Page 199 - ICSE Math 8

18 Three-dimensional Figures

Key Concepts

• Polyhedron, Prism and Pyramid • Nested or Combined Solids
• Visualization of 3D shapes Through Nets • Views of 3D Shapes

In previous classes, you have learnt about one-dimensional, two-dimensional and three- (a) Line
dimensional figures. A figure having only length, for example a line segment, [Fig. 18.1(a)]
is known as one-dimensional (1D) figure. A figure having length and breadth, for example
a rectangle, [Fig. 18.1(b)] is called two-dimensional (2D) figure. Three-dimensional (3D)
figures have length, breadth and height, for example a cuboid [Fig. 18.1(c)]. (b) Rectangle
Visualization of 2D shapes and their representation on paper is very convenient. However,
the tricky part is visualizing a 3D shape and representing it on a 2D sheet of paper. In this
lesson we will learn how to visualize 3D shapes.
(c) Cuboid
Fig. 18.1
Polyhedron
A three-dimensional solid made up of polygons is called a polyhedron (Fig. 18.2).
Let’s define some terms associated with a polyhedron.
Faces: The polygons forming a polyhedron are called its faces.
Edges: The line segments where the faces of a polyhedron meet are called its edges. Fig. 18.2
Vertices: The corners where the edges of a polyhedron intersect are called its vertices. In fact, the vertices of
the polygonal faces are the vertices of the polyhedron. Three or more edges meet at the vertex.
Euler’s relation: The Euler’s relation states that the number of faces (F), the number vertices (V) and the number
of edges (E) of a simple convex polyhedron maintain the following relationship:
F + V = E + 2
Let us verify this relationship for the following polyhedrons.
Some common examples of polyhedrons are given below (Fig. 18.3 to 18.7) where F is for faces, E for edges.
Number of Number of Number of
Polyhedron F + V = E + 2
vertices (V) faces (V) edges (E)
Cuboid E F
Vertex
H
G Face
D Edge 8 6 12 6 + 8 = 12 + 2
C
A B
Fig. 18.3
Cube E F Vertex

H Face
G
Edge 8 6 12 6 + 8 = 12 + 2
D C

A B
Fig. 18.4

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