In mathematics and geometry, a cross-section is the shape created when an object is sliced by a plane. If the formation and dimension of the cross-section are the same at every point throughout the length/ breadth/ height of the solid then the cross-section will always be uniform.

Instance like when a tree is cut it shows a ring shape throughout the surface. This is a real-life example of cross-sections. With this article, you will learn the cross section’s meaning, its type and cross sectional area of geometric figures with examples.

Cross Section Definition 

In geometry, a cross-section is the shape you get when you cut a 3D solid with a flat surface or plane. It shows a 2D shape formed by the slice, usually made parallel to the base of the solid.

For example, If you cut a sphere in any way, the cross-section will always be a circle.

If you cut a cone straight down (vertically), the shape you get is a triangle. If you cut it flat across (horizontally), the shape is a circle.

If you cut a cylinder straight down (vertically), the shape is a rectangle. If you cut it flat across (horizontally), the shape is a circle.

The cross-section in general symbolizes the intersection of a plane with a three-dimensional object or shape. In the previous header we read that depending on the orientation of the plane, we can receive several cross-sections from the identical entity or object. The three common types of orientations are;

• Vertical Cross section
• Horizontal Cross section
• Inclined Cross Section

Learn the different Properties of Rectangles.

Vertical Cross Section

In a vertical or perpendicular cross section, the given plane cuts the solid shape in the vertical orientation i.e., perpendicular to the base of the solid shape, in such a way that it constructs a perpendicular cross-section.

OR

When an object is sliced by the plane perpendicular to the base of the object, that is, at an angle of 90 degrees we get a vertical cross-section of the object.

For example, if we consider a cylindrical bar. The vertical or perpendicular cross section of the cylindrical bar will be a rectangle.

Horizontal Cross Section

In a horizontal or parallel cross section, the given plane cuts the solid shape in the horizontal orientation i.e., parallel to the base of the solid shape, in such a way that it constructs a parallel cross-section.

OR

When an object is sliced by the plane parallel to the base of the object, that is, at an angle of 0 degrees we get a horizontal cross-section of the object.

For example, if we consider a cylindrical bar. The horizontal or parallel cross section of the cylindrical bar will be a circle.

Inclined Cross Section

Other than vertical and horizontal when the cross-section is created and the plane intersects or crosses the object at an angle that is greater than 0 degrees and less than 90 degrees; such a cross section is called an inclined cross section.

Cross Section of Geometric Figures

Now that we know the cross-section definition as well as its different types, let us understand the concepts of the cross sectional area of different geometric shapes.

Cross Sections of Cones

A cone is a type of pyramid but with a circular cross-section or it is a pyramid-like structure having a circular base and a triangular top. Relying upon the position of the plane which encounters the cone and the angle of intersection, different types of conic sections are formed namely; circle, ellipse, parabola and hyperbola.

• A circle is obtained when the cutting face is parallel to the base of the cone. In other words, if the plane is perpendicular to the axis of revolution, the conic section received is a circle.
• An ellipse is generated when a plane touches the cone at a certain angle. In other words, when a cone is cut by a plane that is inclined at a small angle to the base of the cone an ellipse is formed. The angle of inclination is less than 90 degrees for the ellipse formation.
• When the plane is parallel to the generating line, then the cross section formed is said to be a parabola. In other words, a parabola is created when the plane intersecting the cone is parallel to one lateral side of the cone.
• A hyperbola is formed when the interesting plane is parallel to the axis of the cone and connects both the nappes of the double-sided cone.

Read more about the Mensuration 2D here.

Consider the figure below to understand the various cross section of a cone.

Check out this article on Parabola Ellipse and Hyperbola.

Cross Sections of Cube

A cube is a three-dimensional figure having six faces and each of the faces links with the other four faces. Also, all the sides of a cube are of equal length. Let us understand the various cross sections of a cube.

• When the intersecting plane intersects the given cube in such a way that it is parallel to one of its 6 faces, then the cross-section obtained will be a square.
• Next, if the cutting plane crosses any three edges of the cube, then the cross-section obtained will be a triangle.
• If the plane slices the cube in such a way that it crosses the diagonals of the faces, the cross-section obtained is rectangular.
• When the plane slices the cube in such a way that it meets all faces of the cube a hexagonal cross-section is obtained.
• Similarly, other cross sections like a parallelogram, trapezium as well as pentagon can also be obtained by aligning the cutting plane.

The figure below shows the various cross sections of a cube.

Learn about the surface area of a cube here.

Cross Sections of Cuboid

A cuboid is a 3D shape. In a cuboid, the sides are of distinct lengths and are named length, breadth, and height. Let us understand the various cross sections of a cuboid.

• When the given cuboid is sliced along the longer side, that is the side having a smaller base, a square cross-section is obtained.
• When the given cuboid is sliced along the smaller side that is having a larger base a rectangle cross-section is obtained.

Consider the below diagram to understand the same.

Know more about the Mensuration 3D.

Cross Sections of Cylinder

A cylinder is a 3D tube-like structure including two parallel circular bases bound by a curved surface at a particular length from the center.

As per the cutting line cut, the cross-section of a cylinder can be a circle, a rectangle, or an oval. Let us understand them one by one.

• If the plane cuts the cylinder in such a way that is parallel to one of its bases(the cutting plane is perpendicular to the axis of rotation), the cross-section so obtained will be a circle.
• If the same plane slices the cylinder in such a way that it is perpendicular to the bases(the cutting plane is parallel to the axis of rotation), the cross-section so obtained will be a rectangle.
• If the plane slices the cylinder making an angle that is greater than 0° and less than 90° concerning the base, the cross-section so obtained will be an oval. In general, a plane that is either parallel or perpendicular to the base with slight variation in its angle with given an oval shape.

The below represents all the cross sections so formed in a cylinder.

Cross Sections of Sphere

A sphere represents a perfectly round 3D formation such that every point on its surface is at an equal distance from its centre. We can be considered our earth as an example of the same. Also, it has the smallest surface area for its volume.

Let us understand the cross-section of a sphere. Any cross-section of a sphere will always be a circle irrespective of the orientation of the plane. The figure below shows the same.

Cross Sectional Area of Geometric Figures

In the previous section, we read about the various cross sections formed by the different figures as per the orientation of the cutting line. Moving ahead, learn about the cross sectional area formula for different shapes.

When a plane crosses a solid object, an area is projected onto the given plane. Its projection is called the cross-sectional area. Depending on the figure and formation of the cross section, we can calculate the area by recognizing its dimensions.

Consider the case of a cube. When the cutting plane is parallel to the bases of the cube or any side of the cube the cross-section obtained will be a square.

Thus the area of a square= \(a^{2}\)

In the case of a cone; when the cutting front is parallel to the base of the cone a circle is obtained. Thus the cross sectional area of a circle= \(\pi r^{2}\).

Next, consider the case of a cylinder, wherein the cutting plane slices the cylinder such that it is perpendicular to the bases, the cross-section received will be a rectangle.

The cross sectional area of a rectangle=length × breadth.

Learn the concepts of Three Dimensional Geometry here.

Properties of cross section

A cross section is always a two-dimensional shape formed by slicing a three-dimensional object. Its shape depends on how and where the object is cut. For symmetrical solids, cross sections taken parallel to the base have the same shape and size as the base.

1. 2D Shape from a 3D Object:
   A cross-section is always a flat (2D) shape formed when a 3D object is cut by a plane.
2. Shape Depends on the Cut:
   The shape of the cross-section depends on the direction and angle of the cut. For example, cutting a cylinder horizontally gives a circle, while cutting it vertically gives a rectangle.
    
3. Parallel Cuts to Base:
   When a solid is cut parallel to its base, the cross-section has the same shape as the base. For example, a cube cut parallel to the base will give a square.
4. Cross-sections Help in Understanding Solids:
   Studying cross-sections helps us understand the inner structure and dimensions of solid objects.
5. Used in Volume and Area Calculations:
   Cross-sectional areas are often used to calculate the volume or surface area of 3D shapes.
6. Symmetry:
   Some cross-sections show the symmetry of the original shape. For instance, a cone’s vertical cross-section shows a symmetrical triangle.

Applications of Cross Sections:

Cross sections are used in various fields like engineering, architecture, medicine, and science to study the inner structure of objects. They help in understanding shapes, measuring areas, and designing strong, efficient structures or parts by showing internal details of 3D objects in 2D form.

1. Engineering and Design

• Engineers use cross-sections to design and analyze the strength of materials like beams, pipes, and bridges.
• Helps them understand how a structure will bear loads or forces.

 2. Architecture and Construction

• Cross-sections show what a building or structure looks like inside.
• Architects use them in building plans to show walls, floors, and spaces.

 3. Medical Imaging

• In CT scans or MRIs, cross-sectional images of the body help doctors see organs, bones, or tumors clearly.
• These images are like thin "slices" of the body.

 4. Manufacturing

• Factories use cross-sections to check the shape and quality of machine parts.
• Helps in making parts that fit perfectly.

 5. Geography and Geology

• Cross-sections of the earth’s layers help scientists study mountains, rocks, and soil.
• Useful in locating oil, minerals, or understanding earthquakes.

 6. Mathematics and Geometry

• Used to teach and understand 3D shapes better.
• Helps in calculating areas, volumes, and understanding symmetry.

Solved Examples of Cross Section

Now that we know what is the cross sectional area? how to find the same with detailed images. Let us go through some solved examples related to the topic for more practice.

Solved Example 1: Determine the cross-sectional area of a cylinder whose height is 20cm and diameter is 6cm. Given that the plane that cuts the cylinder is parallel to one of its bases.

Solution: Given;

Height of the cylinder =20cm

Diameter of the cylinder = 6cm

Radius=3 cm

The plane when cutting the cylinder that is parallel to any of its bases will give a circle.

Thus, the area of the circle is\( πr^{2}\), where r is the circle’s radius.

Substituting the values we get;

Area of circle =\( \pi\times3\times3=\frac{22}{7}\times3\times3=28.285\text{ cm}^2\) .

Also, learn about the Circumference of a Circle.

Example 2: Obtain the cross-sectional area of a plane parallel to the base of a cube. Given the volume of the cube is equal to 64 \(cm^{3}\).

Solution: Here we need to find the cross-sectional area of the cube as the plane is parallel to the base. Thus the first obtain the measure of the side.

The volume of a cube is calculated using the formula; \(side^{3}\)

Given that;

\(side^{3} = 64\)

Thus each side =4 cm

In our case, the cross-section of the cube will be a square, as the plane is parallel to the base of the cube.

The length of the side of a square=4 cm

The area of a square= \(side^{2}\) = \(4^{2}\) sq.cm=16 sq.cm.

Solved Example 3: Determine the cross sectional area of a cone whose height is 22cm and radius is 5cm. Given that the cutting face is parallel to the base of the cone.

Solution: Given;

Height of the cone=22cm

Radius of the cone = 5cm

When the cutting face is parallel to the base of the cone a circle is obtained.

Thus, the area of the circle is\( πr^{2}\), where r is the circle’s radius.

Substituting the values we get;

Area of circle =\( \pi\times5\times5=\frac{22}{7}\times5\times5=78.571\text{ cm}^2\).

We hope that the above article is helpful for your understanding and exam preparations. Stay tuned to the Testbook App for more updates on related topics from Mathematics, and various such subjects. Also, reach out to the test series available to examine your knowledge regarding several exams.