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The vertex Bg of the upper base of the prism is projected into the center of a circle of radius r inscribed in the lower base. A plane is drawn through the side AC of the base and the vertex Br, which is inclined to the plane of the base at an angle a.

One of the vertices of the upper base of the prism is equidistant from all the vertices of the lower base. Find the volume of the prism if the side edge makes an angle equal to a with the plane - g of the base.

One of the vertices of the upper base of the prism is equidistant from all the vertices of the lower base.

A right circular cone is described near a prism if all the vertices of the upper base of the prism lie on the lateral surface of the cone, and the lower base of the prism lies in the plane of the base of the cone. In this case, the base of the prism is a polygon around which a circle can be described. Note that the lower base of the prism is not inscribed in the base of the cone.

A prism is inscribed in a right circular cone if all the vertices of the upper base of the prism lie on the lateral surface of the cone, and the lower base of the prism lies on the base of the cone. The base of the prism is a polygon around which a circle can be circumscribed (but the lower base of the prism is not inscribed in the circle of the base of the cone.

P BI and P CI determine the frontal projections L, B and C of the combined tops of the upper base of the prism. By connecting successively aligned vertices with broken lines, we obtain a development of the lateral surface of the prism. Adding to it the natural values ​​​​of both bases, we get a complete sweep.

From points 1 - 6 of the horizontal projection of the lower base, direct projections of the ribs are carried out parallel to the x axis, and six points are found on them using vertical communication lines - horizontal projections of the tops of the upper base of the prism.

From points / - 6 of the horizontal projection of the lower base, straight lines are drawn - projections of the ribs - parallel to the axis l: and six points are found on them using vertical communication lines - horizontal projections of the tops of the upper base of the prism.

The base of an inclined prism is an isosceles triangle, in which AB a, AC a and LCAB a. The vertex BI of the upper base of the prism is equidistant from all sides of the lower base, and the edge BI.

The base of an inclined prism is an isosceles trapezoid, in which the lateral side is equal to the smaller base and is equal to a, and the acute angle is equal to a. One of the vertices of the upper base of the prism is equidistant from all the vertices of the lower base.

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Let K be the orthogonal projection of the vertex A of the inclined prism ABCA1B1C1 onto the plane of the base A1B1C1, AB = BC = AC = AA1 = BB1 = DD1 = a. By the condition of the problem AA1K = 60 From the right triangle AKA1 we find that
AK = AA1 sin AA1K = a sin 60o = $$ a\sqrt(3)/2 $$, and since AK is the height of the prism ABCA1B1C1, then
Vprisms = SΔABC AK =$$ a^2\sqrt(3)/4\cdot a\sqrt(3)/2 $$

Answer: $$ 3a^3/8 $$

Related tasks:

1. The base of the prism is a triangle, in which one side is 2 cm, and the other two are 3 cm each. The lateral edge is 4 cm and makes an angle of 45 with the base plane. Find the edge of an equal cube.

2. The base of the inclined prism is an equilateral triangle with side a; one of the side faces is perpendicular to the plane of the base and is a rhombus whose smaller diagonal is c. Find the volume of the prism.

3. In an inclined prism, the base is a right triangle, the hypotenuse of which is equal to c, one acute angle is 30, the side edge is equal to and makes an angle of 60 with the base plane. Find the volume of the prism.

; b) the area of ​​the base of the prism.
its longest diagonal is 7 cm. Find: a) the height of the prism;

13. The side of the base of a regular quadrangular prism is 4 cm. The diagonal of the prism forms an angle of 60 0 with the base plane. Find: a) the height of the prism; b) lateral surface area; c) total surface area; d) area of ​​the diagonal section of the prism; e) the cross-sectional area of ​​the lower base passing through the midpoints of adjacent sides parallel to the diagonal section.

14. Side of the base of a regular triangular prism 2
cm, and the height of the prism is 4 cm. Find the cross-sectional area passing through the side edge of the prism and the height of the base of the prism.

1. The base of a rectangular parallelepiped is a square. The diagonal of the parallelepiped is 4 cm and makes an angle of 30 0 with the side face. Find the side of the base of the parallelepiped, its height and lateral surface area.

four . The base of a right parallelepiped is a rhombus with diagonals 6cm and 8cm. The large diagonal of the parallelepiped is 10cm. Find a) the smaller diagonal of the parallelepiped,

B) total surface area.
5. Diagonal of a rectangular

The parallelepiped makes up with

The base plane angle is 45 0 .

Base sides 3cm and 4cm.

B) the total surface area of ​​the parallelepiped.

B) the area of ​​the side face passing through the unknown leg;

C) the angle of inclination of this face to the plane of the base.

5 . The base of the pyramid is a rhombus with a side of 8 cm and an angle of 30 0 . The side faces form angles of 60 0 with the base plane. Find the total surface area of ​​the pyramid.

No. 228. The base of the inclined prism ABCA1B1C1 is an isosceles triangle ABC, in which AC = AB = 13cm, BC = 10cm, and the lateral edge of the prism forms an angle of 450 with the base plane. The projection of the vertex A1 is the intersection point of the medians of the triangle ABC. Find the area of ​​face CC1B1B. A1. C1. B1. 13. A. C. 13. 10. B.

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Polyhedron

"Problems on polyhedra" - Polyhedron. Diagonal. Triangle. The height of a regular quadrangular prism. Trapeze. Parallelepiped. Side rib. Lateral surface area. Non-convex polyhedron. The edge of an oblique quadrangular prism. Section. Rhombus. The sum of the areas of all faces. Cross-sectional area. Base sides. direct prism.

"Cascades of polyhedra" - Unit tetrahedron. Octahedron and tetrahedron. Octahedron and icosahedron. The edge of the icosahedron. Cascades of regular polyhedra. Tetrahedron and cube. Dodecahedron edge. Polyhedron. Icosahedron and cube. Tetrahedron and dodecahedron. Tetrahedron and octahedron. Edge of a cube. Dodecahedron and tetrahedron. Icosahedron and tetrahedron. Icosahedron and octahedron. Cube and dodecahedron.

"The geometric body is a polyhedron" - Euclid. Let's look at crystals. Geometric shapes. Prisms. Polyhedra. Any diagonal square. Memphis. The first wonder of the world. Edge. Great Pyramid. City buildings. Polyhedra. triangular pyramid. base of the prism. A bit of history. Scientists and Philosophers of Ancient Greece. Side edges. Mausoleum in Halicarnassus.

"The concept of a polyhedron" - Polyhedra. What is a tetrahedron. quadrangular prism. Edges are sides of faces. What is a rectangular parallelepiped. The height of the prism is the perpendicular. Theorem. The sum of the areas of all its faces. Facets. Prism. Definition. A straight prism is called a right prism. What is a parallelepiped. The concept of a polyhedron.

""Polyhedra" stereometry" - Historical reference. Archimedean bodies. Epigraph of the lesson. Do the geometric shapes and their names match. Section of polyhedra. "Game with spectators". Give a name to the polyhedron. Great Pyramid at Giza. Specify the correct section. Fix the logic chain. Polyhedra in architecture. Problem solving.

"Five Platonic solids" - Firstly, all faces of such a body are equal in size. Tetrahedron. By connecting the centers of the faces of the icosahedron, we again obtain a dodecahedron. According to the Mayan tradition, the Tree of Life grew out of a cube. In general, a polyhedron is one of the three-dimensional geometric shapes. A cube has an angle of 90 degrees. cube. Therefore, the cross generated by the development of the cube also denotes limitation, suffering.

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