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

Sol. perimeter of square=4* sides
=4*4 (sides is 4)
=16.

Finding Volume of a Cuboid

Given: length = 8 cm, width = 3 cm, height = 5 cm

To find: Volume of the cuboid

Formula: Volume of a cuboid = length x width x height

Substituting the given values in the formula, we get:

Volume = 8 cm x 3 cm x 5 cm

Volume = 120 cm3

Therefore, the correct answer is option C, 120 cm3.

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.

1/2(10)×2 =10 cm

Given information:
- Heights of two cones are in the ratio of 1:2
- Diameters of their bases are in the ratio of 2:1

To find:
Ratio of their volumes

Solution:
Let the height of the first cone be h1 and the height of the second cone be h2.
Let the diameter of the base of the first cone be d1 and the diameter of the base of the second cone be d2.
Let the radius of the base of the first cone be r1 and the radius of the base of the second cone be r2.

From the given information, we know that:

h1 : h2 = 1 : 2

d1 : d2 = 2 : 1

We also know that the volume of a cone is given by:

V = (1/3)πr^2h

where r is the radius of the base and h is the height of the cone.

We can use the information about the diameters of the bases to find the ratio of the radii:

r1 = (d1/2) and r2 = (d2/2)

r1 : r2 = (d1/2) : (d2/2) = d1 : d2 = 2 : 1

Now, we can use the information about the heights to find the ratio of the volumes:

V1/V2 = (1/3)πr1^2h1 / (1/3)πr2^2h2

V1/V2 = r1^2h1 / r2^2h2

V1/V2 = [(d1/2)^2h1] / [(d2/2)^2h2]

V1/V2 = (d1^2/4)(h1/h2)(1/d2^2)

Substituting the given ratios:

V1/V2 = (2^2/4)(1/2)(1/1^2)

V1/V2 = 1/2

Therefore, the ratio of the volumes of the two cones is 2:1. Hence, option B is the correct answer.

Step1: solve it like a prisum and find the hight of prisum
step2: to get total height add radius of bottom spear to high of the prisum

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.

Surface area of a solid is the sum of the areas of its faces.

The surface area of a solid is a measure of the total area of all the surfaces or faces of the solid. It can be thought of as the amount of material needed to cover the entire surface of the solid.

Explanation:

A solid object is three-dimensional and has length, width, and height. It is made up of multiple faces, which are two-dimensional shapes that form the boundaries of the solid. The surface area of a solid is the sum of the areas of all these faces.

For example, consider a cube. A cube has six faces, each of which is a square. To find the surface area of the cube, we need to calculate the area of each face and then sum them up.

The formula for finding the area of a square is side length squared. So, if the side length of the cube is 's', then the area of each face is s^2. Since there are six faces, the total surface area of the cube is 6s^2.

Similarly, for other solid objects such as rectangular prisms, cylinders, spheres, etc., the surface area is calculated by finding the area of each face and adding them together.

The surface area is an important concept in geometry and has practical applications in fields such as architecture, engineering, and manufacturing. It helps in determining the amount of material required to construct a solid object or the amount of paint needed to cover its surface.

In summary, the surface area of a solid is the sum of the areas of all its faces. It is a measure of the total area of the surface of the solid and is calculated by finding the area of each face and adding them together.

Step 1: Find the semi-perimeter of the triangle
- The semi-perimeter of a triangle is calculated by adding all three sides of the triangle and dividing by 2.
- In this case, the sides of the triangle are 8 cm, 15 cm, and 17 cm.
- The semi-perimeter, s, is given by s = (8 + 15 + 17) / 2 = 20 cm.

Step 2: Use Heron's Formula to find the area of the triangle
- Heron's formula states that the area of a triangle with sides a, b, and c and semi-perimeter s is given by the formula:
Area = √(s(s - a)(s - b)(s - c))
- Substituting the values, we get Area = √(20(20 - 8)(20 - 15)(20 - 17)) = √(20*12*5*3) = √(3600) = 60 cm².

Step 3: Use the formula for the radius of the inscribed circle
- The radius of the inscribed circle in a triangle with sides a, b, and c and area A is given by the formula:
Radius = A/s
- Substituting the values, we get Radius = 60/20 = 3 cm.
Therefore, the radius of the circle inscribed in the triangle with sides 8 cm, 15 cm, and 17 cm is 3 cm. Hence, the correct answer is option 'C'.

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.

To solve this question, we can start by setting up the equation based on the given information.

Let's assume the common ratio is x. Then the length would be 4x, the breadth would be 3x, and the height would be 2x.

The total surface area of the rectangular block is given as 8788 square cm. The surface area of a rectangular block is the sum of the areas of all its faces.

The formula for the surface area of a rectangular block is given by:
Surface Area = 2(length * breadth + breadth * height + length * height)

Substituting the values, we have:
8788 = 2(4x * 3x + 3x * 2x + 4x * 2x)
8788 = 2(12x^2 + 6x^2 + 8x^2)
8788 = 2(26x^2)
8788 = 52x^2
Dividing both sides by 52, we get:
x^2 = 169
Taking the square root of both sides, we have:
x = 13

Since the length is given as 4x, we can calculate it as:
Length = 4 * 13 = 52 cm

Therefore, the length of the rectangular block is 52 cm, which corresponds to option B.

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.

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.

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.

Solve your question with thw help of this example:-

Understanding the Problem
To find the radius of the bicycle wheel, we need to relate the distance traveled to the number of revolutions made and the circumference of the wheel.
Key Facts:
- Distance traveled = 11 km = 11,000 m (since 1 km = 1000 m)
- Number of revolutions = 5000
Finding the Circumference:
The distance traveled in terms of the circumference can be expressed as:
Distance = Number of Revolutions × Circumference
Using this, we can calculate the circumference of the wheel:
Circumference = Distance / Number of Revolutions
Substituting the known values:
Circumference = 11,000 m / 5000 = 2.2 m
Calculating the Radius:
The circumference \( C \) of a circle is given by the formula:
C = 2πr
Where \( r \) is the radius. We can rearrange this formula to find the radius:
r = C / (2π)
Substituting the circumference we found:
r = 2.2 m / (2π)
Now we convert meters to centimeters (1 m = 100 cm):
r = (2.2 × 100 cm) / (2π) = 220 cm / (2π)
Using the approximate value of π (3.14):
r ≈ 220 cm / 6.28 ≈ 35 cm
Conclusion:
Thus, the radius of the wheel is approximately 35 cm, which confirms that the correct answer is option D.

Volume of the Cube
To find the volume of the cube, we use the formula:
- Volume = side × side × side
For a cube with each side measuring 24 cm:
- Volume = 24 cm × 24 cm × 24 cm = 13,824 cm³
Volume of the Cuboid
The volume of the cuboid is also given to be equal to the volume of the cube, which is 13,824 cm³.
Dimensions of the Cuboid
We know the following dimensions of the cuboid:
- Breadth = 32 cm
- Height = 12 cm
Let the length of the cuboid be denoted as 'l'. The formula for the volume of the cuboid is:
- Volume = length × breadth × height
Substituting the known values:
- 13,824 cm³ = l × 32 cm × 12 cm
Calculating the Length
Now, we need to solve for 'l':
1. First, calculate the product of the breadth and height:
- 32 cm × 12 cm = 384 cm²
2. Now, substitute this into the volume equation:
- 13,824 cm³ = l × 384 cm²
3. To find 'l', divide both sides by 384 cm²:
- l = 13,824 cm³ / 384 cm²
- l = 36 cm
Conclusion
Thus, the length of the cuboid is 36 cm.
The correct answer is option 'A'.