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A cross section shows how an object intersects with a plane along its axis. When a solid (such as a cone, cylinder, or sphere) is divided by a plane, a cross-section is formed. A circle is formed when a cylinder-shaped object is divided in half with a plane parallel to its base. As a result, things are beginning to fall into place. A cross section is the non-empty intersection of a solid body in three-dimensional space with a plane, or the equivalent in higher-dimensional spaces, in geometry and physics
Definition of cross-section
A cross-section is the shape created by crossing a solid with a plane in geometry. A two-dimensional geometric form is a cross-section of a three-dimensional shape. In other terms, a cross-section is the shape created by cutting a solid parallel to the base. A cross section is the non-empty intersection of a solid body in three-dimensional space with a plane, or the equivalent in higher-dimensional spaces, in geometry and physics. A cross section is the non-empty intersection of a solid body in three-dimensional space with a plane, or the equivalent in higher-dimensional spaces, in geometry and physics
Examples of cross-sections
Examples of cross-sections for various shapes include:
Any sphere’s cross-section is a circle.
A triangle forms the vertical cross-section of a cone, whereas a circle forms the horizontal cross-section.
A cylinder’s vertical cross-section is a rectangle, while its horizontal cross-section is a circle.
Cross Sections Types
The following are the two types of cross-sections:
Horizontal cross-section
Transverse cross-section
Cross Section: Horizontal or Parallel
To make a parallel cross-section, a plane cuts a solid shape horizontally (i.e., parallel to the base).
Cross Section: Vertical or Perpendicular
To make a perpendicular cross-section, a plane cuts a solid form vertically (i.e., perpendicular to the base).
Cross-sectional Geometry
Here are some cross sectional area examples for various solids. Consider the cross sections of a cube, sphere, cone, and cylinder.
Cross-Sectional Area
When a plane slices a solid object, an area is projected onto the plane. The plane aligns itself with the symmetry axis. The size of the projection is the cross-sectional area.
Cross Sections of Cone
A pyramid having a circular cross-section is called a cone. The cross-section, also known as conic sections (for a cone), can be a circle, a parabola, an ellipse, or a hyperbola, depending on the relationship between the plane and the slant surface.
Cross sections of Cylinder
A cylinder’s cross-section can be either a circle, rectangle, or oval, depending on how it was cut. The form obtained is a circle if the cylinder has a horizontal cross-section. The form obtained if the plane slices the cylinder perpendicular to the base is a rectangle. The oval form is achieved by cutting the cylinder parallel to the base with a minor angle change.
Cross Sections of Sphere
We know that a sphere has the smallest surface area for its volume of all the forms. A circle is formed when a plane figure meets a sphere. A sphere’s cross-sections are all circles.
Problems Related to cross-section area
Calculate the cross-sectional area of a plane with a volume of 64 cm3 that is perpendicular to the cube’s base.
Solution:
We already know that,
Volume of cube = side3
Therefore,
Side3 = 64 [Given]
Thus,
Side = 4 cm
The square’s side will be 4 cm since the cube’s cross-section will be square.
As a consequence, a2 =16 sq.cm is the cross-sectional area.
Calculate the cross-section area of a cylinder with a 2 cm radius and a 10 cm height.
Solution:
Given:
Radius = 2 cm
Height = 10cm
As we all know, a circle is generated when the plane splits the cylinder parallel to the base.
As a result, the area of a circle is A = πr2 square units.
Using π = 3.14
Changing the values in the formula we get,
A = 3.14 (2)2 cm2
A = 3.14 (4) cm2
A = 12.56 cm2
As a consequence, the cross-section area of the cylinder is 12.56 cm2.
Conclusion
Many parallel cross-sections are created when an item is cut into slices. A contour line is the boundary of a three-dimensional cross-section that is parallel to two of the axes, that is, parallel to the plane determined by these axes; for example, if a plane cuts through mountains in a raised-relief map parallel to the ground, the result is a contour line in two-dimensional space showing points on the surface of the mountains of equal elevation.
A cross-section is a typical tool used in technical drawing to portray the internal layout of a 3-dimensional item in two dimensions. It is a projection of an object onto a plane that crosses it. It is generally crosshatched, with the crosshatching technique frequently identifying the materials employed.
When a plane crosses a solid (a three-dimensional object), the region between the plane and the solid is referred to as a cross-section of the solid. A cutting plane is defined as a plane that contains a cross-section of a solid. The orientation of the cutting plane to the solid may affect the form of the solid’s cross-section.
