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Volume of Solid

- Calculus
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Do you like to bake? Each time you measure the ingredients in your recipe you are using volume calculations without even realizing it! Have you ever wondered how much water is needed to fill a pool? You can use a volume calculation to find out how much you will need.

Solids are three-dimensional (3D) shapes. They can be found everywhere in everyday life and sometimes you will need to find the volume of these shapes. There are many different types of solids and each is recognizable based on the way they look. Here are some examples:

Fig. 1 - Examples of solids

It can be helpful to find the volume of these solids. When measuring the volume of a solid you are calculating the amount of space that the solid takes up. For example, if a jug can hold 500ml when it is full, the volume of that jug would be 500ml.

In order to find the volume of a solid, you need to think about the shape itself. To find the **surface area of a solid** you will use the **length** along with the **width**, this gives you the **square units**. To find the **volume of a solid**, you also need to consider the **height** of the solid, this will then give you the **cubic units**.

To find out more about the surface area of a solid, visit Surface of solids.

There are different formulas that can be used to find out the volume of a solid. These formulas are related to the formulas that may be used to find the surface area of a solid.

Let's take the formula to find the surface area of a circle as an example,\[A=\pi r^2.\]

Doing this calculation will give you the surface area of a two-dimensional (2D) shape.

Now, let's relate it to the formula for a cylinder, a 3D shape that involves two circles joined with a curved face.

Since this is now a 3D shape, to find its volume you can take your surface area formula given and multiply it by the height \(h\) of the curved face of the cylinder, which gives you the formula \[V=\pi r^2h.\]

Since each different solid has a different formula to help you find the volume, it is important that you can identify each shape and recognize the formula that is needed.

A **prism** is a type of solid that* has two bases that are parallel to one another*. There are different types of prism and they are named after the shape of the base;

Rectangular prism

Triangular prism

Pentagonal prism

Hexagonal prism

Prisms can either be right prisms or slant prisms.

A **right prism** is a prism in which the joining edges and faces are perpendicular to the base faces.

The prisms in the picture below are all right prisms.

Fig. 2 - Examples of prisms

It helps to have labels for the parts of a prism. So call:

\( B\) the area of the base of the prism;

\(h\) the height of the prism; and

\(V\) the volume of the prism,

Then the formula for the **volume of a right prism** is

\[ V = B\cdot h.\]

Let's take a look at how to use the formula.

Find the volume of the following solid.

Fig. 3 - Volume of a prism example

**Answer****:**

Notice that this is a right prism, so you can use the formula to find the volume.

First, you can start by looking at the formula and writing down what you know from the diagram above. You know that the height of the prism is \(9\, cm\). That means in the formula for the volume of a right prism, \(h = 9\).

You need to calculate the area of the base. You can see that the triangle that makes up the base has one side of length \(4\, cm\) and another side of length \( 5\, cm\).

To do this you can use the formula to find the area of a triangle;

\[\begin{align} B&=\frac{h\cdot b}{2}\\ \\ B&=\frac{5\cdot 4}{2}\\ \\ B&=10 \end{align}\]

Now that you can find the area of the base of the prism, you can put that into the formula to find the volume of the prism;

\[\begin{align} V&=(10)(9)\\ \\ V&=90\,cm^3 \end{align}\]

What about a slant prism?

In a **slant prism**, one base is not directly above the other, or the joining edges are not perpendicular to the base.

Here is an example of what a solid slant prism may look like.

Fig. 4 - Slant prism

When you have been given a slant prism, you can use the slanted height of the solid to find the volume.

To find out more about prisms, visit Volume of Prisms.

A **cylinder** is a type of solid that *has two bases and a curved edge*. They tend to look like those in figure 5.

Fig. 5 - Example of a solid cylinder

It helps to have labels for the parts of a cylinder. So call:

\( B\) the area of the base of the cylinder;

\(h\) the height of the cylinder; and

\(r\) the radius of the cylinder.

A cylinder can be thought of as a prism with a circular base, however, a different formula can also be used to find the **volume of a cylinde****r**;

\[V=Bh=\pi r^2h.\]

To find out more about cylinders, visit Volume of Cylinders.

A **pyramid** is a type of solid that *has one base*. The shape of the base determines the type of pyramid you have. In a pyramid, all of the faces are triangles that come to one vertex. Some different types of pyramids include:

Square pyramid

Rectangular pyramid

Hexagonal pyramid

Here is an example of a square pyramid.

Fig. 6 - An example of a square pyramid

The labels of pyramids are:

\( B\) the area of the base of the pyramid;

\(h\) the height of the pyramid; and

\(V\) the volume of the pyramid,

There is a formula that can be used to help you find the **volume of a pyramid**;

\[V=\frac{1}{3}Bh.\]

You may observe that a pyramid and a cone are two very similar shapes, with a cone being a type of pyramid that has a circular base. This is why you can also see similarities in the formula that can be used to find the volume of the shapes.

To find out more about pyramids, visit Volume of Pyramids.

Similar to a pyramid, a solid **cone** *only has one base*: a circle. A cone only has one face and a vertex. They look like this;

Fig. 7 - A solid cone

The labels of a cone are:

\(h\) the height of the cone;

\(r\) the radius; and

\(V\) the volume of the prism,

There is a formula that can be used to help you find the **volume of a cone**;

\[V=\frac{1}{3}Bh=\frac{1}{3}\pi r^2h.\]

To find out more about cones, visit Volume of Cones.

A **sphere** is a type of solid that *has no bases*. It is like a 3D ball, for example, a football. A sphere has a center point; the distance between the center point and the outer edge gives the radius of the sphere.

Fig. 8 - Example of a solid sphere

It helps to have labels for the parts this solid. So call:

\(r\) the radius; and

\(V\) the volume of the prism,

There is a formula that can be used when trying to find the **volume of a sphere**;

\[V=\frac{4}{3} \pi r^3.\]

To find out more about spheres, visit Volume of Spheres.

A **rectangular solid** is a type of 3D shape where *all of the bases and faces of the shape are rectangles*. They can be considered a special type of right prism.

Fig. 9 - Example of a rectangular solid

To find the **volume of a rectangular solid you can multiply the length by the width by the height of the shape**. This can be written into the following formula:

\[V=L\cdot W\cdot H.\]

Let's take a look at an example using the formula.

Find the volume of the following solid.

Fig. 10 - Worked example

**Answer:**

To start identify each of the labels of the shape so that you know where to input the variable into the formula.

\[L=5cm, \space \space W=7cm, \space \space H=10cm\]

Now you can input the variables into the formula to find the volume of a rectangular solid.

\[\begin{align} V&=L\cdot W\cdot H\\ \\ V&=5\cdot 7\cdot 10\\ \\ V&=350cm \end{align}\]

A **composite solid** is a type of 3D solid that is *made up of two or more solids*. Take a house, for example, the building can be considered a composite solid, with a prism base and a pyramid roof.

Fig. 11 - An example of a composite solid

To find the volume of a composite solid you need to break the shape down into its separate solids and find the volume for each of them.

Going back to the house example, you could first find the volume of the prism and then the volume of the pyramid. To find the volume of the entire house, you would then add the two separate volumes together.

Let's take a look at some more examples.

Calculate the volume of a pyramid that has a square base, with the side lengths measuring \(6\,cm\) and a height of \(10\,cm\).

**Answer:**

To start with you need to find the correct formula to use, since it is a pyramid you will need that specific formula:

\[V=\frac{1}{3}Bh\]

Now you need to find each part of the formula to calculate the volume. Since the base of the pyramid is a square with a side length of \(6\,cm\), to find the area of the base \((B)\) you can multiply \(6\) by \(6\):

\[B=6\cdot 6=36\]

You now know the area of the base and you know the height of the pyramid from the question which means you can now use the formula:

\[\begin{align} V&=\frac{1}{3}(36)(10) \\ \\ V&=120\,cm^3 \end{align}\]

Here is another example.

Calculate the volume of a sphere that has a radius of \(2.7cm\).

**Answer:**

To start with you need to find the correct formula to use, since it is a sphere you will need that specific formula:

\[V=\frac{4}{3}\pi r^3\]

You have been given the radius, so all you need to do is input that value into the formula:

\[\begin{align} V&=\frac{4}{3}\pi (2.7)^3 \\ \\ V&\approx82.45\,cm^3 \end{align}\]

Let's look at a different type of example.

Draw a cone with a height of \(10\,cm\) and a radius of \(9\,cm\).

**Answer:**

To answer this type of question, you will need to draw out the solid according to the given measurements.

In this question, you have been asked to draw a cone that is \(10\,cm\) in height and has a radius of \(9\,cm\). This means it will be \(10\,cm\) tall and the circular base will have a radius of \(9\,cm\), meaning it will be \(18\,cm\) wide.

Fig. 12 - Worked example with a cone

When drawing your own diagram, don't forget to label it with the measurements!

Let's look at one more.

Calculate the volume of a cone that has a radius of \(9\,m\) and a height of \(11\,m\).

**Answer:**

To start with you need to find the correct formula to use, since it is a cone you will need that specific formula:

\[V=\frac{1}{3}\pi r^2h\]

You have been given both the radius and the height of the cone which means you can put the values straight into the formula:

\[\begin{align} V&=\frac{1}{3}\pi (9)^2(11) \\ \\ V&\approx933\,m^3 \end{align}\]

- A solid is a 3D shape, there are many different types of solids and each solid has its own formula to find the volume;
- Prisms - \(V=Bh\)
- Cylinders - \(V=\pi r^2h\)
- Pyramids - \(V=\frac{1}{3}Bh\)
- Cones - \(V=\frac{1}{3}\pi r^2h\)
- Spheres - \(V=\frac {4}{3}\pi r^3\)

- A rectangular solid is a 3D shape where all of the faces and bases are rectangles, you can find the volume of the solid by using the formula, \(V=L\cdot W\cdot H\).
- A composite solid is a 3D shape made up of two or more solids, to find the volume you can break the shape down into its separate solids and find their volumes individually before adding them together.

The volume of a solid describes the cubic units that fit inside the 3D shape.

^{4}/_{3}×π×3^{3} ≈ 113.04cm^{3}.

There are different formulas that can be used to calculate the volume of a solid.

More about Volume of Solid

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