2.025 Properties of 3D solids

3D shapes

 

Vertex (plural is vertices)

A vertex is a point where two or more straight lines meet. Otherwise known as a corner.

This rectangular based pyramid has $5$5 vertices.  

 

Edge

An edge is a line segment that joins two vertices. 

This rectangular based pyramid has $8$8 edges. 

 

Face

A face is any of the individual surfaces of a solid object. This rectangular based pyramid has $5$5 faces. 

 

Types of 3D shapes

 

Polyhedra

A polyhedron is the 3D equivalent of a polygon.  Remember how a polygon is a many angled (many straight sided) figure.  Well a polyhedron is a many sided flat faced figure. A polyhedron has no curved edges and each face is a polygon.  

Here are some examples of 3D solids that are polyhedra:

These ones are not.  These all have curved faces or edges.  

 

Prisms

A prism is defined as a solid geometric figure whose two end faces are similar, equal, and parallel.  These are both prisms.  Prisms need to have all straight edges and faces, no curves.

A prism has a base and a uniform cross-section.  The base is one of the two parallel identical ends.  The bases for these prisms are shaded blue.  

A uniform cross-section means that if we slice the solid (imagine cutting it liked sliced bread) at any point parallel to the base then it will have the exact same shape as the base.

Prisms occur very commonly in the packaging of grocery items, and finding the volume of these contributes to the design, shape and size of packaging and product.

 

Practice question

Question 1

 

Cube

A rectangular prism with all 6 faces being squares is called a cube. 

 

Cylinder

A cylinder is similar to a prism, it has two parallel circular faces.  It also has a uniform cross-section.  It is not classified as a prism because of the circular edge and face.    

 

Pyramids

A pyramid can be made in the following way:

• Use any polygon as a base. We can have square bases, triangular bases or even hexagonal bases.
• Then connect every vertex of the base to an apex point above the base, and you have a pyramid.

We can name a pyramid by the shape of the base. Square and rectangular based pyramids, are the most common in mathematics.

         

 

Practice question

Question 2

 

Cones

A cone is made by connecting a circular base to an apex.  

Cone shapes appear everywhere in the real world.

     

 

Spheres

A sphere is made by having a central point and then creating a surface where all points on the surface are the same distance from the centre.  

We mostly recognise spheres as balls.

Now that we know how to name a solid, the next step is to understand some of the language that is used to describe them.

 

Classifying 3D shapes

 

Regular solids

Regular solids have faces that are all identical. They even have a special name: the Platonic solids.  The most familiar regular shape is a cube, where each face is a square. (notice the names of these normally have -hedron at the end.

 

Convex or concave

As we saw for 2D shapes, describing flat shapes as convex or concave came down to identifying if there was a cave-like shape to the edges.  

For 3D solids, the idea of concave and convex is the same.

 

Right or oblique

A right solid is one that has its axis at right angles to its base.  

• Right cone, has its apex at right angles to the centre of the base.
• Right pyramid, has its apex at right angles to the centre of the base.
• Right prism has the central line (axis) at right angles to the centre of the base.
• Right cylinder has the central line (axis) at right angles to the centre of the circular base. 

These solids are all right solids and notice the red axis line that shows it is perpendicular to the base.

These solids are all oblique solids, notice the red axis line that shows its angle to the base.

 

Practice questions

Question 3

Question 4