Recognition of 3D Shapes (Solids)

(i) A cuboid (or a cube) is bounded by 6 faces, which form three pairs of opposite faces.

Number of faces = 6

Two adjacent faces meet in an edge. A cuboid has 4 edges at the bottom, 4 edges at the top and 4 vertical edges.

Number of edges = 4 + 4 + 4 = 12

Three edges meet at each vertex. A cuboid has 4 vertices at the bottom and 4 vertices at the top.

Number of vertices = 4 + 4 = 8

(ii) A triangular prism has 2 triangular faces (top and bottom) and 3 rectangular lateral faces.

Number of faces = 2 + 3 = 5

It has 3 edges on each of the two triangular faces and 3 edges joining their corresponding vertices.

Number of edges = 3 + 3 + 3 = 9

It has 3 vertices on each of the two triangular faces.

Number of vertices = 3 + 3 = 6

(iii) A hexagonal prism has 2 hexagonal faces (top and bottom) and 6 rectangular lateral faces.

Number of faces = 2 + 6 = 8

It has 6 edges on each of the two hexagonal faces and 6 edges joining their corresponding vertices.

Number of edges = 6 + 6 + 6 = 18

It has 6 vertices on each of the two hexagonal faces.

Number of vertices = 6 + 6 = 12

(iv) A triangular pyramid has 1 triangular base and 3 triangular lateral faces.

Number of faces = 1 + 3 = 4

It has 3 edges of the base and 3 slant edges meeting at the common vertex.

Number of edges = 3 + 3 = 6

It has 3 vertices at the corners of the base and 1 vertex at the top.

Number of vertices = 3 + 1 = 4

(v) A square pyramid has 1 square base and 4 triangular lateral faces.

Number of faces = 1 + 4 = 5

It has 4 edges of the base and 4 slant edges meeting at the common vertex.

Number of edges = 4 + 4 = 8

It has 4 vertices at the corners of the base and 1 vertex at the top.

Number of vertices = 4 + 1 = 5

(vi) A pentagonal pyramid has 1 pentagonal base and 5 triangular lateral faces.

Number of faces = 1 + 5 = 6

It has 5 edges of the base and 5 slant edges meeting at the common vertex.

Number of edges = 5 + 5 = 10

It has 5 vertices at the corners of the base and 1 vertex at the top.

Number of vertices = 5 + 1 = 6

The completed table is shown below:

Solid	Number of Faces	Number of Edges	Number of Vertices
(i) Cuboid/Cube	6	12	8
(ii) Triangular Prism	5	9	6
(iii) Hexagonal Prism	8	18	12
(iv) Triangular Pyramid	4	6	4
(v) Square Pyramid	5	8	5
(vi) Pentagonal Pyramid	6	10	6

(i) Two adjacent faces of a cuboid meet in a line segment called an edge of the cuboid.

(ii) A triangular prism has two triangular faces.

(iii) A square pyramid has four triangular faces.

(iv) The six lateral faces of a hexagonal prism are rectangular.

(v) The triangular pyramid is also known as tetrahedron.

(vi) A cuboid has three pairs of opposite faces.

Number of vertices of a cuboid = 8

Number of vertices of a triangular prism = 6

Number of vertices of a triangular pyramid = 4

Number of vertices of a square pyramid = 5

So, the cuboid has the highest number of vertices.

Hence, option 1 is the correct option.

A triangular prism has 3 lateral faces, which are rectangular.

A triangular pyramid, a square pyramid and a pentagonal pyramid have all their lateral faces triangular, as these faces meet at a common vertex.

So, in a triangular prism, the lateral faces are not triangular.

Hence, option 1 is the correct option.

A triangular prism has 2 triangular faces.

A square pyramid has 4 triangular faces and a pentagonal pyramid has 5 triangular faces.

A hexagonal prism has 2 hexagonal faces and 6 rectangular faces, so none of its faces is triangular.

Hence, option 4 is the correct option.

A net of a cube is made up of six equal squares, which on folding form the six faces of the cube in such a way that no two squares overlap and no face of the cube is left open.

On folding the nets (i), (ii) and (iv), each of the six squares forms a different face of the cube and a cube is obtained.

In net (iii), the two squares attached to the column of four squares lie on the same side of the column. On folding, both these squares form the same face of the cube, so they overlap each other and one face of the cube is left open. So, a cube cannot be formed from it.

Hence, net (iii) is not a net for a cube.

A net of a cube is made up of six equal squares, which on folding form the six faces of the cube in such a way that no two squares overlap and no face of the cube is left open.

On folding the net (ii), each of the six squares forms a different face of the cube and a cube is obtained.

On folding the nets (i), (iii) and (iv), two of the squares overlap each other and one face of the cube is left open. So, a cube cannot be formed from them.

Hence, net (ii) is a net for a cube.

A cylinder has a curved surface and two circular faces at its ends, namely the top and the base.

In the net of a cylinder, the rectangle forms the curved surface when rolled. The two circles must be attached to the opposite sides of the rectangle that form the circular rims. In the given figures, these are the two longer sides.

In net (ii), the two circles are attached to the two opposite longer sides of the rectangle. On folding, they form the top and the base of the cylinder.

In net (i), one circle is attached to a shorter side of the rectangle. In net (iii), both the circles are attached to the same side of the rectangle. In net (iv), both the circles are attached to the shorter sides of the rectangle. So, a cylinder cannot be formed from them.

Hence, net (ii) is a net for a cylinder.

A cuboid has 6 rectangular faces, which form three pairs of equal opposite faces.

On folding the nets (i), (ii) and (iii), the four rectangles in the row form the four lateral faces and the two rectangles attached above and below the row exactly cover the top and the bottom of the cuboid.

In net (iv), the two rectangles attached above and below the row are too small. On folding, they do not cover the top and the bottom of the cuboid completely, so the cuboid is left open. So, a cuboid cannot be formed from it.

Hence, net (iv) is not a net for a cuboid.

Cuboid X has 6 rectangular faces, which form three pairs of equal opposite faces.

On folding the nets (i), (iii) and (iv), the four rectangles in the row form the four lateral faces and the two remaining rectangles exactly cover the two ends of cuboid X.

In net (ii), the two rectangles attached above and below the row are larger than the end faces of cuboid X. On folding, they do not fit the ends of the cuboid and overlap the adjacent faces. So, cuboid X cannot be formed from it.

Hence, net (ii) is not a net for cuboid X.

The net of a 3-dimensional figure is the shape that can be cut out of a plane piece of paper and folded to make the 3-dimensional shape.

On folding net (i), the four triangles meet at a common vertex and the square forms the base. So, it is a net of a square pyramid.

On folding net (ii), the six rectangles form the faces of a cuboid. So, it is a net of a cuboid.

On folding net (iv), the six equal squares form the faces of a cube. So, it is a net of a cube.

In net (iii), four triangles are attached to the four sides of the square and four more triangles are attached to the top and the bottom triangles. On folding, the four triangles attached to the square meet at a common vertex, and the remaining four triangles overlap these faces. So, no polyhedron can be formed from it.

Hence, net (iii) is not a net for any polyhedron.

A square pyramid has 1 square base and 4 triangular lateral faces having a common vertex. So, its net has 1 square and 4 triangles, which on folding form the four sides of the pyramid without overlapping.

On folding the nets (i), (ii) and (iii), each of the four triangles forms a different lateral face and the triangles meet at a common vertex.

In net (iv), only two triangles are attached to the square, at its top and bottom sides. The other two triangles are attached to these triangles on the same (right) side. On folding, both these triangles form the lateral face on the same side of the pyramid, so they overlap each other and the lateral face on the other side is left open. So, a square pyramid cannot be formed from it.

Hence, net (iv) is not a net for a square pyramid.

A pentagonal prism has 2 pentagonal faces (top and bottom) and 5 rectangular lateral faces. So, its net has 2 pentagons and 5 rectangles, with each rectangle forming a different lateral face.

On folding the nets (ii), (iii) and (iv), each of the five rectangles forms a different lateral face and the two pentagons form the top and the bottom of the prism.

In net (i), the squares attached to the two pentagons lie on the corresponding sides of the two pentagons. On folding, the squares attached to one pentagon fall on the squares attached to the other pentagon, so they overlap each other and two lateral faces are left open. So, a pentagonal prism cannot be formed from it.

Hence, net (i) is not a net for a pentagonal prism.

A pentagonal pyramid has 1 pentagonal base and 5 triangular lateral faces having a common vertex. So, its net has 1 pentagon and 5 triangles, all of which meet at a common vertex on folding.

On folding the nets (ii), (iii) and (iv), the five triangles form the five lateral faces and meet at a common vertex.

In net (i), only four of the five triangles have a common vertex. The last triangle at the right end is attached to the base side of the fourth triangle, so on folding it does not meet the other triangles at the common vertex. So, one lateral face is left open and a pentagonal pyramid cannot be formed from it.

Hence, net (i) is not a net for a pentagonal pyramid.

(i) The given net is made up of one rectangle and four triangles attached to its four sides. On folding, the rectangle forms the base and the four triangles meet at a common vertex.

A pyramid whose base is a rectangle and whose side faces are triangles having a common vertex is a rectangular pyramid.

Hence, the 3D shape is a rectangular pyramid.

(ii) The given net is made up of eight equal triangles. On folding, they form eight triangular faces.

A solid with eight triangular faces is an octahedron.

Hence, the 3D shape is an octahedron.

(iii) The given net is made up of three rectangles and two triangles. On folding, the rectangles form three rectangular lateral faces and the two triangles form the top and the bottom faces.

A triangular prism has two identical and parallel triangular faces and three rectangular faces.

Hence, the 3D shape is a triangular prism.

(iv) The given net is made up of one square and four triangles. On folding, the square forms the base and the four triangles meet at a common vertex.

A solid whose base is a square and side faces are triangles having a common vertex is a square pyramid.

Hence, the 3D shape is a square pyramid.

(v) The given net is made up of six rectangles. On folding, they form six rectangular faces, with the opposite faces being equal.

A solid bounded by 6 rectangular faces is a cuboid.

Hence, the 3D shape is a cuboid.

A triangular prism has 2 triangular faces and 3 rectangular lateral faces.

Number of faces = 2 + 3 = 5

Hence, option 1 is the correct option.

A triangular prism has 3 edges on each of the two triangular faces and 3 edges joining their corresponding vertices.

Number of edges = 3 + 3 + 3 = 9

Hence, option 2 is the correct option.

A triangular prism has 3 vertices on each of the two triangular faces.

Number of vertices = 3 + 3 = 6

Hence, option 3 is the correct option.

A hexagonal prism has 2 hexagonal faces and 6 rectangular lateral faces.

Number of faces = 2 + 6 = 8

Hence, option 2 is the correct option.

A tetrahedron (triangular pyramid) has 3 vertices at the corners of the triangular base and 1 vertex at the top.

Number of vertices = 3 + 1 = 4

Hence, option 1 is the correct option.

A tetrahedron (triangular pyramid) has 1 triangular base and 3 triangular lateral faces.

Number of faces = 1 + 3 = 4

Hence, option 2 is the correct option.

A tetrahedron (triangular pyramid) has 3 edges of the base and 3 slant edges meeting at the common vertex.

Number of edges = 3 + 3 = 6

Hence, option 1 is the correct option.

A square pyramid has 4 vertices at the corners of the square base and 1 vertex at the top.

Number of vertices = 4 + 1 = 5

Hence, option 3 is the correct option.

A square pyramid has 4 edges of the base and 4 slant edges meeting at the common vertex.

Number of edges = 4 + 4 = 8

Hence, option 4 is the correct option.

A pentagonal pyramid has 5 vertices at the corners of the pentagonal base and 1 vertex at the top.

Number of vertices = 5 + 1 = 6

Hence, option 2 is the correct option.

A square pyramid, a pentagonal pyramid and a triangular pyramid have all their lateral faces triangular, as these faces meet at a common vertex.

A triangular prism has 3 lateral faces, which are rectangular.

So, in a triangular prism, the lateral faces are not triangular.

Hence, option 3 is the correct option.

A square pyramid has 4 triangular faces and a pentagonal pyramid has 5 triangular faces.

A triangular prism has 2 triangular faces.

A hexagonal prism has 2 hexagonal faces and 6 rectangular faces, so none of its faces is triangular.

Hence, option 4 is the correct option.

(i) A triangular pyramid is called a tetrahedron.

(ii) A pentagonal pyramid has 6 vertices.

(iii) A hexagonal prism has 12 vertices and 18 edges.

(iv) A triangular prism has 6 vertices and 9 edges.

(v) A cuboid has 6 faces, 12 edges and 8 vertices.

(i) The solid has a square base and triangular side faces having a common vertex.

Hence, the shape is a square pyramid.

(ii) The solid has two identical and parallel triangular faces (top and bottom) and rectangular side faces.

Hence, the shape is a triangular prism.

(iii) The solid has a triangular base ABC and triangular side faces having a common vertex D.

Hence, the shape is a tetrahedron (triangular pyramid).

(iv) The solid has two identical and parallel hexagonal faces (top and bottom) and rectangular side faces.

Hence, the shape is a hexagonal prism.

(i) A triangular prism has two triangular faces.

True

(ii) A square pyramid has four triangular faces.

True

(iii) A cube has three pairs of opposite sides.

True

(iv) A pentagonal pyramid has no triangular face.

False

A pentagonal pyramid has five triangular faces.

(v) A hexagonal prism has six rectangular faces.

True

A solid whose base is a triangle and the side faces are triangles, having a common vertex, is called a triangular pyramid or tetrahedron.

A triangular prism has two triangular faces and three rectangular faces, so it is not a tetrahedron.

So, Assertion (A) is false.

A tetrahedron is a triangular pyramid, so its base is a triangle.

So, Reason (R) is true.

Hence, option 4 is the correct option.

A pentagonal pyramid has 5 vertices at the corners of the pentagonal base and 1 vertex at the top.

Number of vertices = 5 + 1 = 6

So, Assertion (A) is true.

A pyramid with a polygonal base of n sides has n vertices at the corners of the base and 1 vertex at the top. So, it has (n + 1) vertices.

So, Reason (R) is true.

For a pentagonal pyramid, n = 5, so the number of vertices = 5 + 1 = 6. So, Reason (R) is the correct explanation of Assertion (A).

Hence, option 1 is the correct option.

A square pyramid has 1 square base and 4 triangular lateral faces.

Number of faces = 1 + 4 = 5

Hence, option 4 is the correct option.

A triangular pyramid has 3 edges of the base and 3 slant edges meeting at the common vertex.

Number of edges = 3 + 3 = 6

Hence, option 1 is the correct option.

A triangular based pyramid has a triangular base and three triangular lateral faces. So, each of its surfaces is a portion of a triangular surface.

Hence, option 2 is the correct option.

A football is in the shape of a sphere and a sphere has a curved surface.

So, the surface of a football is a curved surface.

Hence, option 2 is the correct option.

In the given net, the column of four squares has the faces τ, ∂, w and ÷. On folding, the first and the third squares of this column come opposite to each other and the second and the fourth squares come opposite to each other.

So, τ is opposite to w and ∂ is opposite to ÷.

Also, the striped face to the left of τ comes opposite to the face ○ to the right of τ.

Two opposite faces of a die cannot be seen together.

In option 3, the faces τ and w are seen together, but τ and w are opposite faces.

So, the die in option 3 cannot be made from the given net.

Hence, option 3 is the correct option.

In the given net, the column of four squares has the faces θ, β, U and +. On folding, the first and the third squares of this column come opposite to each other and the second and the fourth squares come opposite to each other.

So, θ is opposite to U and β is opposite to +.

Also, the face ∀ to the left of θ comes opposite to the face α to the right of θ.

Two opposite faces of a die cannot be seen together.

In option 1, the faces θ and U are seen together, but they are opposite faces.

In option 3, the faces ∀ and α are seen together, but they are opposite faces.

In option 4, the faces β and + are seen together, but they are opposite faces.

In option 2, the faces α, θ and β are seen together and no two of them are opposite faces.

In the given net, α is immediately to the right of θ, and β is immediately below θ. On folding and suitably rotating the cube, these three faces can be arranged as shown in option 2.

Hence, option 2 is the correct option.

When the die is rotated horizontally to the right by 90°, the top face 2 remains on the top. The front face moves to the right side and the face on the left side, which is hidden in the first die, comes to the front.

In the first die, the face 3 is in the front and the face 4 is on the right side. The hidden left face of this die is opposite to the right face 4.

In the second die, the face 3 has moved to the right side and the face 5 has come to the front. So, the face 5 is the hidden left face of the first die.

So, the face 5 is opposite to the face 4.

So, the number on the face opposite to the face showing 4 is 5.

Hence, option 3 is the correct option.