Nisha has a large solid cube made from 27 small cubes. Since 3 × 3 × 3 = 27, the large cube is a 3 × 3 × 3 cube.
She paints the entire large cube red.
(a) The large solid cube has 8 corners small cubes, and each corner small cube is painted red.
So, 8 small cubes are three faces painted red.
(b) Since a cube has 12 edges or sides, and 1 cube in the middle of each edge is painted red.
So, there are 12 small cubes with two faces painted red.
(c) These are the small cubes located in the center of each face of the large cube. A cube has 6 faces. Each face of the large cube is a 3 × 3 square of small cubes. The center small cube of each 3 × 3 face has only one face exposed to the outside, or painted red. So, there are 6 small cubes with one face red.
(d) For a 3 × 3 × 3 = 27 cube, if we remove the outer layer of cubes, we are left with an inner cube. This means there is only 1 small cube right in the very center of the large cube that has no faces painted red.
Puzzle
Tanu arranged 7 shapes in a line. She used 2 squares, 2 triangles, 1 circle, 1 hexagon, and 1 rectangle.
Find her arrangement using the following clues:
(a) The square is between the circle and the rectangle.
(b) The rectangle is between the square and the triangle.
(c) The two triangles are next to the square.
(d) The hexagon is to the right of the triangle
(e) The circle is to the left of the square.
Solution:
The arrangement is: Triangle, Circle, Square, Rectangle, Triangle, Hexagon.
NCERT Textbook Page 103
Icosahedron and Dodecahedron
What shapes do you see in an icosahedron and a dodecahedron?
Solution:
Icosahedron: Equilateral triangles, Dodecahedron: Regular pentagons
Do all the faces look the same?
Solution:
Icosahedron: Yes, Dodecahedron: Yes
How many faces meet at a vertex (point)?
Solution:
Icosahedron: 5 faces (Equilateral triangles), Dodecahedron: 3 faces (Regular pentagons)
Do the same number of faces meet at each vertex?
Solution:
Icosahedron: Yes, Dodecahedron: Yes
How many edges do you see?
Solution:
Icosahedron: 30, Dodecahedron: 30
How did you count them such that you do not miss out any edge or count an edge twice?
Solution:
Do it yourself.
Can you think of any other solid shapes that have faces that look the same?
Solution:
Yes, there are a few other types of shapes where all faces are identical. These are known as platonic solids. Besides the icosahedron and dodecahedron, the other platonic solids are:
Tetrahedron: 4 faces, all are equilateral triangles.
Cube: 6 faces, all are squares.
Octahedron: 8 faces, all are equilateral triangles.
Do the same number of faces meet at each common vertex?
Solution:
Yes.