Hey guys. Welcome to this video on the volume of three-dimensional objects.
Let’s start off by defining volume. Volume is the measurement of how much space a liquid or gas
takes up, or how much space a liquid or gas takes up within a given object.
You may not know it, but people use volume everyday.
Volume is used to calculate drinking amounts.
The amount of water you can hold in a cup is dependent on the volume of the cup.
There are several other ways that volume is used.
Now, let’s look at how to calculate the volume of a triangular prism,
a rectangular prism, a sphere, and a cone.
Volume of a Triangular Prism: The area of a triangle is
A = ½ b x h.
Essentially, to find to the volume of the triangular prism, you're multiplying the area of the triangle
times the length or the depth.
The formula for the volume of a triangular prism would be
V = ½ b x h x l.
Let’s take a look:
We have a triangular prism with a height of 8m, a base of 13m, and a length of 4m.
All we have to do is plug our numbers into our formula then solve.
We have volume = ½, times base, times height, times length,
and once we solve we get our answer, which is 208m^3.
It’s important to know that, when dealing with volume, we will always have cubic units
because we're multiplying the units by themselves 3 times.
Volume of a Cube or Rectangular Prism:
To find the same volume of a cube or rectangular prism, you will use the same formula.
Just like with the triangular prism, you want to find the area of 1 side, then multiply it times the length.
However, it’s important to know that the formula you use to find the area of a triangle is not the same formula
used to find the area of a square or a rectangle.
The formula for the area of both a square and a rectangle is A = b x h.
To find the volume of a cube or rectangular prism, you would find the area of the square or the rectangle
then multiply it times its length.
Which, makes the formula V = b x h x l.
Here is an example:
Here we have a cube, which is a rectangular prism, but all the sides are perfect squares.
Because it’s a cube, we know that all of the sides are the same distance,
so all we need to do is multiply 10 times itself 3 times. This gives us 1,000 meters cubed.
Let’s try another:
Here we have a rectangular prism with sides that are different in distance.
We have a base of 12cm, a height of 8cm, and a length of 6cm.
Now all we need to do is plug those numbers into our formula, and once we solve we get 576cm cubed.
Volume of a Sphere:
Now if you remember the area of a circle is A = Pi x r^2. That is Pi times the radius squared.
Well, to find the volume of a sphere you will use a similar formula,
but multiply it times 4/3 and switching the r^2 to make it r^3.
Making the formula for the volume of a sphere
V = 4/3 x Pi x r^3.
Now, you can do what is called a proof, to prove that this is the formula,
but for now we'll just go ahead and plug the numbers into the given formula.
The sphere has a diameter of 20 meters.
This is all the information we need to plug in and solve our equation.
We are looking for the radius, and we know that the radius is equal to half the diameter,
which means that our radius is equal to 10m.
When we plug 10 into our formula, and solve, we get 4,188.9 meters cubed.
Volume of a Cone: The formula for volume of a cone is very similar to the formula for the area of a circle.
However, there are 2 things added to the formula. To find the volume of a cone,
you multiply times ⅓, and times the height, because now you have a height
(because you're working with a three-dimensional shape).
This makes the formula for the volume of a cone
V = ⅓ x h x Pi x r^2.
Here is an example:
Here we can see that we have a height of 5cm and a radius of 2cm.
Once we plug in all of our numbers, we have
V = ⅓ x 5cm x Pi x 2^2.
When solved, we have V = 20.94cm^3.
Great job you guys. Learning new formulas can be hard.
The important thing is to keep practicing, so that you are able to recognize which formula you need to use
and to memorize the formulas.
I hope this was helpful. Be sure to check out more of our videos by subscribing below. See you next time!