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Cross-section

 

A cross-section is the shape of an object when cut through a plane along a particular axis. It is the shape formed when a plane intersects its longitudinal or parallel axis. For example, if a sphere is cut along its axis (can take any axis because a sphere is symmetrical from all its sides), the shape we get is a circle. Hence, the cross-section of a sphere is a circle. A cross-section is possible only in the case of 3D objects and not 2D figures.

Types of cross-section

  1. Horizontal cross-section – It is a parallel cross-section. In this, the object is cut along its parallel axis, such that only the cross-section occurring parallel is visible. For example, the parallel cross-section of a cylinder will be square.
  2. Vertical or perpendicular cross-section – If the object is cut along its perpendicular axis, the shape that we get is the vertical cross-section. For example, the perpendicular cross-section of a cylinder is a circle, i.e., if the cylinder is cut along its base, we will get a circular cross-section.

Cross-sectional area

The area of the cross-section of an object is called the cross-sectional area. So when an object is cut along the perpendicular plane, if we get a plane along the axis of symmetry, the section that we get will be called a Cross-sectional.

Types of shapes and their cross sections

Shape Cross section Figure
Cube We get different cross sections in a cube, if it is cut along different axes. We get a trapezium, a hexagon, a square, and even a triangle, depending upon the axis of the plane. IMAGE 1
Cuboid A cuboid will give a square cross section, if it is cut along the longer side. We will get a rectangular section, if the cuboid is cut along the smaller side. IMAGE 2
Cone A cone is a pyramid-like structure with a circular base and a triangular top. The cross-section of a cone is called a conic section. This is achieved by depending on the relationship between the plan and slant surface. It could be a circle, a parabola, an ellipse. IMAGE 3
Cylinder A cylinder can produce a circle, rectangle, or oval if cut along different axes of symmetry to get the cross-section.

If the cylinder is cut along its horizontal cross-section, then the shape we get is a circle. On the other hand, if the plane of the base of the cylinder is perpendicular to its base, then the shape would be a rectangle.

When the plane intersects the cylinder parallel to the base, we get an oval shape as the cross-section.
IMAGE 4
Sphere A sphere is a round solid figure whose surface, with every point on its surface equidistant from its centre.

A sphere has the littlest area for its volume. The intersection of a plane in a sphere produces a circle. Likewise, all cross-sections of a sphere are circles.
IMAGE 5

Example

Find the cross sectional area of a cylinder whose height is 25cm and radius is 4cm.

Solution

Let us consider, the cylinder is cut along its perpendicular axis. We know, the cross section of a cylinder is a circle in that case.

Therefore, the area of the circle is πr², where r is the radius of the circle.

Area of circle = π x 4 x 4 = 50.24 cm².

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