Advice 1: How to find apofema

The apofema in the pyramid is called a line segment, drawn from its top to bottom one of the sides if the cut is perpendicular to the ground. The side faces of such a volume always has a triangular shape. So if you need to calculate the length of apodeme acceptable use of the properties of a polyhedron (pyramid), and polygon (triangle).
How to find apofema
You will need
  • - geometrical parameters of the pyramid.
In the triangle side faces apofema (f) is the height, so if you known the length of a side edge (b) and angle (γ) between him and the edge to which omitted apofema, you can use the known formula for calculating the height of the triangle. Multiply the given edge length by the sine of the known angle: f = b*sin(γ). This formula applies to any pyramids (right or wrong) shape.
To calculate each of the three abovem (f) the right of a triangular pyramid is enough to know only one parameter, the edge length (a). This is because the faces of the pyramid have the shape of equilateral triangles of equal size. To find the heights of each of them calculate the half works the length of the edges on the square root of three: f = a*√3/2.
If you know the area (s) of the lateral faces of the pyramid, in addition to it is sufficient to know the length (a) overall edges of this face with base three-dimensional figure. In this case, the length of apodeme (f) find the doubling of the ratio between the area and edge length: f = 2*s/a.
Knowing the total surface area of the pyramid (S) and the perimeter of its base (p), too it is possible to calculate the apofema (f), but only for the polytope of the correct form. Double the surface area and divide the result by the perimeter: f = 2*S/p. The form of the base in this case does not matter.
The number of vertices or sides of the base (n) need to know if in terms of the edge length (b), the side faces and the angle (α) formed by two adjacent lateral edges of a regular pyramid. Under these initial conditions calculate the apofema (f) by multiplying the number of sides of the base to the sine of the known angle and the squared length of a side edge, followed by dividing the values in half: f = n*sin(α)*b2/2.
In a right pyramid with a square base to find the length of apodeme (f) you can use the height of the polytope (H) and the edge length of the base (a). Extract the square root of the sum of squared height and a quarter of the squared lengths of the edges: f = √(H2+a2/4).

Advice 2: How to find apofema in the pyramid

Apofema - the height of the side face held in the right of the pyramid from its top. It can be found in a regular right pyramid, and truncated. We will consider both cases
How to find apofema in the pyramid
All lateral edges are equal, the side faces are isosceles triangles are equal, and the base is a regular polygon. Because all apodeme of a regular pyramid are equal, then it is sufficient to find one in any triangle. The triangles are isosceles, and apofema is the height. Height conducted in an isosceles triangle from the vertex to the base is the median and the angle bisector. The median divides the side in half, and bisects the angle into two equal angles. Height – the perpendicular drawn from the apex to the base.
For example, know all the sides of an isosceles triangle and the median, which divides the base into two equal segments. Since the median is an altitude, it is perpendicular, i.e. the angle between the midpoint and the base is 90 degrees. So, it turns out a right triangle. Side is the hypotenuse, half the base and height(median) is the legs. The Pythagorean theorem States: the square of the hypotenuse is equal to the sum of the squares of the other two sides. This way you can find the height.
Let a known angle, lying opposite the base. And any one side (either the side or the base). Bisector drawn from the vertex to the base, is the height. So again, is a right triangle. Known angle and one side. Using sine, cosine and tangent to find the height. Sine – the ratio of the opposite leg to the hypotenuse, the side - ratio of the adjacent leg to the hypotenuse tangent is the ratio of sine to cosine, or opposite over adjacent. Substituting the known sides, calculate the height.

The lateral surface area of a regular pyramid equals half the product of the perimeter of the base to the apofema.
The right truncated pyramid
Lateral faces – regular trapezoid. The lateral edges are equal. Apofema – height held in the line. Let there are two bases and the lateral edge. From the top are height so that on a larger basis, they cut off a rectangle. Then, if you mentally remove the rectangle will remain isosceles triangle, the height of which can be found by the first method. If you know the obtuse angles of the trapezoid, then during the height, subtract the angle equal 90 degrees(because the altitude is the perpendicular)from stupid. Then will be known the acute angle in the triangle. Height or apofema again, you can find 1 way.
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