Advice 1: How to find lengths of edges of the parallelepiped diagonally

Parallelepiped – a multi-faceted geometric shape, has some interesting properties. Knowledge of these properties helps in solving problems. For example, there exists a certain relationship between its linear and diagonal measurements, which can be used to find lengths of edges of the parallelepiped diagonally.
How to find lengths of edges of the parallelepiped diagonally
Instruction
1
The box has one feature that is not characteristic of other figures. Its faces are pairwise parallel and have equal dimension and numerical characteristics, such as area and perimeter. Any pair of these faces could be mistaken for the base, then the rest will be be its side.
2
You can find the edge length of the parallelepiped diagonally, but one of this magnitude is not enough. First, notice what this kind of spatial figures you are given. It can be a right parallelepiped, with right angles and equal dimensions, i.e. cubic In this case is enough to know the length of one diagonal. In all other cases, must be at least one known parameter.
3
Diagonals and length of sides of the parallelepiped associated to a particular ratio. This formula is derived from the spherical law of cosines and represents the equality of the sum of squares of diagonals and the sums of the squares of ribs:
d12 + d22 + d32 + d42 = 4•A2 + 4•b2 + 4•c2, where a – length b – width c - height.
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For a cube the formula is simplified:
4•d2 = 12•A2
a = d/√3.
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Example: to find the length of a side of a cube if the diagonal is 5 cm.
Solution.
25 = 3•A2
a = 5/√3.
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Consider a straight parallelepiped whose lateral edges are perpendicular to the bases, and the bases are parallelograms. Its diagonals are equal and are associated with the lengths of the edges in the following way:
d12 = A2 + b2 + c2 + 2•a•b•cos α;
d22 = A2 + b2 +c2 – 2•a•b•cos α, where α is the acute angle between the sides of the base.
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This formula can be used, if known, for example, one of the sides and angle or these values can be found in other conditions of the problem. The decision is easier when all the angles at the base are straight, then:
d12 + d22 = 2•A2 + 2•b2 + 2•c2.
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Example: find the width and height of a cuboid if the width b is greater than the length and 1 cm in height c – 2 times more, and the diagonal d – 3.
Solution.
Write down the basic formula of the square diagonal (in a rectangular parallelepiped are equal):
d2 = A2 + b2 + c2.
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Express all measurements using a given length a:
b = a + 1;
c = a•2;
d = a•3.
Substitute into the formula:
9•A2 = A2 + (a + 1)2 + 4•A2
10
Solve the quadratic equation:
3•A2 – 2•a – 1 = 0
Find the lengths of all edges:
a = 1; b = 2; c = 2.

Advice 2: How to find the diagonal of a parallelepiped

The parallelepiped is a particular case of the prism, in which all six faces are parallelograms or rectangles. A parallelepiped with rectangular faces is called a rectangular. Of parallelepiped has four diagonals intersect. If the three edges a, b, C, to find all the diagonals of a rectangular parallelepiped, you can do additional build.
How to find the diagonal of a parallelepiped
Instruction
1
Draw a rectangular parallelepiped. Write known data: three edges a, b, C. First, build a single diagonal m. For its determination we use the property of a rectangular parallelepiped, according to which all the corners are straight.
How to find the diagonal of a parallelepiped
2
Construct a diagonal n one of the faces of the parallelepiped. Build a guide so that the edge is known, the required diagonal of the parallelepiped and the diagonal faces together formed a right triangle a, n, m.
How to find the diagonal of a parallelepiped
3
Find built diagonal faces. It is the hypotenuse of another right triangle b, C, n. According to the Pythagorean theorem n2 = S2 + b2. Calculate this expression and take the square root of the obtained values, it will be the diagonal of n faces.
4
Find the diagonal of a parallelepiped m. To do this in a right triangle a, n, m find the unknown hypotenuse: m2 = n2 + a2. Substitute known values, then calculate the square root. The result is the diagonal of a parallelepiped m.
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Likewise consistently spend the remaining three diagonals of a parallelepiped. Also, for each of them will complete additional construction of the diagonals of the adjacent faces. Formed by considering right triangles and applying the Pythagorean theorem, find the values of the remaining diagonals of a rectangular parallelepiped.
How to find the diagonal of a parallelepiped

Advice 3: How to find the edges of the base of the tetrahedron

Four - "Tetra" - to name three-dimensional geometric figures indicates the number of its constituent edges. And the number of faces of the regular tetrahedron, in turn, uniquely determines the configuration of each of them, four surfaces can be three-dimensional shape, having the shape of a regular triangle. The calculation of the lengths of edges is composed of equilateral triangles of the figure does this.
How to find the edges of the base of the tetrahedron
Instruction
1
In the figure, composed of exactly the same faces, the base can be considered as any of them, so the problem is reduced to calculating the length of an arbitrarily chosen edge. If you are aware of the total surface area of the tetrahedron (S), to calculate the length of edge (a) extract from it the square root and divide the result by the cube root of triples: a = √S/3√3.
2
The area of one face (s), must obviously be four times smaller than the full surface area. Therefore, for calculation of the edge length for this parameter is transform the formula from the previous step to this: a = 2*√s/3√3.
3
If the conditions given height (H) of a tetrahedron, to find the length of side (a), each component of the line triple is the only known value, and then divide by the square root of six: a = 3*H/√6.
4
While known because of the problem volume (V) of the tetrahedron to calculate the length of edge (a) will have to extract the cube root of this value, increased twelve times. Calculate this value, divide it, and in the fourth root of two: a = 3√(12*V)/⁴√2.
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Knowing the diameter circumscribed about the tetrahedron of the sphere (D), too it is possible to find the length of its edge (a). To do this, increase the diameter in half and then divide by the square root of six: a = 2*D/√6.
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The diameter of the inscribed in this shape of a sphere (d) the length of the edge is determined by almost the same the only difference is that the diameter should be increased not twice, but six times: a = 6*d/√6.
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Circle radius (r) inscribed in any face of this figure also allows us to calculate the necessary size - multiply it by six and divide by the square root of triples: a = r*6/√3.
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If the conditions of the problem given the total length of all edges of the regular tetrahedron (P), to find the length of each of them just divide this number by six which is the number of edges has this body figure: a = P/6.
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