Volume Formulas

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When you talk about the volume of a given shape or figure, it is simply the measure of space expressed in three dimensions. In finding the volume, you will have to use the values of the base, width, and height. Depending on the kind of shape you have, there are different kinds of formulas you can apply to find the volume of the shape. Think of it this way, when you have a shape, imagine the amount of water, sand, or air that it could hold when filled completely. That would show you the volume the shape has.

Volume is expressed in cubic units, which may be centimeters, inches, feet, or any other unit you may be given. Because there are different kinds of three dimensional shapes, in this article, we will look at the ways of calculating the volume of these shapes. The shapes include cubes, cones, prisms, spheres, cylinders and pyramids.

You may also notice that most of the formulas given for calculating the volume of different types of shapes have some similarities. So if you can figure out those similarities, it will be easy for you to recall the formulas.

Volume of a Cube

A cube consists of six identical square faces or surfaces. You can think of a cube as a box that has all sides equal. A good example is a six-sided die that you see in the house. The letter blocks children use to learn are also examples of cubes provided that they have all the six sides equal.

In finding the volume of a cube, you will want to write down the formula.

Remember that a cube has all the sides with equal lengths and this makes it easy for you to find the volume.

The formula will be:

V = swhere v represents volume and s represents length of the shape or cube.

So if you are given the length of one side of a cube as 6 cm, you can find the volume.

V = s x s x s

V = 6 x 6 x 6

V = 216 cm3

Volume of a Rectangular Solid or Rectangular Prism or a Cuboid

A rectangular solid is also called a rectangular prism or a cuboid. A rectangular solid may also be referred to as a rectangular prism. Another term is a cuboid. So you would want to remember that a cube is not the same as a cuboid. Usually, in a rectangular prism, the width, length, and the height may have different lengths unlike a cube where these have the same length.

A rectangular solid has six sides and all these sides are rectangles. The formula for finding the volume of a rectangular solid is:

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Volume = l × w × h where l represents the length, w represents the width, and h stands for the height.

You may also write down this equation as:

Volume = length x width x height

So if you are given the length of a rectangular solid as 8 cm and the width as 6 cm and the height as 10 cm, you can find the volume.

Volume = l × w × h

Volume = 8 x 6 x 10

Volume = 480 cm3

 Volume of a Cylinder

A cylinder may be defined as a solid consisting of two parallel faces that are congruent circles. By congruent, we mean that the faces are matching or similar in dimension. The faces of the cylinder provide you with the bases of the shape. A cylinder has also one curved surface. Usually, the height of a cylinder is the distance between the congruent circles or the two bases. It is the perpendicular distance from one base of a cylinder to the other base.

You may also define a cylinder as a three-dimensional shape, which consists of two identical flat ends or bases that are circular in their shape, and a single or one curved surface that connect the identical flat ends or the bases.

In finding the volume of a cylinder, you will have to get the area of the base and then multiply by the height. Since the base is a circle, you use the formula of finding the area of a circle and then multiply by the height.

Volume = Area of base x height

Area of base = Area of circle = πr2

Therefore the formula for finding the volume of a cylinder is;

Volume = πr2h

In this formula r represents the radius of the base of the cylinder and h represents the height or simply the length of the cylinder.

Let’s say that you are given the radius of a cylinder as 4 cm and the height as 8 cm, now you can calculate the volume

Volume = π r2h

Volume = 3.14 x 42 x 8

Volume = 3.14 x 16 x 8

Volume = 401.92 cm3

Sometimes, you may be given the circumference of the base of a cylinder and the height, the you are required to find the volume. In this case, you will need to first find the radius. The circumference of a circle is calculated by using this formula: C = 2πr where c stands for the circumference and r the radius.

Let’s say that the circumference is 28 cm and the height of the cylinder is 8 cm. Now find the radius before you calculate the volume.

C = 2πr

Therefore r = C/2π

r = 28/2 x 3.14

r = 14/3.14

r = 4.47 cm

Now you can find the volume of the cylinder;

Volume = π r2h

Volume = 3.14 x 4.472 x 8

Volume = 3.14 x 19.98 x 8

Volume = 501.92 cm3

At other times, you may have a hollow cylinder and you are required to find its volume.

Volume of a Hollow Cylinder

Talking of a hollow cylinder, it can be defined as a cylinder that is empty from inside and consists of different internal and external radii. If you are wondering what a radii is, it is just the plural of radius. Many radius are called radii. In everyday life situations, you have seen examples of hollow cylinders, for example you have straws, tubes, and any other object or item that is cylindrical.

This is how a hollow cylinder would look like:

In finding the volume of a hollow cylinder, you want to find the volume of the outer cylinder and minus the volume of the inner cylinder.

The formula for finding volume of a hollow cylinder is given as:

Volume = πR2h – πr2h

Volume = πh(R2 – πr2)

In this formula, R represents the outer radius of the hollow cylinder and r represents the inner radius of the cylinder.

Let’s say that the outer radius (R) is 6 cm and the inner radius (r) is 4 cm and the height (h) is 10 cm, you can now find the volume of the hollow cylinder.

Volume = πh(R2 – πr2)

Volume = 3.14 x 10 (62 – 42)

Volume = 3.14 x 10(36 – 16)

Volume = 3.14 x 10(20)

Volume = 3.14 x 200

Volume = 628 cm3

Volume of a Cone

A cone has a curved surfaces that tapers or decreases in size as you move to the top or the vertex. Usually, the height is that perpendicular distance running from the center of the base to the vertex.

In other words, a cone is object that has 3 dimensions with a circular base and one single vertex. The single vertex is the point of the cone. You may think of a cone as a kind of pyramid having a circular base. When the vertex or point of the cone is located directly or perpendicular to the center of the base, it is referred to as a right cone. However, if the point of the cone is not directly located or the center of base, it is referred to as an oblique cone.  A very important point to note here is that the formula for finding the volume of the two types of cones (right and oblique cones) is the same. That’s something you should always remember because you may come across the two different types of cones and you are asked to find the volume.

This is how a cone looks like:

The formula for finding the volume of a cone is:

V = 1/3πr2h

In this formula, r stands for radius and h represents the height. If you look at this formula, you will notice that we take the area of the base multiply by the height and then divide by a third.

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Let’s say we are given the radius of a cone as 7 cm and the height as 10 cm, we can use this information to find the volume of that cone.

V = 1/3πr2h

V = 1/3 x 3.14 x 72 x 10

V = 1/3 x 3.14 x 49 x 10

V = 1/3 x 3.14 x 490

V = 1/3 x 1538.6

V = 512.82 cm3

Volume of a Pyramid

A pyramid may be defined as a three-dimensional shape having a polygon for a base as well as lateral faces that are tapering at the apex or point of the pyramid. You may have a pyramid that has a square base or a triangular base.  The lateral faces of a pyramid tend to meet at a common point or vertex and the height is usually the perpendicular line that runs from the center of the base to the tip or vertex of the pyramid.

Also, a pyramid may be named based on the shape it has on the base. For example, you can have a square or regular pyramid if the base is a square. You may have a rectangular pyramid if the base is a rectangle. A triangular pyramid implies that the base is a triangle.

To find the volume of the pyramid, you want to find the area of the base then multiply that with the height.

Volume of pyramid =  Area of base × height

Let’s take for example, a regular pyramid meaning a pyramid that has the base being a regular polygon where all the sides are equal in length. In other words, this pyramid has a square base where all angles and lengths are equal. The formula for finding the volume of a regular pyramid would be:

Volume of a Square or regular pyramid is: Volume = 1/3bh

In this formula, b represents the area and h represents the height. A very important thing to note here is that the formula for finding the volume of a right pyramid and oblique pyramid is the same. A right pyramid is where the apex is located directly, and perpendicularly on top of the center of the base. With oblique pyramids, it is where the apex or the tip of the pyramid is not centered – it is shifted in one side.

Another thing is that when calculating the area of the base, it depends on the number of sides your pyramid’s base has.

So if we have a regular or square pyramid that has a base having a side length of 5 cm and the height is given as 10 cm, we can use this information to find the volume.

Volume = 1/3bh where b represents the area and h represents the height

Now, let’s find the area of the base, which is a square.

Area = s2

Area = 52

Area = 25 cm2

Therefore, the value for b in our equation above is the area of the base of the pyramid, which is 25 cm2

Let’s now calculate the volume.

Volume = 1/3bh

Volume = 1/3 x 25 x 10

Volume = 1/3 x 250

Volume = 83.33 cm3

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