ठोसों के संयोजन का आयतन

The volume of the combination of solids will be equal to the sum of the volume of individual solids.

Volume of a Combination of solids Formula

We get the volume of a combination of a solids 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

Here are the formula for 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 = $l\times b\times h$
• Total surface area = $2(lb+bh+lh)$
• Length of diagonal of cuboid = ${\sqrt {l^{2}+b^{2}+h^{2}}}$

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

• Volume = $a^{3}$
• Total surface area = $6a^{2}$
• Length of diagonal of cube = ${\sqrt {3}}a$

Similarly, for other shapes

• Volume of Sphere = ${\frac {4}{3}}\pi r^{3}$
• Volume of Cone = ${\frac {1}{3}}\pi r^{2}h$
• Volume of Cylinder = $\pi r^{2}h$
• Volume of Hemisphere =${\frac {2}{3}}\pi r^{3}$

Example

A cylinder of volume $150$ cm³ is placed with a cone, whose height is $4$ cm. 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 $V_{1}=150$ cm³

Height of cylinder = Height of cone = $4$ cm

The formula for Volume of cylinder is

$V_{1}=\pi r^{2}h$

$150=\pi \times r^{2}\times 4$

$r^{2}={\frac {150}{4\pi }}........(1)$

The formula for Volume of cone is given by:

$V_{2}={\frac {1}{3}}\pi r^{2}h$

By putting the value of $r^{2}$ from eq. $(1)$

$V_{2}={\frac {1}{3}}\pi \times {\frac {150}{4\pi }}\times 4$

$V_{2}=50$ cm³

Therefore, the total volume of the combined solids, $V=V_{1}+V_{2}=150+50=200$ cm³

Note: Volume of Cone = ${\frac {1}{3}}$(Volume of Cylinder)