In the volume of combination of solids concept, we will come across the different solids and their respective volumes. As we know, all three-dimensional shapes have surface areas and volumes, based on their dimensions. Now, if we see in our real life, there are many shapes that are formed by combining different shapes. For example, a tent is formed by the combination of a cone and cylinder shape. Similarly, an ice cream cone is a combination of a cone and a hemisphere. Hence, **the volume of the combination of solids will be equal to the sum of the volume of individual solids.**

A solid which is bounded by six rectangular faces is known as a cuboid and if the length, breadth and height of the cuboid are equal, then it is a cube. Both cube and cuboid have 8 vertices, 6 faces and 12 edges. The faces of cuboids are in rectangular shape and the faces of cubes are in square shape. Let us learn here the volumes for such shapes and their combination.

## Volume of a Combination of solids Formula

As discussed above, the volume of a combination of a solid is found by adding the volume of the individual solids. Hence, the formula to calculate the volume of a combination of solids is given by the formula,

**V = V _{1}+V_{2}+…**

Where

“V” is the volume of the combination of solids

“V_{1}” and “V_{2}” are the volume of the individual solids such as solid 1, solid 2, and so on.

## Volume of Solids Formula

Before we solve the problems based on the combination of solids, let us have a look at the volumes of all the three-dimensional solid shapes.

For a cuboid which has length (l), breadth (b) and height (h), the formula for volume and surface area is given by:

- Volume = l×b×h
- Total surface area = 2(lb+bh+lh)
- Length of diagonal of cuboid = √(l
^{2}+b^{2}+h^{2})

For a cube having edge length equal to x, the formula for volume and surface area is given by:

- Volume = x
^{3}(because l = b = h = x) - Total surface area = 6x
^{2} - Length of diagonal of cube = √3x

Similarly, for other shapes such as spheres, cones and cylinders, the formulas for volumes are:

- Volume of Sphere = (4/3)πr
^{3} - Volume of Cone = (1/3) πr
^{2}h - Volume of Cylinder = πr
^{2}h - Volume of Hemi-sphere = (2/3)πr
^{3}

### Volume of a Combination of Solids Examples

**Example 1:**

A cylinder of volume 150 cu.cm is placed with a cone, whose height is 4cm. If the height of cone and cylinder is equal, then find the total volume of the shape formed by the combination of cylinder and cone.

**Solution**:

Given, Volume of cylinder = 150 cu.cm

Height of cylinder = Height of cone = 4cm

By the formula of volume of cylinder we know,

V1 = πr^{2}h

150 = πr^{2}(4)

r^{2}= 150/4π …..(1)

Now, volume of cone is given by:

V2 = 1/3 πr^{2}h

By putting the value of r^{2} from eq. 1 we get;

Therefore, V2 = 1/3 π (150/4π) (4)

On solving the above, we get;

V2 = 50 cu.cm.

Therefore, the total volume of the combined solids, V = V1 + V2 = 150 + 50 = 200 cu.cm.

**Note:** Volume of Cone = 1/3 (Volume of Cylinder)

**Example 2:**

A cube has a volume of 343 cm^{3}. Find its surface area.

**Solution**:

Given, volume of 343 cm^{3}

As we know,

The volume of the cube = side^{3 }Cubic units

Therefore,

side^{3} = 343

side = 7 cm

Hence, surface area of the cube = 6(side)^{2}

= 6 (7)^{2}

= 294 sq.cm.

**Example 3:**

A cuboidal water tank is aluminium steel sheet which is 4.5 m thick. The outer dimensions are 1.5 m × 2.5 m × 3 m. Find the internal dimensions and total surface area of the tank.

**Solution**:

External dimensions of the cube are:

l = 150 cm, b = 250 cm and h = 300 cm

As we know that the sheet is 4.5 m thick, the internal dimensions are:

L = (150-9) = 141 cm

B = (250-9) = 241 cm

H = (300-9) = 291 cm

Total surface area of the tank = 2(lb + bh + lh)

= 2(1.5×2.5+2.5×3+3×1.5)

= 31.5 m^{3}.

**Example 4:**

How many tissue boxes of size 10 cm × 8 cm × 9 cm can be adjusted inside a cupboard box of size 36 cm × 40 cm × 100 cm.

**Solution**:

Volume of the tissue box = 10 x 8 x 9 cm^{3}

= 720 cm^{3}

Volume of the cupboard = 36 x 40 x 100 cm^{3}.

= 144,000 cm^{3}

Therefore, the number of tissue boxes that can be adjusted = 144000/720

=200 boxes

Therefore, we can say that 200 tissue boxes can be adjusted in the cupboard box.

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## Frequently Asked Questions on Volume of a Combination of Solids

### What is meant by a combination of solids?

The combination of solids means the shape formed by combining two or more solids shapes.

### How to calculate the volume of a combination of solids?

The volume of a combination of solids can be calculated by adding the volume of the individual solids.

### Give two examples of the combination of solids.

A tent is formed by a combination of solids such as cones and cylinders.

Ice cream is formed by the combination of cone and hemisphere.

### What is meant by the volume of solids?

The volume of solid is defined as the measure of how much space is occupied by three-dimensional objects.

### What is the formula for the volume of combined solids?

The formula for the volume of a combination of solids is given by;

V = V_{1}+V_{2}+ …

Where V is the volume of the combination of solids

V_{1} and V_{2} are the volumes of solid 1 and solid 2, respectively.

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