# Free Visualising Solid Shapes Subjective Test 02 Practice Test - 7th grade

We have 4 congruent equilateral triangles. What else do we need to make a pyramid?  [1 MARK]

#### SOLUTION

Solution :

We need a square with same side length as that of the triangle. The resultant solid is known as a square pyramid.

Is a square prism same as a cube? Give reason.  [2 MARKS]

#### SOLUTION

Solution :

Explanation: 2 Marks

A square prism has a square as its base. However, its height is not necessarily same as the side of the square.

Thus, a square prism can also be a cuboid.

All cubes are square prism but all square prisms need not be cubes.

How are prisms and cylinders as well as pyramids and cones alike? [2 MARKS]

#### SOLUTION

Solution :

Explanation: 1 Mark each

A cylinder can be thought of as a circular prism that is a prism with a circle as its base.

A cone can be thought of a circular pyramid that is a pyramid with a circle as its base.

What is the key feature of a sphere? How many faces, corners and edges does a sphere have?  [2 MARKS]

#### SOLUTION

Solution :

The key feature of a sphere is that every point on the surface of a sphere is equidistant from the centre.

The normal definition of faces, vertices and edges are not appropriate for a sphere because it is not a polyhedron.

(a) There are 9 cans on this shelf. What can be the total number of cans in this shelf?

(b) Name the polygon formed on the front side of the figure.

[2 MARKS]

#### SOLUTION

Solution : Each part: 1 Mark

(a) By visual estimation, it is easy for us to see that 3 more cans can fit in the empty space. Hence, we can say that there will be 12 cans on a full shelf.

(b) The front side of the figure is a pentagon.

What is a 3-D shape and a 2-D shape? Give examples.  [3 MARKS]

#### SOLUTION

Solution :

Definition: 2 Marks
Example: 1 Mark

A shape that has only two dimensions (such as width and height) and no thickness is known as 2-D shape. In 2-D shapes, the sides are made of straight and curved lines.
Example: squares, circles, triangles

A 3-D shape is a solid which encloses volume and has length, breadth and height. 3-D shapes have four properties that set them apart from 2-D shapes: faces, vertices, edges and volume.
Example: sphere, cylinder, cube

Draw the top, front, and side view of the following 3D objects.

(a)

(b)

[3 MARKS]

#### SOLUTION

Solution : (a) Diagram of each view: 0.5 Mark
(b) Diagram of each view: 0.5 Mark

(a)

(b)

(a) How many edges are there in a cylinder?

(b) How many vertices are there in a sphere?

(c) Identify the figure given below?

[3 MARKS]

#### SOLUTION

Solution :

Each part: 1 Mark

(a) Edge is defined as the line segment where two faces of a polyhedron meet. A cylinder has no such line segments. Therefore, it has 0 edges.

(b) There are no vertices in a sphere as there are no sharp corners in a sphere.

(c) The figure which has all the three dimensions, viz. length, breadth and height, equal is called a cube. Therefore, the given figure is a cube.

(a) If two cubes of dimensions 3 cm by 3 cm by 3 cm are placed side by side, then what is the resulting dimension(length (l), breadth (b) and height (h)) of the cuboid?  Find the value of l+b+h (in cm).

(b) A metallic cylinder with volume 1331 cm3 is melted and is transformed into a cube. What will be the length of the sides of the cube formed? (Volume of a cube with side length a is given as a3
[3 MARKS]

#### SOLUTION

Solution :

Each Part: 1.5 Marks

(a) When we put the cubes side by side, the lengths of both the cubes are added, but there will be no change in the breadth and height of the cube.

Length (l) = (3 + 3) = 6 cm.

Height (h) = 3 cm.

Hence, l+b+h=6+3+3=12cm

(b) Since the metallic cylinder is melted to form a cube, the cube will be of the same volume as that of the cylinder.

Volume of cube  =  Volume of cylinder

a3=1331cm3

a=3(1331)=11 cm

There are 18 boxes on a shelf.

(a) How many boxes in total can be kept on the shelf without leaving any space?

(b) If the length and height of the shelf are doubled, how many boxes can be kept on the shelf now without leaving any space?

(c) If the length is halved and height of the shelf is doubled, how many boxes can be kept on the shelf now without leaving any space?

(d) If the length is doubled and height of the shelf is halved, how many boxes can be kept on the shelf now without leaving any space?

[4 MARKS]

#### SOLUTION

Solution : Each part: 1 Mark

From visual estimation, each column contains 4 boxes.

(a)  6 columns can be fit in the shelf.

In the 5th column, there is space for 2 more boxes.

and in the 6th column, there is space for 4 boxes.

Number of boxes on the full shelf = 18 + 2 + 4 = 24 boxes

(b) If the length of the shelf is doubled, the shelf can have 12 columns.

If the height of the shelf is doubled, then each column can have 8 boxes.

Total number of boxes on the full new shelf:

= 12×8

= 96 boxes

(c) If the length of the shelf is halved, the shelf can have 3 columns.

If the height of the shelf is doubled, then each column can have 8 boxes

Total number of boxes on the full new shelf:

= 3×8

= 24 boxes

(d)  If the length of the shelf is doubled, the shelf can have 12 columns.

If the height of the shelf is doubled, then each column can have 4 boxes.

Total number of boxes on the full new shelf:

= 12×4

= 48 boxes

Match the nets with the appropriate solids:  [4 MARKS]

#### SOLUTION

Solution : Each correct match: 1 Mark

(a) Are sphere, cone and cylinder polyhedrons? Give reason for your answer.

(b) Match the 2-D figures with their names.

[4 MARKS]

#### SOLUTION

Solution : Each part: 2 Marks

(a) Sphere, cone and cylinder are not polyhedrons.

Sphere: Sphere has only one curved face and no polygons as faces. So it is not a polyhedron.

Cone: Cone has one circular face and the remaining part is a curved surface. So it is not a polyhedron.

Cylinder: Cylinder has two circular faces and the remaining part is a curved surface. So it is not a polyhedron.

(b) (i) - (b)

(ii) - (a)

(iii) - (e)

(iv) - (c)

(v) - (d)

(a) Can a polyhedron have 10 faces, 20 edges and 15 vertices? Justify your answer.

(b) A glowing bulb is kept above the following solids. Name the shape of the shadows obtained in each case. Attempt to give a rough sketch of the shadow.

[4 MARKS]

#### SOLUTION

Solution : Each part: 2 Marks

(a)  By Euler's formula,

F + V - E = 2

Substituting the given values

10 + 15 - 20 = 35 - 20 =15 which is not equal to 2.

A solid needs to satisfy the Euler's formula to be called a polyhedron.

A polyhedron cannot have 10 faces, 20 edges and 15 vertices.

(b)  The shapes of the shadows of these figures will be as follows.

(i) A ball

The shape of the shadow of a ball will be a circle.

(ii) A cylindrical pipe

The shape of the shadow of a circular pipe will be a rectangle.

(iii) A book

The shape of the shadow of a book will be a rectangle.

Find the number of faces, edges and vertices of the following polyhedrons and verify them by using Euler's formula.  [4 MARKS]

(i) Tetrahedron

(ii) Octahedron

#### SOLUTION

Solution : (i) Tetrahedron

Number of faces = 4

Number of vertices = 4

Number of edges = 6

Clearly, F + V = E + 2 i.e., 5 + 6 = 9 + 2

(ii) Octahedron

Number of faces = 8

Number of vertices = 6

Number of edges = 12

Clearly, F + V = E + 2 i.e., 8 + 6 = 12 + 2