3D solids
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.
Cube
A rectangular prism with all $6$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. Here are some examples of square and rectangular based pyramids in real life.
Cones
A cone is made by connecting a circular base to an apex. Such as the right-cone shown below:
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.
Sports balls and planets may come to mind when we think of spheres.
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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 and the same number of faces meet at each vertex. There are only $5$5 regular solids and they are given the special name: the Platonic solids. The most familiar regular shape is a cube, where each face is a square and three squares meet at each vertex. Notice the names of these commonly have -hedron at the end. All other polyhedrons are known as irregular solids.
The $5$5 Platonic solids
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.
- If the solid has a uniform cross-section it is a prism. We name it by identifying the base, (or the shape of the cross-section) and the word prism. eg rectangular prism, triangular prism, hexagonal prism, trapezoidal prism
- If the solid has $2$2 parallel circles as edges, we call this a cylinder.
- If the solid has $6$6 sides and all of them are squares, we call this a cube.
- If the base is a polygon, and all the vertices are connected to a common apex then it is a pyramid and we name it by using the name of the shape that the base is and using the word pyramid. e.g. rectangular pyramid, triangular pyramid, octagonal pyramid
- If the base is a circle, and it connects to a point we call this a cone.
- If each point on the surface is equally distant from the centre, we call this a sphere.
Practice questions
Question 1
Select the right name for each solid:
Square pyramid
ATriangular pyramid
BTriangular prism
CSquare prism
D
Rectangular prism
ASquare pyramid
BSquare prism
CRectangular pyramid
D
Question 2
Classify the following solids as convex or non-convex:
Does the shape have a uniform cross-section ?
No
AYes
BIs the solid convex or non-convex?
Convex
ANon-convex
B
Question 3
Here is a rectangular prism.
How many faces does it have?
How many vertices does it have?
How many edges does it have?