Visualising The Solid Shapes

Option (a) is a polyhedron because it satisfies its definition which is the solid is made up of polygonal region called faces and these faces meet at edges called line segments and the edges meet at vertices are called point.

(b) is a polyhedron because it satisfies its definition which is the solid is made up of polygonal region called faces and these faces meet at edges called line segments and the edges meet at vertices are called point.

(c) is not a polyhedron because it does not satisfy its definition. The figure contains circular base.

(d) is a polyhedron because it satisfies its definition which is the solid is made up of polygonal region called faces and these faces meet at edges called line segments and the edges meet at vertices are called point.

Option (a) will not form a polyhedron because it must have more than four faces. So, it is not possible in 3 triangles which have 3 faces only.

(b) 2 triangles and 3 parallelograms form a polyhedron which satisfies its definition. The solid is made up of polygonal region called faces and these faces meet at edges called line segments and the edges meet at vertices are called point.

(c) 8 triangles can form a polyhedron because it has 8 faces.

(d) 1 pentagon and 5 triangles form a polyhedron which satisfies its definition. The solid is made up of polygonal region called faces and these faces meet at edges called line segments and the edges meet at vertices are called point.

Option (a) cuboid is not a regular polyhedron because its vertices are not formed by the same number of faces.

(b) Triangular prism is not a regular polyhedron which has two triangular face and three rectangular sides. so, vertices are not formed by the same number of faces.

(c) cube is a regular polyhedron because its vertices are formed by the same number of faces.

(d) Square prism is not a regular polyhedron because its vertices are not formed by the same number of faces.

Option (a) is a two-dimensional figure because it has two dimensions length and breadth

(b) Rectangular prism is a three-dimensional figure it has length, width and height

(c) Square Pyramid is a three-dimensional figure, its faces have 4 triangles and one square

(d) Square prism is a three-dimensional figure, it has two square bases and flat sides

Option (a) cannot be the base of a pyramid because pyramid is a polyhedron and its base is polygon.

(b) Circle cannot be the base of a pyramid because pyramid is a polyhedron and its base is polygon.

(c) Octagon can be the base of a pyramid because it has a polygon as a base and lateral faces are triangle.

(d) Oval cannot be the base of a pyramid because pyramid is a polyhedron and its base is polygon

Option (a) The point at which two or more edges meets is called vertex. Pyramid has atleast 4 vertices.

(b) The point at which two or more edges meets is called vertex. Prism has atleast 6 vertices.

(c) The point at which two or more edges meets is called vertex. Cone has one vertex.

(d) The point at which two or more edges meets is called vertex. Sphere does not have any vertex and edge.

Option (a) a polyhedron is the solid is made up of polygonal region called faces and these faces meet at edges called line segments and the edges meet at vertices are called point. So it has only line segments as its edge.

(b)Cone does not have only line segment. It is connecting with line and a half line also.

(c) Cylinder having two plane surfaces meeting at the point.

(d) Polygon also contain plane surface not only the line segment.

According to Euler’s formula, F + V-E = 2 where F = face, V = vertices and E = edge

So, from the given F = V = 5

⇒ 5 + 5-E = 2

⇒ 10-E = 2

⇒ E = 10-2

⇒ E = 8

Option (a) is not match to our solution. so it is not a correct answer.

(b) is not match to our solution. so it is not a correct answer.

(c) is the correct answer and matches with our solution.

(d) is not match to our solution. so it is not a correct answer.

The top view of the shape is determined by the view from the top of a given picture. Here the given shape is

The view from the top of this shape is

Option (a) is the correct answer

(b) is not match to our solution

(c) is not match to our solution

(d) is not match to our solution

The shape of a cube is follows

If the given picture is folded into cube, N will opposite to P,Q is on the top and O in the bottom. so, L faces opposite to M.

Option (a) is the correct answer matches the solution.

(b) is not match to our solution

(c) is not match to our solution

(d) is not match to our solution

Option (a) cone is a three-dimensional shape which have circular base and it tapers from a base to a point called vertex. This figure have circular base so it is the correct answer.

(b) this figure does not have any circular base so it is not the correct answer.

(c) this figure does not have any circular base so it is not the correct answer.

(d) this figure does not have any circular base so it is not the correct answer.

Option (a) is a prism because the sides are congruent.

(b) is not a prism because top and bottom faces are not congruent in the picture. so it is the correct answer.

(c) is a prism because the sides are congruent.

(d) is a prism because the sides are congruent.

A pyramid is a polyhedron whose base is a polygon and its lateral faces are triangles. we have 4 congruent equilateral triangle, as per definition of pyramid if we add a square with same side length of a triangle we get a polygon as its base and it will make pyramid.

Option (a) is not enough to make pyramid as per solution.

(b) is the correct answer and it is enough to make pyramid.

(c) is not enough to make pyramid as per solution.

(d) is not enough to make pyramid as per solution.

Given, the side of a square = 30m

The formula for perimeter of the square = 4 × side

∴ the perimeter of the square = 4 × 30 = 120m

The scale used to draw its picture is 1 cm : 5 m

Hence the perimeter of a square =

= 24 cm

Option (a) is not match to our solution. so it is not a correct answer.

(b) is the correct answer and matches with our solution.

(c) is not match to our solution. so it is not a correct answer.

(d) is not match to our solution. so it is not a correct answer.

Option (a) is not correct because vertex is a point where two or more edges meet. this picture does not have edges.

(b) is not correct because vertex is a point where two or more edges meet. this picture does not have edges.

(c) is a correct answer because vertex is a point where two or more edges meet. this picture have edge.

(d) is not correct because vertex is a point where two or more edges meet. this picture does not have edges.

In the given map, the scale 1 cm: 0.5 km

The distance between the between school and the book shop in the map is 2.2cm

So, the actual distance between school and the book shop = 2.2 × 0.5 km

= 1.1 km

Option (a) is not match to our solution. so it is not a correct answer.

(b) is not match to our solution. so it is not a correct answer.

(c) is not match to our solution. so it is not a correct answer.

(d) is the correct answer and matches with our solution.

Option (a) V = 4, F = 4, E = 6

According to Euler’s formula, F + V-E = 2 where F = face, V = vertices and E = edge

So, from the given

⇒ 4 + 4-6 = 2

⇒ 8-6 = 2

⇒ 2 = 2

⇒LHS = RHS. it is true for a polyhedron

(b) V = 6, F = 8, E = 12

According to Euler’s formula, F + V-E = 2 where F = face, V = vertices and E = edge

So, from the given

⇒ 8 + 6-12 = 2

⇒ 14-12 = 2

⇒ 2 = 2

⇒LHS = RHS. it is true for a polyhedron

(c) V = 20, F = 12, E = 30

According to Euler’s formula, F + V-E = 2 where F = face, V = vertices and E = edge

So, from the given

⇒ 12 + 20-30 = 2

⇒ 32-30 = 2

⇒ 2 = 2

⇒LHS = RHS. it is true for a polyhedron

(d) V = 4, F = 6, E = 6

According to Euler’s formula, F + V-E = 2 where F = face, V = vertices and E = edge

So, from the given

⇒ 6 + 4-6 = 2

⇒ 10-6 = 4 = LHS

⇒ RHS = 2

⇒ LHS ≠ RHS. it is not true for a polyhedron

From the given, height of the room in blueprint = 33 cm

The actual height of the room = 330 cm

Then,

⇒

So, the scale is 1:10

Option (a) is not match to our solution. so it is not a correct answer.

(b) is the correct answer and matches our solution.

(c) is not match to our solution. so it is not a correct answer.

(d) is not match to our solution. so it is not a correct answer.

From the map, denotes the hospital.so count the in the map, we get 2

Option (a) is not match to our solution. so it is not a correct answer.

(b) is the correct answer and matches our solution.

(c) is not match to our solution. so it is not a correct answer.

(d) is not match to our solution. so it is not a correct answer.

From the map, we can count the number of general stores is 6 and the number of the ground is 4.

∴

⇒

⇒

Option (a) is not match to our solution. so it is not a correct answer.

(b) is not match to our solution. so it is not a correct answer.

(c) is not match to our solution. so it is not a correct answer.

(d) is the correct answer and matches our solution.

From the map, denotes the school.so count the in map, we get 5

Option (a) is not match to our solution. so it is not a correct answer.

(b) is not match to our solution. so it is not a correct answer.

(c) is the correct answer and matches with our solution.

(d) is not match to our solution. so it is not a correct answer.

A square prism is a three-dimensional solid object who’s base and sides are all squares. Since all the angles are right angles and all the sides are equal, it can also be called a cube.

A rectangular prism is a three-dimensional solid object who’s base is a square but sides are all rectangles. Since all the angles are right angles and the opposite sides are equal, it can also be called a cuboid.

Let us rename the edges in the figure. Now we see that there are 6 faces in total, namely ABC, ABD, EBC, EBD, CDE and CDA. The faces meeting at B have the letter B in it. From the figure, it is evident that the faces meeting at B are ABC, ABD, BCE&BED.(4)

Each side in a polygon contributes to 1 face and an the n sided polygon itself, so if there are n sides in a polygon there will be n+1 faces.

According to Euler’s formula, in any solid (Faces)+(Vertices)=(Edges)+2 or F+V=E+2

Therefore, 12+20=x+2 gives x=20

When the net is folded, it gives

Which is a triangular prism.

By definition, A cone is a three-dimensional geometric shape that tapers smoothly from a flat base to a point called the apex or vertex.

From formula, F+V=E+2, since E=8, F+V must be 10. We know that in any pyramid, the number of faces = the number of vertices. Therefore, 2F=10 gives F=5.

From the figure, it is evident that the rectangular prism when unfolded contains 6 rectangles.

[Hint : Every square is a rectangle but every rectangle is not a square.]

If a line segment joins two vertices that lie on the same face, then it is called an edge. However, if a line segment joins two vertices that do not lie on the same face, it is a diagonal.

4km→1 cm

16km→x cm

By the method of cross multiplication, we have x = 16/4 = 4 cm.

In such problems, it is necessary to convert the given quantities into the same unit. For simplicity, let us convert the actual distance given in kilometres to millimetres.

110km = 110000000mm

Therefore Scale = 110000000mm/25mm = 4400000

So, the scale will be 4400000:1

Each side in a prism contributes to 1 face and counting the top and the bottom faces of the prism so if there are 5 sides in a pentagon there will be 5+2=7 faces.

Hexagonal base contains 6 vertices. Since the problem talks about a pyramid, it is constructed on to another vertex. Total, we have 7 vertices.

given figure is the top view of hut.

The front row contains 4 cubes and so does the row in the back. In total, we count 4+4=8 cubes.

From Euler’s formula, we have F+V=E+2 and from the problem, F+V=14 gives E+2=14, or E=12.

Tetrahedron, Cube, Octahedron, Dodecahedron and Icosahedron.

4, 6, 8, 12 or 20

(i) Front View

(ii) Side View

(iii) Top View

(b) (i) Side View

(ii) Top View

(iii) Front View

(c) (i) Front View

(ii) Top View

(iii) Side View

(d) (i) Side View

(ii) Front View

(iii) Top View

The other name for cuboid is square prism.

The minimum faces a polyhedron can have is 4.

The minimum faces a polyhedron can have is 4 and is called a triangular pyramid or a tetrahedron.

A regular octahedron has 8 congruent faces which are equilateral triangles.

A pentagonal prism has 7 faces which include two pentagonal faces.

Even though the cylinder has two opposite faces, the requirement of a prism is the solid to have flat faces. The cylinder does not have all faces flat.

Euler’s formula is only true for a polyhedron that doesn’t intersect itself.

F+V=E+2 formula tells us that 10+15 is not equal to 20+2, so it is not possible for a polyhedron to have 10 faces, 20 edges and 15 vertices.

T

A net is a figure obtained when a three-dimensional object is opened. Since a 3D object can be opened from any edge, the solid shape has more than one unique net.

In a pyramid, it is not possible to have two opposite vertices. So there cannot be a diagonal, which by definition is the line segment joining two opposite vertices.

A cylinder has two circular faces of equal radius.

In a cuboid, the number of diagonals is 4.

A cube has opposite faces which are equal.

A cylinder is a 3D shape having two circular opposite faces of same radii.

(T)

The scale is 10000:1

If the scale is 1:60, then the height 3.5cm will become 3.5x60 cm = 210cm

The scale given is 1:7, so the width of the room in the drawing will be 280cm/7= 40cm.

(a)tetrahedron:-4

(b) Hexahedron:-6

(c) Octagonal Pyramid:-9

(d) Octahedron:-8

The is drawing with top, side, front view given below

(a) Cone:-1

(b) Cylinder:-0

(c) Sphere:-0

(d) Octagonal Pyramid:-1

(e) Tetrahedron:-4

(f) Hexagonal Prism:-12

(a) Cone:-1

(b) Cylinder:-2

(c) Sphere :-0

(d) Octagonal Pyramid:-16

(e) Hexagonal Prism:-18

(f) Kaleidoscope:-9

(a) Edge-9, vetrices:-6, face:-5

By using Euler’s formula

F + V = E + 2

⟹ 5 + 6 = 9 + 2

⟹ 11 = 11. Therefore it is a polyhedral.

(b) Edge-12, vetrices:-8, face:-6

By using Euler’s formula

⟹ F + V = E + 2

⟹ 6 + 8 = 12 + 2

⟹ 14 = 14.therfore it is a polyhedral.

(c) Edge-2, vertices:-0, face:-3

By using Euler’s formula

F + V = E + 2

⟹ 3 + 0 = 2 + 2

⟹ .Hence it is not possible. Therefore it is not a polyhedral.

(d) Edge-15, vetrices:-10, face:-7

By using Euler’s formula

F + V = E + 2

⟹ 7 + 10 = 15 + 2

⟹ 17 = 17

Therefore, it is a polyhedral.

(e) Edge-9, vertices:-6,face:-5

By using Euler’s formula

F + V = E + 2

⟹ 5 + 6 = 9 + 2

⟹ 11 = 11 therefore it is a polyhedral.

(f) Edge-2, vertices:-0, face:-3

By using Euler’s formula

F + V = E + 2

⟹ 3 + 0 = 2 + 2⟹

It is not possible. Therefore, it is not polyhedral.

(g) Edge-20, vertices:-11, face:-11

By using Euler’s formula

F + V = E + 2

⟹ 11 + 11 = 20 + 2

⟹ 22 = 22. Therefore, it is a polyhedral.

(h) Edge-16, vertices:-9,face:-9

By using Euler’s formula

F + V = E + 2

⟹ 9 + 9 = 16 + 2

⟹ 18 = 18. Therefore it is a polyhedral.

(i) Edge-18, vertices:-12, face:-8

By using Euler’s formula

F + V = E + 2

⟹ 8 + 12 = 18 + 2

⟹ 20 = 20. Therefore it is a polyhedral.

(j) Edge-12, vertices:-6,face:-8

By using Euler’s formula

F + V = E + 2

⟹ 8 + 6 = 12 + 2

⟹ 14 = 14. Therefore it is a polyhedral.

(k) Edge-0, vetrices:-1, face:-2

By using Euler’s formula

F + V = E + 2

⟹ 2 + 1 = 0 + 2

⟹ 32. It is not possible. Therefore it is not a polyhedral.

(l) Edge-24, vetrices:-16, face:-10

By using Euler’s formula

F + V = E + 2

⟹ 10 + 16 = 24 + 2

⟹ 26 = 26. Therefore it is a polyhedral.

(m) Edge-1, vetrices:-0, face:-1

By using Euler’s formula

F + V = E + 2

⟹ 1 + 0 = 1 + 2

⟹ 13.it is not possible. Therefore it is not a polyhedral.

(a) number of cube are 10

(b) number of cube are 10

(c) number of cube are 10

(d) number of cube are 9

(e) number of cube are 11

(f) number of cube are 9

(g) number of cube are 11

(h) number of cube are 110

(i) number of cube are 113

(j) number of cube are 66

(k) number of cube are 15

(l) number of cube are 14

(a) The front, side and top view of the given shapes are given below

(b) the front, side and top view of the given shapes are given below

(c) the front, side and top view of the given shapes are given below

(d) the front, side and top view of the given shapes are given below

(e) the front, side and top view of the given shapes are given below

(f) The front, side and top view of the given shapes are given below

(g) The front, side and top view of the given shapes are given below

(h) The front, side and top view of the given shapes are given below

(i) The front, side and top view of the given shapes are given below

(j) The front, side and top view of the given shapes are given below

Given that (i) face = 7, vertices = 10, edge = x

By using Euler’s formula

We know that

Face + vertices = edge + 2

⟹ 7 + 10 = x + 2

⟹ 17 = x + 2

⟹ X = 17-2

⟹ X = 15.

Given that (ii) face = y, vertices = 12, edge = 18

By using Euler’s formula

We know that

Face + vertices = edge + 2

⟹ y + 12 = 18 + 2

⟹ y + 12 = 20

⟹ y = 20-12

⟹ y = 8

Given that (iii) face = 9, vertices = z, edge = 16

By using Euler’s formula

We know that

Face + vertices = edge + 2

⟹ 9 + z = 16 + 2

⟹ 9 + z = 18

⟹ Z = 18-9

⟹ Z = 9.

Given that (iv) face = p, vertices = 6, edge = 12

By using Euler’s formula

We know that

Face + vertices = edge + 2

⟹ P + 6 = 12 + 2

⟹ P + 6 = 14

⟹ P = 14-6

⟹ P = 12.

Given that (v) face = 6, vertices = q, edge = 12

By using Euler’s formula

We know that

Face + vertices = edge + 2

⟹ 6 + q = 12 + 2

⟹ 6 + q = 14

⟹ q = 14-6

⟹ q = 8

Given that (vi) face = 8, vertices = 11,edge = r

By using Euler’s formula

We know that

Face + vertices = edge + 2

⟹ 8 + 11 = r + 2

⟹ 19 = r + 2

⟹ r = 19-2

⟹r = 17

according to the data given in question

Vertices (V) = Face (F) = 9 and Edge (E) = 16

By Euler’s formula we know that

Face + Vertices = Edge + 2

Face + vertices = 9 + 9 = 18

Edge + 2 = 16 + 2 = 18

Therefore Face + Vertices = Edge + 2 = 18

So, the polyhedron is possible for the above data.

according to the data given in question

Vertices (V) = 12, Face(F) = 8 and Edge(E) = 6

By Euler’s formula we know that

Face + Vertices = Edge + 2

So,Face + vertices = 8 + 12 = 20

and Edge + 2 = 6 + 2 = 8

⟹ 208

Therefore Face + VerticesEdge + 2

So, the polyhedron is not possible for the above data

according to the data given in question

Vertices = 40, Face = ? and Edge = 6

By Euler’s formula we know that

Face + Vertices = Edge + 2

⟹ Face + 40 = 60 + 2

⟹ Face + 40 = 62

⟹ Face = 62-40

⟹ Face = 22.

for figure given below:-

number of faces are 14

for the above figure, the number of faces are 10

for the above figure, the no of faces are 16

according to the data given in question

Vertices = 12, Face = 20 and Edge = ?

By Euler’s formula we know that

Face + Vertices = Edge + 2

⟹ 12 + 20 = edge + 2

⟹ 32 = edge + 2

⟹ Edge = 32-2

⟹ Edge = 30.

according to the data given in question

Vertices = ?, Faces = 40 and Edge = 60

By Euler’s formula we know that

Face + Vertices = Edge + 2

⟹ 40 + vertice = 60 + 2

⟹ 40 + vertice = 62

⟹ Vertices = 62-40

⟹ Vertices = 22.

Given below is the net drawing of regular hexahedron with side as 3cm each.

given below is the net drawing of regular tetrahedron with side 6cm.

The net drawing is given below

(a) :-(b)hexagonal.

It has a hexagon at it both the side.

(b):- :-(d)cone

It has a cone like shape.

(c) :-(c) square pyramid

It has square shape base with triangular faces.

(d) :-(a) hexahedron

It is a cone and cone is also called hexahedron.

The net drawing is

the net solid drawing of the above question is

number of cubes in base layer is 6

The figure below is top view of only shaded cubes. The base layer is not seen from top.

Faces = n + 1,vertices = n + 1,edges = 2n

Explanation: in a pyramid ,the number of faces is 1 more than the number of sides of the polygon base i.e faces = n + 1

Also, the number of vertices is 1 more than the number of sides of the polygon base i.e. vertices = n + 1

Also the no of edges of pyramid is 2 times the no of sides of polygon

The book is like the shape of a cuboids. Cuboids is like rectangular Prism

In the first figure hemisphere is mounted on the cylinder. In the second figure the hexagonal prism is mounted with a cone

If We increase height of cube

If We decrease the height of cube

(a)

(b)

(a)The built-up area of Govt. model School I = 2.5acre

(b)Two schools shown in picture Govt model School I &II

(c)Park A is nearest to the dispensary

(d)The main market belongs to block A

(e)6 parks have been represented in the map

The map is not sufficient to answer the question

(a)Flower road, Khel marg,mall road and sneha marg road meet at round

(b)The address of the Stadium sector27,B.Town,India.

(c)The police station is situated on sneha marg

(d)Aneha address H.N-1 Nr.Bank 1(A)

(e)Sector 27 has the maximum no of house

(f)Fire station is located in sector 26

(g)In the map, four sectors have been shown.

Width before editing the photograph = 2 units

Width after editing the photograph = 4 units

So, Scale =

Given, the side of a square board is = 50cm.

So, perimeter of the square board = cm.

On drawing in the notebook, the perimeter of a square board = 40cm.

Scale

Given Scale = 1cm:5km i.e. 1cm in picture = 5km of actual distance

So, 5cm in picture = 5*5 km of actual distance

Hence, the actual distance between the two places is 25 km.

So,5cm represent = distance

Given scale is 1cm:10km i.e.1cm in a picture = 10km of actual distance

(a)The distance between the school and library in the picture = 6cm

Hence, the actual distance between the school and library =

(b)Distance between the college and complex in the picture = 2cm.

Actual distance between the college and complex

(c)Distance between the school and house in the picture = 3.5c.m = 3.5*10 = 35km

The actual length of the painting 2m or 200c.m

Scale used in the painting = 1mm = 20cm

Hence, the length of painting in photograph = scale*Actual size =

(a)Scale =

(b)Scale =

Given height of ice cream sculpture = 360cm

Scale used for ice cream sculpture = 1:30

The height of ice-cream served =

Hence, the height of the ice-cream served is = 12cm