Mensuration

The side of the cube = 5 cm is painted on all sides. Its figure is shown below:

We can say that the side of 5 cm is made up of 5 parts each of 1 cm.

Total number of cubes of side 1 cm = 25 + 25 + 25 + 25 + 25 = 125.

Now, the cubes whose one face is painted are marked in red colour.

As seen above, in one face of cube total 9 small cubes have 9 one side painted.

And, there are 6 faces in a cube.

Hence, total 9 × 6 = 54 faces will have one face painted.

The cube has side = 4 cm, as shown below:

Now, the cube is cut into small cubes of each side cm.

The total number of cubes = 4 × 16 = 64 small cubes.

Numbers of cut-out cubes =

Now, surface area of cut-out cubes = 64 × 6 × 1 cm²

And surface area of the original cube = 6 × 4²

The required ratio = = 1:4

Let a be the side of a square sheet

Then area of bigger square sheet = a² …..1

Now, we make the circle of maximum possible size from it.

Then the radius of circle = ………….2

So its diameter = a

Now, any square in a circle of maximum size will have the length of diagonal equal to the diameter of circle.

i.e. diagonal of square made inside the circle = a

so, the side of this square =

Area of this square =

From eqs.1and 2,

Area of final square is of original square.

Let ABCD be the rectangle of length l and width w.

Now,we construct a triangle of maximum area inside it in all possible ways.

We know that,

Area of triangle = × base × height

So ,for maximum area ,base and height of maximum, length is needed.

Here, maximum base length = l

And maximum height = w

Area (maximum) of triangle = × l × w sq.units.

we know that,

The volume of cylinder having base radius = r

And original height of cylinder = h

Volume of cylinder (v) = π × r² × h

New height H = h and

New radius R = 2r (new radius = 2 times of original radius)

New volume of cylinder(V) = π × 4r² × h = πr²h = v

Hence, the volume of new cylinder = the volume of original cylinder.

And the answer is (b).

according to the question,

Curved surface area of cylinder having radius (r) and the height (h).

Curved surface area of cylinder = 2πrh

And the new curved surface area of cylinder having radius (2r) and the height(h)

And the new curved surface area of cylinder = 2π × 2r × = πrh

Then, multiplying and divide by 2

= × 2πrh

New curved surface area of cylinder = × original curved surface area.

And the answer is (c).

according to the question,

The total surface area of cylinder having radius (r) and the height(h).

Total surface area of cylinder = 2πr(h + r)

And the new total surface area of cylinder having radius(2r) and the height ().

= 2π (2r)[2r + h]

= πr(8r + h)

Volume of the cuboid = lbh

6 = l × b

15 = l × h

10 = b × h

6 × 15 × 10 = l² b² h²

Volume = l × b × h

= √ 6 × 15 × 10 = 30cm²

A regular hexagon comprises 6 equilateral triangles, each of them having one of their vertices at the centre of the hexagon.

The sides of the equilateral triangles are equal to the radius of the smallest circle inscribing the hexagon.

Hence, each side of the hexagon is equal to the radius of the hexagon and the perimeter is 6r.

Given, dimension of a godown are 40m,25m and 10m.

Volume of godown = 40 × 25 × 10 = 10000m³

Now, volume of each cuboidal box = 2 × 1.25 × 1 = 2.5m³

The number of boxes,that can be filled in the godown =

= 4000

Let the side of cube be a . then,

Volume of a cube = a³ = 64

a = 4

now, surface area of the cube = 6 = 96cm²

Let H be the new height.

Curved surface area of a cylinder with radius r and height h = 2πrh

Now, according to the question, radius is tripled

Then,

Curved surface area = 2π × 3r × h = 2πrh

6πrh = 2πrh

H =

H = h

Hence ,the new height will be of the original height.

Volume of cube = (side)³

Volume of each small cube = 20³ = 8000cm³

= 0.008m³

Now, volume of the cubical box = 2³ = 8m³

Number of small cubes, that can just be accommodated in the cubical box

= = 1000

Given, r = h

Then, volume of cylinder = πr² h = π r² × r = πr³

We know that,the volume of a cube = (side)³

= a³

= (3x)³

= 27x³

it is clear from the figure that, quadrilateral ABCD is a parallelogram. The diagonal AC of the given parallelogram ABCD divides it into two triangles of equal areas.

Area of the triangle ABC = × base × height

= × 12 × 3

= 18

Area of parallelogram ABCD = 2 × 18 = 36 cm²

The diagonal AC of the rhombus ABCD divides it into two triangles of equal areas.

Now, area of Δ ABC = × base × height = × 4 × 6 = 12cm²

Area of the rhombus ABCD = 2 × area of Δ ABC

= 2 × 12 = 24cm²

we know that,

Area of a parallelogram = side × altitude

a × h = 60

a × 5 = 60

a = 12cm

Then, sum of its parallel sides = 52-(10 + 10) = 32cm

Area of the trapezium = (a + b)h

= × 32 × 8 = 128cm²

Area of the given quadrilateral = (sum of altitudes) × corresponding diagonal.

20 = (1 + 15)BD

BD = 16cm

Given ,a metal sheet 27cm long, 8cm broad and 1cm thick.

Then the volume of the sheet = 27 × 8 × 1 = 216cm³

Now, since this sheet is melted to form a cube of edge length a(say)

Then, volume of the cube = volume of the metal sheet

a³ = 216cm³

a = 6cm

hence ,the side of the cube is 6cm

Sum of volumes of the three metals cubes = 6³ + 8³ + 10³

= 216 + 512 + 1000

= 1728cm³

Since , a new cube is formed by melting these three cubes.

Let a be the side of new cube .

Volume of the new cube = sum of volumes of three metal cubes

A³ = 1728

a = 12cm

hence ,the edge of the new cubes is 12cm

Since , thickness of the box is 2.5cm,then outer measures will be 115 + 5,75 + 5 and 35 + 5,i.e.120cm,80cm and 40cm

The outer volume = 120 × 80 × 40 = 384000cm³

And the inner volume = 115 × 75 × 35 = 301875cm³

Volume of the wood = outer-inner volume

= 384000-301875 = 82125cm³

Let r,R be radii of two cylinder and h,H be their heights.

Then , and

Now, = = =

=

Hence , v:V = 1:6

According to the question,

Now ,ratio of areas,

= 1:16

Since the solid has rectangular faces.

So , we have lb = 16 ……(i)

bh = 32 …………(ii)

lh = 72 …………….(iii)

on multipiying eqns.i,ii and iii,we get

l × b × b × h × l × h = 16 × 32 × 72

l² × b × h² = 36864

L × b × h = 192

Hence ,the volume is 192cu cm.

(i)The volume of the cylindrical container having radius r and height h = πr²h

(ii)The volume of the cylinder container with radius 2r and height

= π (2r)² × × h = 2πr²h

(iii)The volume of the cuboidal container having dimension

= r² h

From parts i ,ii and iii ,we have the following order C,A,B.

Now total surface area of hate = curved surface area + top surface area + base surface area

= 2πrh + πr² + π(R²-r²)

= 2πrh + πR²

The volume of a cube of side 4cm = 4 × 44 = 64cm³ when it is sliced into 1cm³ cubes, we will get 64small cubes.

In each side of the larger cube, the smaller cubes in the edges will have more than one face painted.

The cube which are situated at the corner of big cube, have three faces painted.

So, to each edge two small cubes are left which have two faces painted. As,the total numbers of edges in a cubes are 12.

Hence, the number of small cubes with two faces painted = 12 × 2 = 24.

Given,

A Cube of side 5cm is cut into 1cm cubes.

Volume of a cube = 5 × 5 × 5 = 125cm³

Now, the big cube is cut into 1cm cubes.

The number of small cubes =

Thus, the volume of big cube = the volume of 125 cubes having an edge 1cm

Hence, there is no change in the volume.

we have, two cubes of side a.

These two cubes are joined face to face , then the resultant solid figure is a cuboid which has same breadth and height as the joined cubes has length twice of the length of a cube, i.e. l = 2a,b = a and h = a

Thus, the total surface area of cuboid = 2(lb + bh + hl)

= 2(2a × a + a × a + a × 2a)

= 2[2a² + a² + 2a²]

= 10a²

We know that,

Area of rhombus = × d₁ × d₂

Where ,d₁ and d₂ are diagonals of the rhombus.

If the diagonals get doubled, then the area = × 2d₁ × 2d₂

=

Hence, the new area becomes 4times its original area.

since, the cube fits exactly in the cylinder with height h.

Then, each side of the cube = h

Now, volume of the cube = (side)³ = h³

And the surface area of cube = 6 × (side)² = 6 × h²

The volume of a cylinder with radius r and height h = πr²h if radius is halved, then new volume = πh = πr²h

Hence, the new volume is of original volume.

The curved surface area of a cylinder with radius r and height h = 2π rh

If the height is halved, then new curved surface area of cylinder = 2 = πrh

Percentage reduction in curved surface area =

= 50%

Since, the cylinder that exactly fits in cube of side a ,has its height equal to the edges of the cubes and radius equal to half the edges of the cube.

Height = a and radius =

Now, volume of the cylinder = πr²h = πa

=

Since , the cylinder that exactly fits in a cube of side b,has its height equal to the edge of the cube and radius equal to half the edges of the cube.

Height = b and radius =

Now, curved surface area of the cylinder = 2π = × b = πb²

Let ABCD be a quadrilateral ,where h1 and h2 are height on the diagonal BD = d

Then , area of quadrilateral ABCD = (h₁ + h₂)BD

= × 2d

= (h₁ + h₂) × d

Perimeter of a rectangle with length l and breadth b = 2(l + b)

if its length and breadth are doubled, then new perimeter = 2(2l + 2b)

= 2[2(l + b)]

Let ABCD is a trapezium, in which

AD = DC = BC = a(say)

And AB = 2a

Draw medians through the vertices D and C on the side AB.

AE = EB = a

Now, in parallelogram ADCE, we have

AD = EC = a and AE = CD = a {opposite side in a parallelograms are equals}

In triangle ADE and DEC

AD = EC

AE = CD

DE = BC

BY SSS,

Thus, triangle ADE and DEC are equilateral triangles having equal sides.

Hence , the trapezium can be divided into 3 equilateral triangles of equal area.

We know that , a cuboid is made of 6rectangular plane regions,i.e.6 rectangular faces ,which have different lengths and breadth. Therefore the area of the rectangular faces are different.

We know that , a cuboid has 6 rectangular faces, of which opposite faces have the same length and breadth. Therefore area of the opposite faces are equal.

We know that ,the curved surface area of a cylinder of radius h and height r.

= 2π × rh = 2πrh

Given , radius of cylinder = r and height of cylinder = h

Total surface area of a cylinder = curved surface area + area of top surface + area of base

= 2πrh + πr² + πr²

= 2πh(r + h)

Volume of a cylinder = πr²h

We know that ,the area of rhombus = half of the product of its diagonals

= (product of diagonals)

We have a rectangular sheet of dimension 20cm × 10cm

If we fold it along its length, which is 20cm,then the resultant figure

is a cylinder with height ,h = 10cm and

base circumference ,2πr = 20cm

r = cm

volume of the cylinder = πr²h

= π × × 10

= cm³ = v(say) (eq..i)

Again ,if we fold the rectangular sheet along its breadth ,which is 10cm ,the figure so obtained is a cylinder with height h = 20cm

And the base circumference 2πr = 10cm

r = =

volume of the cylinder = πr²h

= π × × 20

= cm³ = V(say) (….ii)

i.e. V = 2v

from eqs,(i) and (ii), we see that the volume of A is twice the volume of B.

For cylinder A, h = 10cm and r = cm

Curved surface area of A = 2πrh = 2π × × 10 = 200cm²

Again , for cylinder B, r = cm and h = 20cm

Curved surface area of B = 2πrh = 2π × × 20 = 200cm²

Hence the curved surface area of both the cylinder are same.

We know that ,a solid always occupies some space and magnitude of this space region is known as the volume of the solid.

lateral

We know that, a room is in the shape of a cuboid. Its 4 walls are treated as lateral faces of the cuboid.

Lateral surface area of room = area of 4 walls.

Let r ,R be the radii and h, H be the heights of two cylinders.

Given

Now, according to the question,

πr²h = πR²h

Hence , r:R = 3:1

Let r, R be the radii and h, H be the heights of two cylinders.

Given,

Now, according to the question,

πr²h = πR²h

h:H = 36:1