Enclosed by six identical squaresStereoscopic figureIt is called a regular hexahedron, also called a cube or a cube.A regular hexahedron is a straight body whose sides and bottom are squareParallelepiped, i.eArris lengthAll equal hexahedrons.Hexahedron is specialBox。The dynamic definition of a regular hexahedron is:squareTranslate the side length of the square in the direction perpendicular to the face of the squareStereoscopic figure。
Chinese name
Regular hexahedron
Foreign name
cube
Alias
Cube、cube
Discipline
mathematics
Properties
It belongs to hexahedron and is a special box
Features
6 faces have the same shape and each edge has the same length
(1) A regular hexahedron has eight vertices, each of which connects three edges.
(2) A regular hexahedron has 12 edges, each of which has the same length.
(3) A regular hexahedron has 6 faces, each of which has the same area and shape.
(4) Body diagonal of a regular hexahedron:
, where a is the edge length.
Surface area and volume
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Surface area
Because the six faces of the hexahedron are all equal and square, the surface area of the hexahedron
, where a is the edge length of the regular hexahedron, and S is the surface area of the regular hexahedron.
volume
Cube is a kind of prism, and the volume formula of prism is also applicable, that is, volume=bottom area × height.Since the six faces of the hexahedron are all equal and square, the volume of the hexahedron=edge length × edge length × edge length.
If the edge length of a cube is a, its volume is:
。
Related concepts
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Body diagonal
For example, as shown in Figure 1, set the normal cube ABCD-
The edge length of is a,
Figure 1
(1) First, take the diagonal of the upper and lower surfaces (as the line AC in Figure 1), and calculate the length of AB=
；
(2) The diagonal AC of this face intersects with its edge, which is perpendicular to the upper and lower face
, can form a right triangle, and the hypotenuse of the right triangle
It is the body diagonal. According to Pythagorean theorem, we can get the length of the body diagonal of the regular hexahedron=
。
Unit volume
(1) Hexahedron with edge length of 1cm and volume of 1cm;
(2) Hexahedron with edge length of 1 decimeter and volume of 1 cubic decimeter;
(3) A regular hexahedron with a length of 1 meter and a volume of 1 cubic meter.
Sphere radius
(1) Radius of the outer ball: the radius of the outer ball R=half of the diagonal of the cube;
(2) Radius of inscribed sphere: radius of inscribed sphere r=half of the side length of the cube.
Plane truncated cube
The following triangles, rectangles, squares, pentagons, hexagons, regular hexagons, rhombus and trapezoids can be obtained by cutting a plane into a cube. The specific cutting method is as follows:
(1) Triangle: a line passing through a vertex within the diagonal of the opposite face;
(2) Rectangle: passing through two opposite edges or one edge;
(3) Square: parallel to a face;
(4) Pentagon: passing through a point on four edges and a vertex or a point on five edges;
(5) Hexagon: passing the point on six edges;
(6) Regular hexagon: passing through the midpoint of six edges;
(7) Diamond: passing the relative vertex;
(8) Trapezoid: a parallel and unequal line passing through two opposite faces.
Panorama
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The unfolded figure 2 of the regular hexahedron is as follows:
(1) Type 1, 4, 1:
Figure 2
(2) Type 2, 3, 1:
Regular hexahedron
(3) Type 2, 2, 2:
Regular hexahedron
(4) 3, Type 3:
Regular hexahedron
Beautiful fixed value
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Theorem 1
Theorem 1: If the side length of a square is a, and the radius of a circle with its center as the center is r, then the sum of the squares and the sum of the fourths of the lengths of the line segments connecting any point on the circle and each vertex of the square are fixed values.
(1) Conclusion 1.1: If the side length of square ABCD is a and P is any point on its circumscribed circle, then:
Is a fixed value.
(2) Conclusion 1.2: If the side length of square ABCD is a and P is any point on its inscribed circle, then:
(3) Inference 1.3: If the side length of a positive n-sided shape (n=2k) is a, and the radius of the circle with the center of the positive n-sided shape as the center is r, then the square sum of the lengths of the line segments connecting any point on the circle and each vertex of the positive n-sided shape is:
(fixed value).
(4) Inference 1.4: If the diagonal length of the positive n-sided shape (n=4k) is m, and the radius of the circle with the center of the positive n-sided shape as the center is r, then the sum of the fourth power of the length of the line segments connecting any point on the circle and each vertex of the positive n-sided shape is a fixed value:
Theorem 2
Theorem 2: If the edge length of the cube is a, and the radius of the sphere whose center is the center of the sphere is R, then the square sum of the lengths of the line connecting any point on the sphere and each vertex of the cube is:
(constant value), the sum of the fourth power is:
(fixed value).
(1) Inference 2.1: If the edge length of the cube is a, then the sum of the squares of the lengths of the connecting lines between any point on its circumscribed sphere and each vertex of the cube is
, the sum of the fourth power is
。
(2) Inference 2.2: If the edge length of a cube is a, then the sum of the squares of the lengths of the lines connecting any point on the inscribed sphere and the vertices of the cube is
