Hi, I’m Rob. Welcome to Math Antics! In this lesson, we’re going to learn about another important geometry quantity called Volume. Specifically, we’re going to learn what volume is, and what kind of units we use to measure volume, and how we can calculate the volumes of a few simple geometric shapes. The first thing you need to know is that Volume is a quantity that all 3-Dimensional objects have, but to understand what it means, it will help if we back up just a little and start out with a 1-Dimensional object like this line segment. To measure a 1-Dimensional object we need a 1-Dimensional quantity which we usually call “length”. The length of this line happens to be exactly one centimeter (which is a common unit for measuring length). And in the Math Antics video about area, we saw that if we move (or extend) this 1-Dimensional line in a direction perpendicular to it by a distance of one centimeter, it forms a 2-Dimensional object called a square. 2-Dimensional objects are measured by the 2-Dimensional quantity that we call area. Because the original line was one centimeter long and we extended it a distance of one centimeter, the amount of area that this square occupies is one “square centimeter” (which is a common unit for measuring area.) Okay now, imagine that we take that 2-Dimensional square and move (or extend) it in a direction perpendicular to its surface by a distance of 1 centimeter. It forms a 3-Dimensional object that’s called a cube. And to measure a 3-Dimensional object like this, we use the 3-Dimensional quantity called volume. Volume tells us how much 3-Dimensional space (or 3D space) an object occupies. Alright, but how much volume does this cube have? Well, since it was made by extending 1 square centimeter a distance of one centimeter in the 3rd dimension, we say that its volume is exactly one “cubic centimeter” (which is a common unit for measuring volume.) So “square units” are used to measure area, and “cubic units” are used to measure volume. Since square units are made by multiplying TWO 1-Dimensional units together, like centimeter times centimeter, we can abbreviate them using the exponent notation “centimeters to the 2nd power” or “centimeters squared”. And since cubic units are made by multiplying THREE 1-Dimensional units together, like centimeter times centimeter times centimeter, we can abbreviate them using exponent notation: “centimeters to the 3rd power” or “centimeters cubed”. And, just like there were different sizes of square units like square inches, square meters or square miles, there are also different sized cubic units like cubic inches, cubic meters or cubic miles. See how area and square units are related to volume and cubic units? And there’s another similarity too. Do you remember how you can use square units to measure the area of ANY 2D shape, not just squares? Well, you can use cubic units to measure the volume of ANY 3D shape, not just cubes. To see how that works, take a look at this 2D circle and this 3D object called a sphere (which is like a ball). Just like you can uses a bunch of small squares to approximate the area of the circle, you can use a bunch of small cubes to approximate the volume of the sphere. And here’s the really cool part…The smaller the squares you use, the closer their combined area will match the area of the circle. And the smaller the cubes you use, the closer their combined volume will match the volume of the sphere. Okay, so volume is a 3D quantity for measuring 3D objects, but since we’ve also talked a lot about area in this video, I want to point out that 3D objects also have a type of area that you don’t want to confuse with volume. You might remember from our previous videos, that 2D shapes have BOTH a 2D quantity called area and a 1D quantity called perimeter. Well, in a similar way, 3D objects have BOTH a 3D quantity called volume and a 2D quantity called called “surface area”. Surface area is a lot like it sounds… it’s the area of the object’s outer surface or shell. Perimeter and surface area are both outer boundaries of geometric shapes. Perimeter is the 1-Dimensional outer boundary of a 2-Dimensional shape, and surface area is the 2-Dimensional outer boundary of a 3-Dimensional shape. And to help you see the difference between surface area and volume, imagine that you have a perfectly thin box filled with ice. If you unfold the box, you can see the 2D area that surrounds the volume, while the volume itself if the amount of 3D space occupied by the ice inside the box. Alright, so now that you understand what volume is, and you won’t confuse it with surface area, we’re going to spend the rest of the video learning how to calculate the volumes of some basic geometric shapes. But before we do that, I want to quickly mention something important about terminology (or the words we use to describe things in math). Most of the time, people agree on what to call things in math, but not aways. And that’s especially true when it comes to the words that we use to describe the dimensions of geometric objects. Take the words, “length”, “width” and “height” for example. If I have a rectangle, I could name its two dimensions “length” and “width” like we did in the Area video. But I could also name them “width” and “height” if I wanted to. The actual names of the dimensions really aren’t important, so different teachers might use different names. The important thing is to be flexible and realize that the math concepts are the same, even if different words are used to explain them. For example, the area of a rectangle is always found by multiplying its two side dimensions together, no matter what they’re called. Okay… back to calculating volumes… A lot of 3D shapes can be formed by taking a 2D shape and then extending it along the third dimension. For example, if you start with a rectangle, and then extend it along the third dimension, you get a 3D shape called a “rectangular prism”. If you start with a triangle, and then extend it along the third dimension, you get a 3D shape called a “triangular prism”. And if you start with a circle, and extend it along the third dimension, you get a 3D shape called a “cylinder”. From the others, you might have thought it should be called a “circular prism” but technically prisms are shapes that are formed by extending a polygon, and since a circle is not a polygon, the resulting shape is not called a prism. Okay, so the good news is that there’s a general formula for calculating the volume of these types of 3D shape. All you have to do is find the area of the original 2D shape that got extended, and then multiply that by the length (or distance) that it was extended. Usually, that original shape is called the “base” of the object. So let’s start with this rectangular prism and calculate its volume. The base is the original rectangle, and its dimensions are 4 centimeters by 3 centimeters. So to find the area of that base, we need to remember how to calculate the area of a rectangle. To do that, we just multiply the two side dimensions together. 4 centimeters times 3 centimeters gives us 12 centimeters squared as the area. Now… to find the volume, we just need to multiply that area by the length that the base was extended. Our diagram tells us that distance is 10 centimeters, so if we multiply 12 centimeters squared by 10 centimeters, we get 120 centimeters cubed. So this rectangular prism has a volume of 120 cubic centimeters. Okay, let’s try to find the volume of the triangular prism now. Again, the first step is to calculate the area of the base of the prism which is this triangle. The formula for finding the area of a triangle is one-half of the triangle’s base times its height. Be careful not to confuse the base of the triangle with the base of the prism which IS the triangle itself. Our diagram tells us that the base of the triangle is 10 inches and its height is 8 inches, so one-half the base times the height would be one-half of 10 times 8. 10 times 8 is 80, and one-half of 80 is 40, so the area of the triangle is 40 inches squared. Now all we have to do to find the volume is multiply that area by the length that it was extended. We’re told that the length is 50 inches, so we multiple 40 inches squared by 50 inches and we get 2,000 inches cubed. That’s the volume of this prism. Alright, are you ready to try finding the volume of the cylinder now? The cylinder is made by extending a circle, so first we need to calculate the area of that circle. As we learned in a previous video, the area of a circle is found by multiplying pi times its radius squared. The radius of this circle is 2 meters, so if we square that, we get 4 meters squared. Then we multiply by pi (which is approximately 3.14) and we get 12.56 meters squared. Now that we know the area of the circle, to find the volume of the cylinder, we just multiply that area by the length that the circle was extended (which happens to be 10 meters). 10 meters times 12.56 meters squared gives us 125.6 meters cubed. So the volume of this cylinder is 125.6 cubic meters. So, do you see how to find the volume of 3D shapes like these? …shapes that are made by taking a 2D shape an then extending it along a third axis? You just find the area of that shape and then you multiply it by the length that it was extended, and that gives you the volume. And it works for ANY 2D shape… it works for trapezoids, or pentagons, or diamonds, or stars, or hearts… “Man… now that I know how to accurately calculate volumes, playtime is WAY more fun!” Of course, not all 3D shapes are made by extending a 2D shape along a third dimension. Some are made by rotating a shape around an axis …like that sphere we saw earlier in the video for instance. One way to form a sphere is to rotate a 2D circle around one of its diameter lines. And another common 3D shape that can be formed by rotating a 2D shape around an axis is a cone. A cone can be formed by rotating a right triangle around one of its two perpendicular edges. Since the formulas for finding the volumes of spheres and cones are more complicated to explain, we don’t have time to learn how we arrive at them in this video. So for now, it’s best if you just memorize the formulas so you can use them on tests if you need to. To find the volume of a sphere, you use the formula: Volume equals four-thirds times pi times radius cubed. And to find the volume of a cone, you use the formula: Volume equals one-third times height times pi times radius squared. Let’s try one example of each before we wrap up. Here’s a sphere with a radius of 2 centimeters. Fortunately, that’s the only dimension we need to find its volume. Our formula says that the volume of a sphere is equal to: 4/3 x pi x r^3 Cubing the radius mean multiplying it by itself 3 times. So we take 2 cm x 2 cm x 2 cm which equals 8 cm cubed. Next we’ll multiply that by our approximation for pi: 3.14 times 8 is 25.12. And last, we multiply that by four-thirds, which is the same as multiplying by 4 then dividing by 3, which gives us a volume of 33.49 centimeters cubed. Now let’s try a cone. To use our formula to find the volume of a cone, we need to know two things: the radius of the circle that forms the base of the cone, and the heigh of the cone (which is similar to the height of a triangle. It’s the distance from the point at the tip of the cone, straight down to the center of the circular base.) The radius of the base of this cone is 3 feet, and its height is 9 feet. So first let’s plug that radius into our equation and square it: 3 feet x 3 feet is 9 feet squared. Next we multiply that by pi (again using 3.14) and we get 28.26 feet squared. You may notice that that’s just the area of the base circle. But now we need to multiply that base area by the one-third times the height of the cone. One-third times 9 feet is 3 feet, and 3 feet times 28.26 square feet gives us 84.78 cubic feet, which is the volume of the cone. Alright…. so now you know a lot about volume. You know that it’s a 3D quantity of geometric objects, and you know that we measure volume with cubic units. You also learned how to calculate the volume of some basic 3D shapes. Of course, there’s a lot of other 3D shapes that we didn’t have time to cover in this video, but now you know about some of the most common ones. The key is to put what you’ve learned into practice by trying some exercise problems on your own. As always, thanks for watching Math Antics and I’ll see ya next time. Learn more at www.mathantics.com
