Visualising Solid Shapes

The exploration of solid shapes is like embarking on a captivating visual journey. From cubes to spheres, these three-dimensional wonders add depth and dimension to our understanding of geometry. Unlike flat shapes, solid shapes have depth, width and height. Developing this skill allows you to imagine how these shapes look from different angles and perspectives. In this chapter, we'll take a journey into the fascinating realm of visualizing solid shapes.

Plane Figures

Shapes that have length and width (breadth) are called plane figures or two-dimensional figures. These shapes can be made using straight lines, curved lines or a combination of both.

Examples of plane figures are shown as:

Solid Figures

Shapes that have length, width and height are called solid figures or three-dimensional figures. These objects can have flat surfaces or curved surfaces. Examples of solid figures are shown as:

Polyhedron

A polyhedron is a three-dimensional shape with flat faces, straight edges and sharp vertices.

It has three main parts:

1. Face: A face is the flat surface of the shape.
2. Edge: An edge is a straight line (side) where one flat side of the polyhedron meets another.
3. Vertex: A vertex is a sharp corner where the edges come together.

Examples of polyhedrons are shown as:

Euler's Formula

If the number of faces in any polyhedron is ‘F’, the number of vertices is ‘V’ and the number of edges is ‘E’, then Euler's polyhedron formula is given by:

V + F = E + 2

Net of a Three-Dimensional Figure

A net of a three-dimensional figure is a two-dimensional representation of a three-dimensional figure that is unfolded along its sides. This way, you can see each side of the shape in two dimensions. It is like unfolding a folded paper model to see all the flat parts of a 3D object.

Visualising Solid Shapes

You can see different parts of a 3D shape in various ways:

a. Look at it from different angles — from the front, the side or the top. These views are the front view, the side view and the top view.

b. Watch the 2D shadow the 3D shape makes.

c. Cut or slice the shape which shows a cross-section like cutting a cake to see what is inside.

The front view, side view and the top view of the given 3D object is shown as:

The front view, side view and the top view of the given 3D object is shown as:

Symmetry

Symmetry is a quality that describes an object or shape that looks the same when certain operations are performed on it such as reflection, rotation or translation. If you were to draw a line or fold the object, then both sides would be nearly identical.

Line of Symmetry

The line of symmetry is an imaginary line that divides a shape into two identical halves. If you were to fold the shape along this line, both sides would match perfectly.

The line of symmetry of some 2D figures is shown as:

Rotational Symmetry

If you spin a shape around its centre and it still looks the same after each turn, we say the shape has rotational symmetry.

Imagine a shape that when rotated around its centre looks the same at certain angles. This characteristic is what we call rotational symmetry. It is like a merry-go-round where each turn reveals the same delightful view.

Degrees of Rotational Symmetry: Shapes can have different degrees of rotational symmetry which depends on how many times they look the same after a full rotation.

1. Order-1 Rotational Symmetry: A shape with order-1 rotational symmetry looks the same after a full 360° rotation. The letter Z is an example of order-1 symmetry.
2. Order-2 Rotational Symmetry: A shape with order-2 rotational symmetry looks the same after a full 180° rotation. A rectangle shape is an example of order-2 symmetry.
3. Order-3 Rotational Symmetry: A shape with order-3 rotational symmetry looks the same after a full 120° rotation. An equilateral triangle is an example of order-3 symmetry.

Example: A square has rotational symmetry of order 4 because when you turn it by 90°, 180°, 270° and 360°, it looks identical each time.

Note: For n-sided regular polygons, the angle of rotation is equal to 360°/n.