Definition types and solved examples of sections of solids in slicing and shadow problems
In our day to day life, we come across various objects having different shapes and sizes which are based on parameters like physical properties such as length, breadth, diameter, etc and sometimes it also depends on the material. But no matter how different their dimensions are, all the objects are matter and occupies space. These objects are also referred to as three-dimensional or solid shapes which can be viewed from different sections. Visualization of solid shapes helps us to understand the solid object.
What are solid Shapes?
Solid shapes are the objects having three-dimensional shapes such that the position of any point can be explained by using three coordinate axes known as x-axis, y-axis, and z-axis. Many objects that we see in your day to day life as a bed, cylinder, cupboard, etc are three-dimensional objects occupying some shape and having length, breadth, height, and depth.
Properties Of 3-D Shapes
There are four properties that set three-dimensional shape apart from two-dimensional shapes and these properties are faces, vertices, edges, and volume. These properties not only allow you to determine whether the shape is 2D or 3D but it also helps you to understand which type or division of solids it belongs to.
Faces, Edges and Vertices
A face is a two-dimensional surface as one of the surfaces of a three-dimensional solid. An edge is the meeting line of two faces just like how sky and land appear to meet at the horizon. Vertex is the point or tip of the corner of three-dimensional geometric shapes. Thus, a solid figure has faces (sides/ surfaces having areas), edges (the meeting line of two surfaces) and vertices (corners/ tips).
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Faces
A face is a flat or curved surface of a solid shape. For example, a cube has six faces whereas a cylinder has three faces and a sphere has only one face.
Edges
An edge is where two faces meet it appears to be a straight line. For example, a cube has 12 edges, a cylinder has two edges and a sphere has no edges at all.
Vertices
A vertex is a corner of the solid where the edges meet. A lot of vertexes together is known as vertices. For example, a cube consists of eight vertices, a cone consists of one vertex and a sphere has no vertex.
Cross Sections Of Solid Shapes
When we cut a solid object, we get a surface which is called cross-section and it has an area too. In other words, a cross-section is a shape we get when after cutting an object straight through. It is more like a view into the inside of the object by cutting through it. A cross-section is the intersection of a three-dimensional figure with a plane that is more like a face you obtain by slicing through a solid object. A cross-section is always two-dimensional and the area of the face of the cross-section depends on the orientation (angle) of the plane while cutting the object. Cross-sections are usually either parallel or perpendicular to the base but it can be in any direction.
Difference Between Section And Cross Section
The main difference between a Section and a Cross Section is that a section is the cutting of a solid by a plane, whereas a cross-section is actually the surface or the face having an area which is exposed when we cut the object. A section refers to a closeup of a particular section or part of the design that can be any angle but a cross-section refers to a view of something that has been cut across to show the interior of the object.
Ways to View The Sections of solids
There are three ways to view the section of a Solid Shape:
• Viewing the cross-sections
• Using shadows
• Viewing at certain angles
A solid can be viewed from different angles such as from the front, side and top. On the other hand, cutting or slicing a solid will show the cross-section of the object. Observing the two-dimensional shadow of a three-dimensional solid is also another way of viewing a solid. Shadows of three-dimensional solids are of different sizes depending on its position and the source of light.
Cutting or Slicing
We have already read about the cross-section of solids which is basically the exposed surface of a solid that you get when you make a cut through it. The original face cannot be retained once the object is cut therefore the cross-section is a surface “inside” the object.
To view the cross-section of 3D objects you can cut or slice the object from any place at any angle. You can cut an object horizontally, vertically or from any angle.
Shadow Play
You can view the cross-section of a solid by using shadow which requires a source of light a, for example, a torch or sun or bulb, etc. You can view solids such as cuboid, cone, sphere, etc by keeping them in front of a screen and bring the torch in front of the solid the opposite side of the screen. You can view the shadow of the object on the screen. The size of the shadow depends on the angle and distance of the light source and the screen from each other and the object.
FAQs on Sections Of Solids Through Slicing And Shadows
1. What are sections of solids in geometry?
A section of a solid is the flat shape formed when a solid is cut by a plane. In solid geometry, when a plane slices a 3D object like a cube, cone, or cylinder, the shape seen on the cut surface is called the cross-section.
- Cutting a cube parallel to its base gives a square cross-section.
- Cutting a cylinder parallel to its base gives a circle.
- Cutting a cone at an angle can give an ellipse.
2. What is a cross-section of a solid?
A cross-section is the two-dimensional shape obtained when a plane cuts through a three-dimensional solid. It represents the intersection between the solid and the slicing plane.
- Sphere → always a circle.
- Cylinder → circle or rectangle (depending on the cut).
- Cone → circle, ellipse, parabola, or triangle.
3. What shapes can be formed by slicing a cube?
Slicing a cube can produce different cross-sectional shapes depending on the angle of the cut. Common sections include:
- Square – when cut parallel to a face.
- Rectangle – when cut parallel to an edge.
- Triangle – when cut through three vertices.
- Hexagon – when cut diagonally through six edges.
4. What is meant by shadows of solids in maths?
A shadow of a solid is the two-dimensional shape formed when light falls on a 3D object and projects its outline onto a surface. In geometry, shadows help understand projections and views of solids.
- A sphere can cast a circular shadow.
- A cube can cast a square or rectangular shadow.
- A cylinder may cast a rectangle or circle.
5. What is the difference between a cross-section and a shadow?
The main difference is that a cross-section is formed by cutting a solid, while a shadow is formed by projecting light onto a solid.
- Cross-section → physical slice through the solid.
- Shadow → outline projection without cutting.
- Cross-section shows internal shape; shadow shows external view.
6. How do you find the cross-section of a cylinder?
The cross-section of a cylinder depends on how the plane cuts it.
- If cut parallel to the base → cross-section is a circle with area πr².
- If cut perpendicular to the base and through the axis → cross-section is a rectangle with area 2r × h.
7. What cross-section do you get when a cone is sliced?
When a cone is sliced, the cross-section can be a circle, triangle, or ellipse depending on the cut.
- Parallel to base → circle.
- Through the vertex → triangle.
- Angled cut (not parallel) → ellipse.
8. Can a sphere have different cross-sections?
A sphere always has a circular cross-section when cut by any plane. However, the size of the circle changes depending on the distance from the center.
- If cut through the center → largest circle called a great circle.
- If cut away from center → smaller circle.
9. How are sections of solids used in real life?
Sections of solids are used to understand internal structures and shapes in practical applications.
- Engineering drawings show cross-sections of machines.
- Architecture uses sectional views of buildings.
- Medical imaging (like CT scans) shows cross-sections of the human body.
10. What are common mistakes when drawing sections and shadows of solids?
Common mistakes include incorrect orientation of the slicing plane and misunderstanding projection direction.
- Assuming all cross-sections are the same shape.
- Forgetting that a sphere’s section is always a circle.
- Drawing shadows without considering the direction of light.
