Mensuration Extra Questions

Explanation:

When the 5 cm cube is sliced into 1 cubic centimetre cubes, it will be divided into 5x5x5 = 125 small cubes. Each face of the original cube is made up of 5x5 = 25 square units. Therefore, each face is composed of 25 small cubes, and the entire cube is made up of 6x25 = 150 small cubes.

To count the number of small cubes that have exactly one face painted, we need to count the number of small cubes that have exactly one face exposed. The small cubes on the corners and edges have less than one face exposed, and the small cubes in the centre have all faces hidden. Therefore, we need to count the number of small cubes that are on the faces but not on the edges or corners.

Each face has 3x3 = 9 small cubes that are not on the edges or corners. There are six faces, so there are 6x9 = 54 small cubes that are not on the edges or corners.

Therefore, the answer is (c) 54.

Explanation:

The surface area of the original cube is given by 6 x (side)^2 = 6 x 4^2 = 96 square cm.

When the 4 cm cube is cut into 1 cm cubes, it will be divided into 4x4x4 = 64 small cubes. Each small cube has a surface area of 6 x (1 cm)^2 = 6 square cm. Therefore, the total surface area of all the small cubes is 6 x 64 = 384 square cm.

The ratio of the surface areas of the original cube to the small cubes is therefore:

96 : 384 = 1 : 4

Therefore, the answer is (c) 1 : 4.

Explanation:

Let the side of the original square be "x" units. When a circle of maximum possible size is cut from the square, the diameter of the circle will be equal to the side of the square, which means the radius of the circle will be (1/2)x units. Therefore, the area of the circle will be:

π(1/2 x)2 = (π/4)x2 square units.

Now, a square of maximum possible size is cut from the resultant circle. The diagonal of this square will be equal to the diameter of the circle, which is x units. Therefore, the side of the square will be (1/√2)x units, and the area of the square will be:

[(1/√2)x]2 = (1/2)x2 square units.

Therefore, the area of the final square is (b) 1/2 of the original square.

Explanation:

The area of the largest triangle that can be fitted into a rectangle of length l units and width w units is actually (a) lw/2.

The largest triangle that can be fitted into the rectangle will be a right triangle, with one vertex at the corner of the rectangle and the hypotenuse along the diagonal of the rectangle. The base of the triangle will be one of the sides of the rectangle, and the height of the triangle will be the other side of the rectangle.

Therefore, the area of the largest triangle will be:

(base x height)/2 = (lw)/2 = lw/2

Hence, the answer is (a) lw/2.

Explanation:

The volume of a cylinder is given by the formula:

V = πr^2h

If the height becomes 1/4 of the original height, and the radius is doubled, then the new volume of the cylinder will be:

V' = π(2r)2(h/4) = 4πr2(h/4) = πr2h = V

Therefore, the volume of the cylinder will remain unchanged, and the answer is (b) Volume of the cylinder will remain unchanged.

Explanation:

The curved surface area of a cylinder is given by the formula:

A = 2πrh

If the height becomes 1/4 of the original height, and the radius is doubled, then the new curved surface area of the cylinder will be:

A' = 2π(2r)(h/4) = πrh/2

So, the new curved surface area will be half of the original curved surface area.

Therefore, the answer is (c) Curved surface area of the cylinder will be halved.

Explanation:

The total surface area of a cylinder is given by the formula:

A = 2πr(r + h)

If the height becomes 1/4 of the original height, and the radius is doubled, then the new total surface area of the cylinder will be:

A' = 2π(2r)(2r + h/4) = 4πr(2r + h/4) = 4πr(2r + h)/4 = π(4r^2 + rh)

So, the new total surface area will be π(4r^2 + rh), which is not equal to the original total surface area 2πr(r + h).

Therefore, the answer is (d) None of the above.

Explanation:

Let the three coterminous faces of the cuboid be of dimensions l × b, b × h, and h × l. Then, we have:

2(lb + bh + hl) = 6 + 15 + 10 = 31 lb + bh + hl = 31/2

And, the volume of the cuboid is given by:

V = lbh

Now, we need to use the given information to find the value of V. From the given data, we can write:

lb = 6 bh = 15 hl = 10

Multiplying these equations, we get:

(lb)(bh)(hl) = (6)(15)(10) (lbh)^2 = 900 lbh = √900 = 30

Substituting this value of lbh in the equation V = lbh, we get:

V = 30

Therefore, the volume of the cuboid is 30 cubic cm, and the answer is (a) 30 cm3.

Explanation:

regular hexagon is inscribed in a circle, which means that each vertex of the hexagon touches the circle. The distance from the center of the circle to a vertex of the hexagon is the radius of the circle, denoted by r.

The hexagon can be divided into six equilateral triangles, each with side length r. The perimeter of the hexagon is equal to the sum of the side lengths of the six triangles:

Perimeter = 6 × r = 6r

Therefore, the answer is (b) 6r.

Explanation:

The volume of each box is:

2 × 1.25 × 1 = 2.5 cubic meters

The total volume of the godown is:

40 × 25 × 10 = 10,000 cubic meters

The number of boxes required to fill the godown is therefore:

10,000 / 2.5 = 4,000

Therefore, the answer is (c) 4000.

Explanation:

The volume of a cube is given by:

Volume = s3, where s is the length of each side of the cube.

So, if the volume of the cube is 64 cm3, then we have:

S3 = 64

Taking the cube root of both sides, we get:

s = 4 cm

The surface area of a cube with side length s is given by:

Surface area = 6s2

Substituting s = 4 cm, we get:

Surface area = 6 × 42 = 96 cm2

Therefore, the answer is (c) 96 cm2.

Explanation:

The answer is (d) one third.

When the radius of a cylinder is tripled, its curved surface area remains the same.

Let's assume the original radius of the cylinder is r and the original height is h. The original curved surface area of the cylinder would be 2πrh.

When the radius is tripled, the new radius becomes 3r. The curved surface area remains the same, so:

2πrh = 2π(3r)h

Simplifying this equation gives:

r = h/3

This means that the original height of the cylinder is three times the original radius. When the radius is tripled, the new radius becomes 3r and the new height becomes 3h.

Therefore, the new height is one third of the original height.

Explanation:

Each edge of the large cube is 2 m = 200 cm. Each edge of the small cube is 20 cm.

Therefore, the number of small cubes that can be accommodated along one edge of the large cube is given by:

200 cm / 20 cm = 10

Since there are three edges to consider, the total number of small cubes that can be accommodated in the large cube is:

10 x 10 x 10 = 1000

Therefore, the answer is (c) 1000.

Explanation:

The volume of a cylinder is given by V = πr^2h, where r is the radius and h is the height.

If the radius r is equal to the height h, then we can write V = πr 2h = πr3.

Therefore, the answer is (c) πr3.

Explanation:

The volume of a cube with edge length a is given by V = a3

Here, the edge length is 3x.

So, the volume of the cube is V = (3x)3 = 27x3.

Therefore, the answer is (a) 27x3.

Explanation:

From the given figure, it is clear that a quadrilateral ABCD is a parallelogram. Here, the diagonal AC divides the parallelogram into two equal triangles.

Hence, the area of a triangle ABC = (1/2) bh

Here, b = 12 and h = 3

= (1/2) (12)(3)

= 18

The correct answer is option (B) 36 cm2

Explanation:

To find the area of the rhombus, we can use the formula:

Area of rhombus = 1/2 x d1 x d2

where d1 and d2 are the lengths of the diagonals.

Since AC = 6 cm and BE = 4 cm, we can use the fact that the diagonals of a rhombus bisect each other at right angles to find the lengths of the diagonals. Let the intersection of the diagonals be O. Then, we have:

AO = CO = AC/2 = 6/2 = 3 cm

 BO = DO = BE/2 = 4/2 = 2 cm

Using the Pythagorean theorem, we can find the length of the diagonal BD:

BD2 = BO2 + DO2 BD2 = 22 + 32 BD2 = 13 BD = sqrt(13)

Now, we can find the area of the rhombus:

Area of rhombus = 1/2 x d1 x d2 

= 1/2 x AC x BD 

= 1/2 x 6 cm x sqrt(13) cm 

= 3 sqrt(13) cm2 ≈ 11.42 cm2

Therefore, the area of the rhombus is approximately 11.42 cm2, which corresponds to option (c).

Explanation:

The area of a parallelogram is given by the formula:

Area = base x height

where the base is the length of any one of its sides and the height is the length of the perpendicular from that side to the opposite side.

Let the base of the parallelogram be b and the height be h. Then, we have:

Area = b x h = 60 cm2 h = 5 cm

Substituting h = 5 cm and Area = 60 cm2, we get:

b x 5 = 60

Solving for b, we get:

b = 12 cm

Therefore, the length of the corresponding side of the parallelogram is 12 cm, which corresponds to option (a).

Explanation:

Given:

The perimeter of a trapezium = 52 cm

The sum of its parallel sides = 52 – (10+10) = 32 cm

We know that, the area of a trapezium = (1/2) (a+b) h

A = (1/2) (32) (8)

A = 128 cm2

Therefore the correct option is c.

Explanation:

Let h1 and h2 be the lengths of the perpendiculars on BD from vertices A and C, respectively. Then, we have:

Area of quadrilateral ABCD = (1/2) x BD x (h1 + h2)

Substituting the given values, we get:

20 cm2 = (1/2) x BD x (1 cm + 1.5 cm) BD = 20 cm2 / (0.5 cm x 2.5 cm) BD = 16 cm

Therefore, the length of BD is 16 cm, which corresponds to option (c).

Explanation:

The volume of the metal sheet is given by its length times breadth times thickness:

Volume of metal sheet = 27 cm x 8 cm x 1 cm = 216 cm3

When this metal sheet is melted and used to make a cube, the volume of the cube will be the same as the volume of the metal sheet. Let x be the length of a side of the cube. Then, the volume of the cube is:

Volume of cube = x3

Equating the volumes of the metal sheet and the cube, we get:

X3 = 216 cm3

Taking the cube root of both sides, we get:

x = 6 cm

Therefore, the side of the cube is 6 cm, which corresponds to option (a).

Explanation:

The volumes of the three cubes are:

Volume of first cube = (6 cm)3 = 216 cm3

 Volume of second cube = (8 cm)3 = 512 cm3 

Volume of third cube = (10 cm)3 = 1000 cm3

The total volume of the three cubes is:

Total volume = 216 cm3 + 512 cm3 + 1000 cm3 = 1728 cm3

When these cubes are melted and used to make a single cube, the volume of the new cube will be the same as the total volume of the three cubes. Let x be the length of a side of the new cube. Then, the volume of the new cube is:

Volume of new cube = x3

Equating the volumes of the new cube and the three cubes, we get:

X3 = 1728 cm3

Taking the cube root of both sides, we get:

x = 12 cm

Therefore, the edge of the new cube is 12 cm, which corresponds to option (a).

Explanation:

The volume of the wooden box can be calculated by subtracting the volume of the inner box from the volume of the outer box. The volume of the inner box is:

Volume of inner box = 115 cm x 75 cm x 35 cm = 302,625 cm3

The dimensions of the outer box can be obtained by adding twice the thickness of the wood to each dimension of the inner box. Thus, the outer box has dimensions:

(115 + 22.5) cm x (75 + 22.5) cm x (35 + 2x2.5) cm = 120 cm x 80 cm x 40 cm

Therefore, the volume of the outer box is:

Volume of outer box = 120 cm x 80 cm x 40 cm = 384,000 cm3

The volume of the wood used in making the box is the difference between the volumes of the outer and inner boxes:

Volume of wood = Volume of outer box - Volume of inner box 

= 384,000 cm3 - 302,625 cm3

 = 81,375 cm3

Therefore, the volume of the wood used in making the box is 81,375 cm3, which corresponds to option (c).

Explanation:

The ratio of the radii of the two cylinders is 1:2 and the ratio of their heights is 2:3. Let the radii of the cylinders be r and 2r and their heights be 2h and 3h, respectively.

Then, the volumes of the two cylinders are:

Volume of cylinder 1 = πr2(2h) = 2πr 2 h 

Volume of cylinder 2 = π(2r)2(3h) = 12πr 2h

Therefore, the ratio of their volumes is:

Volume of cylinder 1 : Volume of cylinder 2 

= 2πr2h : 12πr2h = 1:6

Hence, the ratio of the volumes of the two cylinders is 1:6, which corresponds to option (a).

Explanation:

Let the edge of the first cube be a, then its volume is a3. Since the ratio of the volumes of the two cubes is 1:64, the volume of the second cube is 64a3. The edge of the second cube can be found by taking the cube root of 64a3, which is 4a.

The area of a face of the first cube is a2, and the area of a face of the second cube is (4a)2 = 16a2. Therefore, the ratio of the area of a face of the first cube to that of the second cube is:

A2 : 16a2 = 1:16

Hence, the ratio of the area of a face of the first cube to that of the second cube is 1:16, which corresponds to option (c).

Explanation:

Let the length, width, and height of the rectangular solid be a, b, and c, respectively. Then, we have:

2ab + 2bc + 2ca = 16 + 16 + 32 + 32 + 72 + 72

2ab + 2bc + 2ca = 240

Dividing both sides by 2, we get:

ab + bc + ca = 120

The volume of the rectangular solid is given by:

V = abc

We can also express the surface areas in terms of the dimensions:

2ab + 2bc + 2ca = 2a(b + c) + 2bc = 2b(a + c) + 2ac = 2c(a + b) + 2ab

Equating the expressions for 2ab + 2bc + 2ca, we get:

2a(b + c) + 2bc = 2b(a + c) + 2ac = 2c(a + b) + 2ab

Simplifying, we get:

a(b + c) = b(a + c) = c(a + b)

This implies that a:b:c = 1:2:3 (since the ratios a:b, b:c, and a:c are all equal to 1:2).

Substituting a:b:c = 1:2:3 into ab + bc + ca = 120, we get:

1(2 + 3) + 2(1 + 3) + 3(1 + 2) = 2 + 4 + 6 + 6 + 8 + 9 = 35

So, abc = V = 1 × 2 × 3 = 6.

Therefore, the volume of the rectangular solid is 6 cubic centimetres, which corresponds to option (a).

Explanation:

 (i)If the cylinder has radius r and height h, then the volume will be πr2h

(ii) If the cylinder has radius 2r and height (1/2)h, then the volume will be 2πr2h

(ii) The volume of the cuboidal container with dimensions is r2 h

Then, the arrangement of the containers in the increasing order of their volumes is C, A, B

Thus, the correct option is c.

Explanation:

The total outer surface area of the hat is given by the sum of the lateral surface area and the area of the base. The lateral surface area is the curved surface area of a cone with radius R and height h, which is given by πRh. The area of the base is πR2. Therefore, the total outer surface area of the hat is 2πRh + πR2, which is option (c).

Explanation:

The number of such cubes that will have exactly two of their faces painted is 24.

Explanation:

The given cube has a side of 5 cm, so its volume is 5^3 = 125 cubic cm.

When this cube is cut into 1 cm cubes, it will be divided into 125 small cubes.

The volume of each small cube is 1^3 = 1 cubic cm.

The total volume of all the small cubes is 125 cubic cm.

So, the new volume after cutting is also 125 cubic cm.

The percentage increase in volume is given by:

increase in volume/original volume x 100%

= (new volume - original volume)/original volume x 100%

= (125 - 125)/125 x 100%

= 0%

Therefore, the percentage increase in volume after cutting is 0%.

Explanation:

Let the side of each cube be 'a'. Then, the length, width, and height of the cuboid formed by joining two cubes of side face to face are as follows:

Length = 2a (since the cuboid is formed by joining two cubes face to face, its length will be twice the length of one cube)

Width = a (since the width of the cuboid will be the same as the width of one cube)

Height = a (since the height of the cuboid will be the same as the height of one cube)

The surface area of the cuboid is given by the formula:

Surface area = 2(lw + lh + wh)

Substituting the values of length, width, and height, we get:

Surface area = 2(2a x a + 2a x a + a x a)

= 2(4a^2 + 2a^2)

= 2(6a^2)

= 12a^2

However, we need to remember that two faces of the cuboid are covered by the cubes, and are not a part of the surface area of the cuboid. Therefore, we need to subtract the area of these two faces from the total surface area calculated above.

The area of each face of the cube is a^2, and there are two such faces. Therefore, the total area of the two faces is 2a^2.

Subtracting this from the total surface area calculated above, we get:

Surface area = 12a^2 - 2a^2

= 10a^2

Therefore, the surface area of the cuboid formed by joining two cubes of side face to face is 10a^2.

Explanation:

Let us assume that the diagonals of the rhombus are AC and BD, and let their lengths be d1 and d2, respectively.

The area of the rhombus can be expressed as (d1xd2)/2.

When the diagonals are doubled, their new lengths become 2d1 and 2d2, respectively.

The area of the new rhombus can be expressed as (2d1 x 2d2)/2 = 2d1d2.

So, the ratio of the area of the new rhombus to the original rhombus is:

(2d1d2) / (d1d2/2) = 4

Therefore, the area of the rhombus becomes 4 times its original area when the diagonals are doubled.

Explanation:

Each side of a cube = h

Thus, the volume of cube = h3

Surface area of a cube = 6 (h2)

So, the answer is h3 and 6h2

Explanation:

The volume of a cylinder is given by the formula V = πr2 h, where r is the radius of the base of the cylinder, h is the height of the cylinder, and π is the constant pi.

If the radius of the cylinder becomes half of the original radius, then the new radius of the cylinder is r/2. The height of the cylinder remains the same.

The new volume of the cylinder can be calculated by substituting r/2 for r in the formula for the volume of the cylinder:

New volume = π(r/2)2 h = (πr2h)/4

Therefore, the new volume of the cylinder is one-fourth of the original volume. In other words, if the radius of the cylinder becomes half of the original radius, the volume of the cylinder becomes one-fourth of the original volume.

So the answer is 1/4.

Explanation:

If the height of a cylinder is reduced to half of the original height, the curved surface area is reduced by 50%.

The curved surface area of a cylinder is given by the formula A = 2πrh, where r is the radius of the base of the cylinder, h is the height of the cylinder, and π is the constant pi.

If the height of the cylinder is reduced to half of the original height, the new height is h/2. The radius of the base of the cylinder remains the same.

The new curved surface area of the cylinder can be calculated by substituting h/2 for h in the formula for the curved surface area of the cylinder:

New curved surface area = 2πr(h/2) = πrh

Therefore, the new curved surface area of the cylinder is half of the original curved surface area of the cylinder.

The percentage reduction in the curved surface area is given by:

Reduction in curved surface area = (Original curved surface area - New curved surface area) / Original curved surface area × 100%

= (2πrh - πrh) / 2πrh × 100%

= πrh / 2πrh × 100%

= 50%

Therefore, if the height of a cylinder is reduced to half of the original height, the curved surface area of the cylinder is reduced by 50%.

Explanation:

If a cylinder exactly fits in a cube, then the height of the cylinder is equal to the side length of the cube. Let's assume the side length of the cube is a.

The volume of the cylinder is given by the formula V = πr2 h, where r is the radius of the base of the cylinder, and h is the height of the cylinder.

Since the cylinder fits exactly in the cube, the diameter of the base of the cylinder is equal to the side length of the cube, which is a. Therefore, the radius of the base of the cylinder is a/2.

The height of the cylinder is also equal to the side length of the cube, which is a.

Substituting the values of r and h in the formula for the volume of the cylinder, we get:

V = πr2 h

V = π(a/2)2(a)

V = (πa3)/4

Therefore, the volume of the cylinder which exactly fits in a cube of side a is (πa3)/4.

Explanation:

When the cylinder exactly fits in the cube of side “b”, the height equals to the edges of the cube and the radius equal to half the edges of a cube.

It means that,

h = b, and r = b/2

Then the CSA of a cylinder be = 2πrh

= 2π (b/2)(b)

= πb2

Explanation:

Let ABCD be a quadrilateral with diagonal d and heights h1 and h2 falling on the diagonal.

The area of the quadrilateral is given by the formula A = (1/2)d(h1+h2).

If the diagonal d is doubled and the heights h1 and h2 are halved, then the new area of the quadrilateral, denoted by A', is given by:

A' = (1/2)2d(h1/2 + h2/2)

A' = d x (h1/2 + h2/2)

A' = (1/2)d(h1 + h2)

Comparing this with the formula for the area of the original quadrilateral, we see that A' is equal to half the area of the original quadrilateral. Therefore, the area of the quadrilateral is halved when the diagonal is doubled and the heights are halved.

Explanation:

Let the original length and breadth of the rectangle be l and b respectively. The original perimeter of the rectangle is given by:

P = 2(l + b)

If both the length and breadth are doubled, the new length and breadth become 2l and 2b respectively. The new perimeter of the rectangle, denoted by P', is given by:

P' = 2(2l + 2b)

P' = 4(l + b)

Comparing this with the formula for the original perimeter, we can see that the new perimeter is twice the original perimeter. Therefore, the perimeter of a rectangle becomes twice its original perimeter if both the length and breadth are doubled.

Explanation:

3, equal areas

Explanation:

All six faces of a cuboid are rectangles in shape and of different area.

The areas of the six faces of a cuboid can be calculated as follows:

• Top and bottom faces: length x width

• Front and back faces: height x width

• Left and right faces: length x height

Therefore, the six faces have different areas, except in the special case where the cuboid is a cube (i.e., all sides have equal length), in which case all six faces have the same area.

Explanation:

Opposite faces of a cuboid are equal in area.

Let's say the length, width, and height of the cuboid are l, w, and h, respectively. Then, the areas of the opposite faces are:

• Top and bottom faces: lw each

• Front and back faces: lh each

• Left and right faces: wh each

Therefore, the areas of the opposite faces are equal.

Explanation:

2πrh

where π is the mathematical constant pi (approximately equal to 3.14159).

The formula for the curved surface area of a cylinder can be derived by "unrolling" the curved surface of the cylinder into a rectangle. The length of the rectangle is equal to the circumference of the base of the cylinder, which is 2πr. The height of the rectangle is equal to the height of the cylinder, which is h. Therefore, the area of the rectangle, which is the curved surface area of the cylinder, is 2πrh.

Explanation:

The total surface area of a cylinder of radius r and height h is given by:

2πr(r+h)

where π is the mathematical constant pi (approximately equal to 3.14159).

The formula for the total surface area of a cylinder can be derived by adding the areas of the top and bottom circular faces of the cylinder (which are each equal to πr^2) to the curved surface area of the cylinder (which is 2πrh). Therefore, the total surface area is:

2πr2 + 2πrh

Simplifying this expression gives:

2πr(r+h).

Explanation:

πh2r cubic units

Explanation:

Diagonals

Explanation:

The cylinder A will be formed by rolling the sheet along its length, and its height will be equal to the breadth of the sheet, which is 10 cm. The radius of cylinder A will be equal to half the length of the sheet, which is 10 cm. Therefore, the volume of cylinder A will be:

Volume of A = π(10 cm)2(10 cm) = 1000π cm3

Similarly, the cylinder B will be formed by rolling the sheet along its breadth, and its height will be equal to the length of the sheet, which is 20 cm. The radius of cylinder B will be equal to half the breadth of the sheet, which is 5 cm. Therefore, the volume of cylinder B will be:

Volume of B = π(5 cm)2(20 cm) = 500π cm3

So the volume of cylinder A is two times the volume of cylinder B:

Volume of B = 1000π cm3 = 2(500π cm3) = 2Volume B

Explanation:

 Same

The curved surface area of cylinder A:

CSA of A = 2πrh = 2π(10 cm)(10 cm) = 200π cm2

where r = radius of cylinder A

and h = height of cylinder A

Similarly, the curved surface area of cylinder B will be:

CSA of B = 2πrh = 2π(5 cm)(20 cm) = 200π cm2

where r = radius of cylinder B

and h = height of cylinder B.

Therefore, the curved surface area of cylinder A is equal to the curved surface area of cylinder B, which is 200π cm2.

Explanation:

Volume of a solid is the measurement of the space occupied by it.

Explanation:

Lateral