The cross-section represents the intersection of a plane with a three-dimensional object. The cross-section is the two-dimensional figure obtained by the intersection of a plane with a three-dimensional figure. For example, when a cylinder is cut by a plane that is parallel to one of its bases, the cross-section obtained is a circle. Depending on the orientation of the plane, we can obtain several cross-sections from the same object.

Here, we will learn more details about the cross-sections and we will explore the cross-sections of the most important figures.

GEOMETRY
hexagonal cross section of a cube

Relevant for

Learning about the cross-sections of geometric figures.

See cross-sections

GEOMETRY
hexagonal cross section of a cube

Relevant for

Learning about the cross-sections of geometric figures.

See cross-sections

Types of cross sections

Depending on the orientation of the plane that cuts the object, we can have three types of cross-sections:

  • Horizontal cross section
  • Vertical cross section
  • Inclined cross section

Horizontal or parallel cross-section

This cross-section is formed when a plane intersects an object in a direction parallel to the base of the object.

Vertical or perpendicular cross-section

A vertical cross-section is formed when an object is cut by a plane perpendicular to its base, that is, at an angle of 90°.

Inclined cross-section

This cross-section is formed when the angle of inclination of the plane that intersects the object is greater than 0° and less than 90°.


Cross-section area

When a plane intersects a solid object, an area is projected onto the plane. If the plane is oriented horizontally, it will be perpendicular to the axis of symmetry. Depending on the figure, we can calculate the cross-sectional area by recognizing that the cross-section equals the bases of the figure.

EXAMPLE

Find the cross-sectional area of a plane parallel to the base of a cube that has a volume equal to 125 m³.

Solution: We can recognize that the cross-sectional area will be equal to the area of one of the faces of the cube since the plane is parallel to the bases. Therefore, we have to find the area of one face of the cube.

The volume of the cube is equal to V = {{a}^3}, where, a is the length of one side of the cube. This means that the length of one side is a = 5.

Now, we know that the area of a square is A = {{a}^2}, so the cross-sectional area is A=25.


Cross-sections of a cube

A cube is a three-dimensional figure that has all its sides of the same length. A cube has a total of six square faces. This means that when a plane intersects a cube in a direction that is parallel to one of its faces, the cross-section will always be a square.

square cross section of a cube

However, it is also possible to obtain different cross-sections by cutting the cube with an inclination relative to its base. If the plane intersects three edges of the cube, its cross-section will be a triangle.

triangular cross section of a cube

If the cutting plane cuts the cube in a way that crosses the diagonals of the faces, we will obtain a rectangular cross-section:

rectangular cross section of a cube

Additionally, we can also form a hexagonal cross-section by intersecting the cube with an inclined plane as follows:

hexagonal cross section of a cube

Cross-sections of a cylinder

A cylinder is a 3D figure that has circular bases, which are connected by a lateral surface. Depending on how it is cut, the cross-sections of the cylinder can be a circle, a rectangle, or an oval. If the cylinder is cut with a plane parallel to one of its bases, the cross-section will be a circle.

circular cross section of a cylinder

The oval cross-section is obtained when the plane intersects the cylinder with an angle greater than 0° and less than 90° with respect to the base.

oval cross section of a cylinder

If the plane cuts the cylinder in a direction perpendicular to the bases, the cross-section will be a rectangle.

rectangular cross section of a cylinder

Cross-sections of a cone

A cone can be considered as a pyramid with a circular cross-section. Depending on the relationship between the plane and the cut surface, the cross-sections of a cone can be circles, ellipses, parabolas, and hyperbolas.

A circle is formed when a cone is cut by a plane that is parallel to its base.

circular cross section of a cone

An ellipse is formed when a cone is cut by a plane that is inclined at a small angle (less than the angle of the lateral sides) with respect to the base of the cone.

elliptical cross section of a cone

A parabola is formed when the plane that intersects the cone is parallel to one lateral side of the cone.

parabolic cross section of a cone

A hyperbola is formed when the plane that intersects the cone has a superior angle (greater than the angle of the lateral sides) with respect to the base of the cone.

hyperbolic cross section of a cone

Cross-sections of a sphere

A sphere is a perfectly round three-dimensional figure. The cross-section of a sphere is always a circle regardless of the orientation of the plane.

circular cross sections of a sphere

See also

Interested in learning more about geometric figures? Take a look at these pages:

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