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9. Spatial geometry

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Spatial Geometry is one of the most important parts of mathematics for the Enem test. Spatial Geometry is the study of figures in space, that is, figures that have more than two dimensions. Among the most studied figures, we can highlight the prism, the pyramid, the cylinder, the sphere and the cone.

The prism is a solid figure that has two parallel and equal bases and all of its lateral faces are parallelograms. The total area of a prism is the sum of the areas of its faces. The volume of the prism is given by the product of the area of the base and the height.

The pyramid is a solid figure that has a base and all its lateral faces are triangles that meet at the same point, called the apex of the pyramid. The total area of a pyramid is the sum of the areas of its faces. The volume of the pyramid is given by the product of the area of the base and the height divided by 3.

The cylinder is a solid figure that has two equal circular bases and a side face that is a rectangle. The total area of the cylinder is the sum of the areas of its faces. The volume of the cylinder is given by the product of the area of the base and the height.

A sphere is a solid figure that has all its points at the same distance from a central point, called the center of the sphere. The area of the sphere is given by 4 times the product of pi and the radius squared. The volume of the sphere is given by 4/3 times the product of pi and the radius cubed.

The cone is a solid figure that has a circular base and a side face that is a curved surface. The total area of the cone is the sum of the base area and the lateral area. The volume of the cone is given by 1/3 times the product of the area of the base by the height.

In addition to the calculation of areas and volumes, Spatial Geometry also involves the study of plane sections of solid figures, called conic sections, which include the ellipse, parabola and hyperbola. These sections are generated by the intersection of a plane and a cone.

The ellipse is the conic section obtained when the plane intersects the cone obliquely, not parallel to the axis of the cone. The parabola is the conic section obtained when the plane intersects the cone parallel to one of the cone's generatrix. The hyperbola is the conic section obtained when the plane intersects the two parts of the cone.

These are the main topics of Spatial Geometry that are required in the Enem test. However, it is important to remember that Spatial Geometry is a part of Geometry, which also includes Plane Geometry and Analytic Geometry. Therefore, for a complete preparation for the Enem test, it is important to study all these parts of Geometry.

Furthermore, Spatial Geometry, like all other parts of mathematics, requires a lot of practice. Therefore, in addition to understanding the concepts, it is important to do a lot of exercises to become familiar with the types of problems that may appear on the test and to develop problem-solving skills.

In summary, Spatial Geometry is a fundamental part of mathematics for the ENEM exam and, with study and practice, can be mastered by any student.

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