area (S) it is easy to calculate, knowing the length of any edge (a) three-dimensional figures. In this rectangle one of the sides is the height, which coincides with the length of the edge. The length of the other diagonal - calculate the Pythagorean theorem for the triangle in which it is the hypotenuse, and the two edges of the base - the legs. In General it can be written as: a*√2. The area of the diagonal section find by multiplying the two sides, the length of which you found out: S = a*a*√2 = a2*√2. For example, if the edge length of 20 cm the area of the diagonal section of the cube should be approximately equal 202*√2 ≈ 565,686 cm2.
To compute the area of the diagonal cross section of the parallelepiped (S) proceed the same way, but note that the Pythagorean theorem in this case involved the legs different lengths - length (l) and width (w) volumetric shapes. The length of the diagonal in this case is equal to √(l2+w2). Height (h) may also vary from the edge lengths of the bases, so in General the formula for the area of the cross section can be written as: S = h*√(l2+w2). For example, if the length, width and height of a cuboid are equal, respectively, 10, 20 and 30 cm square of its diagonal section approximately 30*√(102+202) = 30*√500 ≈ 670,82 cm2.
The diagonal cross-section of the quadrangular pyramid has a triangular shape. If the height (H) of the polytope are known, and at its base lies a rectangle, the lengths of connected edges (a and b) which is also given in the conditions, the calculation of the area of the cross section (S) begin with computing the length of a diagonal of the base. As in previous steps, use the triangle of the two edges of the base and diagonal, where the Pythagorean theorem the length of the hypotenuse is equal to √(a2+b2). The height of the pyramid in this polyhedron coincides with the height of the triangle diagonal cross-section, is lowered to the side whose length you just determined. So to find the area of a triangle, find half of the product height on the length of the diagonal: S = ½*H*√(a2+b2). For example, at a height of 30 cm and lengths of adjacent sides of the base 40 and 50 cm the area of the diagonal section should be approximately equal to ½ *30*√(402+502) = 15*√4100 ≈ 960,47 cm2.

Advice 2 : How to determine the cross-sectional area

If the cross section of the object has a complex shape, calculate its area to be split into areas of simple shapes. After that, it will be possible to calculate the areas of these sections by appropriate formulas, and then to add them up.
How to determine the cross-sectional area
Divide the cross section of the object on a region having the shape of triangles, rectangles, squares, sectors, circles, semi-circles and quarters of circles. If the result of the division will be diamonds, divide each of them into two triangles, and parallelograms if two triangle and one rectangle. Measure the dimensions of each of these areas: hand, radii. All measurements are carried out in the same units.
A right triangle can be represented in the form of half of a rectangle split in half diagonally. To calculate the area of a triangle multiply each other the length of those sides which are adjacent to the right angle (called the legs), and then multiplying the result divide by two. If the triangle is not rectangular, calculate its area first swipe it from any angle height. It will be divided into two different triangles, each of which is rectangular. Measure the lengths of the legs of each of them, and then the measurement results calculate their area.
To calculate the area of a rectangle, multiply on top of each other the lengths of its two adjacent to each other of the parties. A square they are equal, so the length of one side multiplied on itself, that is, to erect it in the square.
To determine the area of a circle divide erect its radius squared, then multiply the result by the number π. If the figure is not a circle, semi-circle, divide the square into two, and if a quarter of the circle four. The sector measure the angle between the center of the imaginary center and the ends of the arc, move it from degrees to radians, multiply the square of the radius, and then divide by two.
Add up all the resulting square with each other, and work area, expressed in units of the same order as the original data. For example, if the lengths of the sides and radii measured to you in millimeters, the area of work in square millimeters.
Greatly facilitate the measurement of area of a complex figure will help the device, called a planimeter. Set its scale to zero, and then slide the probe along the contour of the figure. Read the scale. The accuracy of such measurements are relatively small.
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