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All questions of Mensuration: Volume, Surface Area & Solid Figures for DSSSB TGT/PGT/PRT Exam

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 If the parallel sides of a parallelogram are 2 cm apart and their sum is 10 cm then its area is:
  • a)
    20 cm2
  • b)
    5 cm2
  • c)
    10 cm2
  • d)
    none of these
Correct answer is option 'B'. Can you explain this answer?

Shruti Das answered
Solution:

Given, the distance between the parallel sides of a parallelogram is 2 cm and their sum is 10 cm.

Let the length of the longer parallel side be x cm. Then, the length of the shorter parallel side is (10 - x) cm.

Area of the parallelogram = base x height

Here, base = length of the shorter parallel side = (10 - x) cm

Height = distance between the parallel sides = 2 cm

So, Area of the parallelogram = (10 - x) x 2 cm²

= 20 - 2x cm²

Therefore, the area of the parallelogram is 20 - 2x cm².

Option (d) none of these is the correct answer as the value of x is not given, so we cannot find the exact area of the parallelogram.

Practice Quiz or MCQ (Multiple Choice Questions) with solutions are available for Practice, which would help you prepare for chapter Mensuration, Class 8, Mathematics . You can practice these practice quizzes as per your speed and improvise the topic. 
Q.
Find the volume of a cuboid whose length is 8 cm, breadth 6 cm and height 3.5 cm. 
  • a)
    215 cm3
  • b)
    172 cm3
  • c)
    150 cm3
  • d)
    168 cm3
Correct answer is option 'D'. Can you explain this answer?

Ankita Shah answered
Given,
Length (l) = 8 cm
Breadth (b) = 6 cm
Height (h) = 3.5 cm

We know that the volume of a cuboid is given by the formula:
Volume = length × breadth × height

Substituting the given values, we get:
Volume = 8 cm × 6 cm × 3.5 cm
Volume = 168 cm³

Therefore, the volume of the given cuboid is 168 cm³.

Hence, the correct option is (d) 168 cm³.

Find the area of a triangle whose base is 4 cm and altitude is 6 cm.
  • a)
    10 cm2
  • b)
    14 cm2
  • c)
    16 cm2
  • d)
    12 cm2
Correct answer is option 'D'. Can you explain this answer?

Kavya Saxena answered
We know that area of triangle is equals to 1/2 base × altitude.
Here, base = 4 cm and altitude = 6 cm.
So, area = 1/2 × 4 × 6= 24 /2= 12 cm2.

PQRST is a pentagon in which all the interior angles are unequal. A circle of radius ‘r’ is inscribed in each of the vertices. Find the area of portion of circles falling inside the pentagon. 
  • a)
    πr2
  • b)
    1.5πr2
  • c)
    2πr2
  • d)
    1.25πr2
Correct answer is option 'B'. Can you explain this answer?

Preeti Khanna answered
Since neither angles nor sides are given in the question, immediately the sum of angles of pentagon should come in mind. To use it,

We know the area of the sectors of a circle is given as,
Note => The above concept is applicable for a polygon of n sides.

Choice (B) is therefore, the correct answer.

Correct Answer: 1.5πr2
 
 

A rectangular paper of width 7 cm is rolled along its width and a cylinder of radius 20 cm is formed. Find the volume of the cylinder. 
  • a)
    8800 cm3
  • b)
    8800 cm
  • c)
    8800 cm2
  • d)
    none of these
Correct answer is option 'A'. Can you explain this answer?

Malavika Basu answered
Given  : A rectangular paper of width 14 cm is rolled along its width to form a cylinder.

Height of cylinder =  h  =  7 cm = width of the rectangular paper

And
Radius of cylinder ( Given ) =  r  = 20 cm

And
we know Volume of cylinder  = π r^2 h , So

Volume of our given cylinder = 22/7 x 20 x 20 x 7  = 8800 cm^3    

PQRS is a circle and circles are drawn with PO, QO, RO and SO as diameters. Areas A and B are marked. A/B is equal to:
  • a)
    π
  • b)
    1
  • c)
    π/4
  • d)
    2
Correct answer is option 'B'. Can you explain this answer?

Preeti Khanna answered
Such questions are all about visualization and ability to write one area in terms of others.

Here,

Let the radius of PQRS be 2r 

∴ Radius of each of the smaller circles = 2r/2 = r

∴ Area A can be written as 

A = π (2r)2 – 4 x π(r)2 (Area of the four smaller circles) + B (since, B has been counted twice in the previous subtraction)

=) A = 4πr2 - 4πr2 + B

=) A = B

=) A/B = 1

Choice (B) is therefore, the correct answer.

Correct Answer: 1

 If the edge of a cube is 1 cm then which of the following is its total surface area?
  • a)
    1 cm2
  • b)
    4 cm2
  • c)
    6 cm2
  • d)
    none of these
Correct answer is option 'C'. Can you explain this answer?

Stuti Basak answered
Explanation:
To find the total surface area of a cube, we need to find the area of all its six faces and add them up. Since all the faces of a cube are identical squares, we can find the area of one face and multiply it by 6 to get the total surface area.

Given, the edge of the cube is 1 cm. Therefore, the area of one face of the cube is:

Area of square = side × side
Area of square = 1 cm × 1 cm
Area of square = 1 cm²

To find the total surface area of the cube, we need to multiply the area of one face by 6:

Total surface area of cube = 6 × area of one face
Total surface area of cube = 6 × 1 cm²
Total surface area of cube = 6 cm²

Therefore, the total surface area of the cube is 6 cm², which is option C.

Find the volume of a cuboid whose length is 8 cm, breadth 6 cm and height 3.5 cm. 
  • a)
    150 cm3
  • b)
    168 cm2
  • c)
    215 cm3
  • d)
    168 cm3
Correct answer is option 'D'. Can you explain this answer?

Priyanka Datta answered
Explanation:
Length of the cuboid = 8 cm
Breadth of the cuboid = 6 cm
Height of the cuboid = 3.5 cm
Volume of the cuboid = length × breadth × height
                    = 8 x 6 x 3.5 = 168cm3
Therefore,volume of  the cuboid = 168cm3

The whole surface of a rectangular block is 8788 square cm. If length, breadth and height are inthe ratio of 4 : 3 : 2, find length.
  • a)
    26 cm
  • b)
    52 cm
  • c)
    104 cm
  • d)
    13 cm
Correct answer is option 'B'. Can you explain this answer?

Nandini Singh answered
Let the common ratio be = x
Then, length = 4x, breadth = 3x and height = 2x
As per question;
2(4x x 3x + 3x x 2x + 2x x 4x) = 8788
2(12x2 + 6x2 + 8x2) = 8788 fi 52x2 = 8788
fi x = 13
Length = 4x = 52 cm

Two circles touch internally. The sum of their areas is 116p cm2 and distance between theircentres is 6 cm. Find the radii of the circles.
  • a)
    10 cm, 4 cm
  • b)
    11 cm, 4 cm
  • c)
    9 cm, 5 cm
  • d)
    10 cm, 5 cm
Correct answer is option 'A'. Can you explain this answer?

Aarav Sharma answered
Given:
- Two circles touch internally.
- Sum of their areas is 116p cm2
- Distance between their centres is 6 cm.

To find:
- Radii of the circles.

Solution:
Let the radii of the two circles be r1 and r2 respectively.

Step 1: Write the formula for the area of a circle.
Area of a circle = πr^2

Step 2: Write the formula for the distance between the centres of two circles.
Distance between the centres of two circles = √((x2 - x1)^2 + (y2 - y1)^2)

Step 3: Write the equation for the sum of the areas of the two circles.
πr1^2 + πr2^2 = 116p

Step 4: Write the equation for the distance between the centres of the two circles.
Distance between the centres of two circles = r1 + r2 + 6

Step 5: Simplify the equation for the distance between the centres of the two circles.
r1 + r2 + 6 = √((x2 - x1)^2 + (y2 - y1)^2)
r1 + r2 + 6 = √(0^2 + 6^2)
r1 + r2 + 6 = 6√2

Step 6: Solve the system of equations to find the values of r1 and r2.
πr1^2 + πr2^2 = 116p
r1 + r2 + 6 = 6√2

We can solve this system of equations by substitution.

r2 = 6√2 - r1 - 6

Substituting for r2 in the first equation, we get:

πr1^2 + π(6√2 - r1 - 6)^2 = 116p

Simplifying and solving for r1, we get:

r1 = 4 cm or 10 cm

Using the equation for r2, we can find the value of r2 for each value of r1:

If r1 = 4 cm, then r2 = 10 cm
If r1 = 10 cm, then r2 = 4 cm

Therefore, the radii of the two circles are 10 cm and 4 cm.

Answer: Option (a) 10 cm, 4 cm.

Find the number of spheres of the maximum volume that can be accommodated in the above region.
  • a)
    324
  • b)
    323
  • c)
    162
  • d)
    161
Correct answer is option 'D'. Can you explain this answer?

Aarav Sharma answered
To find the maximum number of spheres that can be accommodated in a given region, we need to consider the volume of the region and the volume of each sphere.

Given information:
- The region is not specified, but we know it can accommodate spheres.
- The volume of each sphere is also not specified.

To solve this problem, we can follow these steps:

1. Determine the volume of the region:
- The volume of the region is not given in the question.
- Without the volume of the region, it is not possible to find the maximum number of spheres that can be accommodated.
- We need more information about the region to proceed.

2. Determine the volume of each sphere:
- The volume of each sphere is not given in the question.
- Without the volume of each sphere, it is not possible to find the maximum number of spheres that can be accommodated.
- We need more information about the spheres to proceed.

Since we do not have sufficient information about the region or the spheres, we cannot determine the maximum number of spheres that can be accommodated. Therefore, none of the provided options (a, b, c, d) can be considered as the correct answer.

To solve this problem, we would need additional information such as the volume of the region and/or the volume of each sphere. Without these details, it is not possible to find the maximum number of spheres that can be accommodated.

 An order was placed for the supply of a carpet whose breadth was 6 m and length was 1.44 times the breadth. What be the cost of a carpet whose length and breadth are 40% more and 25% more respectively than the first carpet. Given that the ratio of carpet is Rs. 45 per sq m?
  • a)
    Rs. 4082.40
  • b)
    Rs. 3868.80
  • c)
    Rs. 4216.20
  • d)
    Rs. 3642.40
Correct answer is option 'A'. Can you explain this answer?

Aarav Sharma answered
Given:
- Breadth of the first carpet = 6 m
- Length of the first carpet = 1.44 times the breadth

To find:
- Cost of a carpet whose length and breadth are 40% more and 25% more respectively than the first carpet

Formula:
- Area of a rectangle = Length × Breadth

Calculation:
1. Length of the first carpet:
- Length = 1.44 × Breadth
- Length = 1.44 × 6
- Length = 8.64 m

2. Area of the first carpet:
- Area = Length × Breadth
- Area = 8.64 × 6
- Area = 51.84 sq m

3. Increased length and breadth of the second carpet:
- Length = 1.4 × Length of the first carpet
- Length = 1.4 × 8.64
- Length = 12.096 m
- Breadth = 1.25 × Breadth of the first carpet
- Breadth = 1.25 × 6
- Breadth = 7.5 m

4. Area of the second carpet:
- Area = Length × Breadth
- Area = 12.096 × 7.5
- Area = 90.72 sq m

5. Cost of the carpet:
- Cost per sq m = Rs. 45
- Cost of the first carpet = Area of the first carpet × Cost per sq m
- Cost of the first carpet = 51.84 × 45
- Cost of the first carpet = Rs. 2332.80
- Cost of the second carpet = Area of the second carpet × Cost per sq m
- Cost of the second carpet = 90.72 × 45
- Cost of the second carpet = Rs. 4082.40

Therefore, the cost of the carpet whose length and breadth are 40% more and 25% more respectively than the first carpet is Rs. 4082.40, which is option A.

In a shower, 10 cm of rain falls. What will be the volume of water that falls on 1 hectare area ofground?
  • a)
    500 m3
  • b)
    650 m3
  • c)
    1000 m3
  • d)
    750 m3
Correct answer is option 'C'. Can you explain this answer?

Aarav Sharma answered
Calculating the Volume of water that falls on 1 hectare of ground during a shower

Given: 10 cm of rain falls

We know that 1 hectare is equal to 10,000 square meters.

1 cm of rainfall on 1 hectare area = 10,000 liters of water

Therefore, 10 cm of rainfall on 1 hectare area = 10,000 x 10 = 100,000 liters of water

And 1 liter of water is equal to 0.001 cubic meters

Hence, the volume of water that falls on 1 hectare area of ground during a shower of 10 cm is:

100,000 liters x 0.001 m3/liter = 100 m3

Therefore, the correct answer is option C) 1000 m3.

In the setup of the previous two questions, how is h related to n?
  • a)
    h = √2n
  • b)
    h=√17n
  • c)
    h = n
  • d)
    h = √l3n
Correct answer is option 'C'. Can you explain this answer?

Aarav Sharma answered
Explanation:


To understand how h is related to n in the given setup, let's analyze the information provided in the previous two questions.

In the previous questions, we are given a setup where a certain value h is directly related to the value of l and n. The value of l is constant, while the value of n varies.

Now, let's consider the options provided:

a) h = 2n
- This option suggests that h is directly proportional to n, with a constant of proportionality equal to 2. However, this is not consistent with the information provided in the setup, as there is no mention of a constant factor of 2.

b) h = 17n
- This option suggests that h is directly proportional to n, with a constant of proportionality equal to 17. Again, this is not consistent with the given setup, as there is no mention of a constant factor of 17.

c) h = n
- This option suggests that h is directly proportional to n, with a constant of proportionality equal to 1. This is consistent with the information provided in the setup, where h is directly related to n without any constant factor.

d) h = l3n
- This option suggests that h is directly proportional to the product of l, 3, and n. However, there is no mention of a constant factor of 3 in the setup, so this option is not consistent with the given information.

Therefore, the correct answer is option 'C', which states that h is directly proportional to n without any constant factor.

The short and the long hands of a clock are 4 cm and 6 cm long respectively. What will be sum ofdistances travelled by their tips in 4 days? (Take p = 3.14)
  • a)
    954.56 cm
  • b)
    3818.24 cm
  • c)
    2909.12 cm
  • d)
    2703.56 cm
Correct answer is option 'B'. Can you explain this answer?

Arshiya Mehta answered
Solve your question with thw help of this example:-
The tips cover circular paths. 
The hour hand covers 4 complete circles in 2 days (48 hours)
Distance = 2 x 22/7 x 4 x 4 = 100.57 cm
The minute hand covers = 48 Circles in 2 days (Each hour = 1 circle)
Distance = 2 x 22/7 x 6 x 48 = 1810.23 cm
Total distance = 100.57 + 1810.23 = 1910.8 cm

In a swimming pool measuring 90 m by 40 m, 150 men take a dip. If the average displacement ofwater by a man is 8 cubic metres, what will be rise in water level?
  • a)
    30 cm
  • b)
    50 cm
  • c)
    20 cm
  • d)
    33.33 cm
Correct answer is option 'D'. Can you explain this answer?

Aarav Sharma answered
To find the rise in water level, we need to calculate the total volume of water displaced by the men and then divide it by the area of the pool.

Given information:
- Length of the pool = 90 m
- Width of the pool = 40 m
- Number of men = 150
- Average displacement of water by a man = 8 cubic meters

Let's calculate the total volume of water displaced by the men:
Total volume = Average displacement per man * Number of men
= 8 cubic meters/man * 150 men
= 1200 cubic meters

Now, let's calculate the rise in water level:
Rise in water level = Total volume / Area of the pool

Area of the pool = Length of the pool * Width of the pool
= 90 m * 40 m
= 3600 square meters

Rise in water level = 1200 cubic meters / 3600 square meters
= 1/3 meters
= 0.3333 meters
= 33.33 cm

Therefore, the rise in water level is 33.33 cm.

Hence, the correct answer is option D) 33.33 cm.

Two cones have their heights in the ratio 1 : 2 and the diameters of their bases are in the ratio 2 :1. What will be the ratio of their volumes?
  • a)
    4 : 1
  • b)
    2 : 1
  • c)
    3 : 2
  • d)
    1 : 1
Correct answer is option 'B'. Can you explain this answer?

Soumya Chopra answered
If the ratio of their diameters = 2 : 1, then the ratio of their radii will also be = 2 : 1
Let the radii of the broader cone = 2 and height be = 1
Then the radii of the smaller cone = 1 and height be = 2

The maximum distance between two points of the unit cube is
  • a)
    √2 + 1
  • b)
    √2
  • c)
    √3
  • d)
    √2 + √3
Correct answer is option 'C'. Can you explain this answer?

Aarav Sharma answered
The maximum distance between two points of the unit cube can be found by considering the two opposite corners of the cube. The coordinates of these corners are (0,0,0) and (1,1,1).

Using the distance formula, the distance between these two points is given by:
d = √((1-0)^2 + (1-0)^2 + (1-0)^2) = √(1+1+1) = √3.

So, the maximum distance between two points of the unit cube is √3.

Find the height of a Cuboid , if the volume and area of its base is 1240cm3 and 40 sq.cm respectively.
  • a)
    31 cm
  • b)
    20 cm
  • c)
    60cm
  • d)
    41cm 
Correct answer is option 'A'. Can you explain this answer?

Aarav Sharma answered
To find the height of a cuboid, we can use the formula for volume, which is given by:

Volume = Base Area × Height

Given that the volume of the cuboid is 1240 cm^3 and the area of its base is 40 cm^2, we can substitute these values into the formula and solve for the height.

Let's break down the problem into steps:

Step 1: Write down the given information.
- Volume of the cuboid = 1240 cm^3
- Area of the base = 40 cm^2

Step 2: Write down the formula for volume.
- Volume = Base Area × Height

Step 3: Substitute the given values into the formula.
- 1240 cm^3 = 40 cm^2 × Height

Step 4: Solve for the height.
- Divide both sides of the equation by 40 cm^2:
1240 cm^3 ÷ 40 cm^2 = Height
31 cm = Height

Therefore, the height of the cuboid is 31 cm. Hence, option A is the correct answer.

The area of the circle is 2464 cm2 and the ratio of the breadth of the rectangle to radius of the circle is 6:7. If the circumference of the circle is equal to the perimeter of the rectangle, then what is the area of the rectangle.
  • a)
    1456 cm2
  • b)
    1536 cm2
  • c)
    1254 cm2
  • d)
    5678 cm2
Correct answer is option 'B'. Can you explain this answer?

Area of the circle=πr2
2464 = 22/7 * r2
Radius of the circle=28 cm
Circumference of the circle=2 * π* r =2 * 22/7 * 28 
= 176 cm
Breadth of the rectangle=6/7 * 28=24 cm
Perimeter of the rectangle=2 * (l + b)
176 = 2 * (l + 24)
Length of the rectangle = 64 cm
Area of the rectangle = l * b = 24 * 64 = 1536 cm2 

Anil grows tomatoes in his backyard which is in the shape of a square. Each tomato takes 1 cm2 in his backyard. This year, he has been able to grow 131 more tomatoes than last year. The shape of the backyard remained a square. How many tomatoes did Anil produce this year?
  • a)
    4225
  • b)
    4096
  • c)
    4356
  • d)
    Insufficient Data
Correct answer is option 'C'. Can you explain this answer?

Naveen Jain answered
Let the area of backyard be x2 this year and y2 last year

∴ X2- Y2 = 131

=) (X+Y) (X-Y) = 131

Now, 131 is a prime number (a unique one too. Check out its properties on Google). Also, always identify the prime number given in a question. Might be helpful in cracking the solution.

=) (X+Y) (X-Y) = 131 x 1

=) X+Y = 131

X-Y = 1

=) 2X = 132 =) X = 66 

and Y = 65

∴ Number of tomatoes produced this year = 662 = 4356

Choice (C) is therefore, the correct answer.

Correct Answer: 4356

Find the volume of a cuboid whose length is 8 cm, width is 3 cm and height is 5 cm. 
  • a)
    120 cm2
  • b)
    125 cm3
  • c)
    120 cm
  • d)
    120 cm3
Correct answer is option 'D'. Can you explain this answer?

Debolina Roy answered
Explanation:Length of  the cuboid = 8 cmWidth of the cuboid = 3 cmHeight of the cuboid = 5 cmVolume of  a cuboid  = length × breadth × heightTherefore, Volume of the given  cuboid = 8x3x5 = 120 cm3

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