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• Hi, I’m Rob. Welcome to Math Antics!

• In this lesson, were going to learn about another important geometry quantity called Volume.

• Specifically, were 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 calllength”.

• 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 onesquare 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 onecubic centimeter” (which is a common unit for measuring volume.)

• Sosquare unitsare used to measure area, andcubic unitsare 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 notationcentimeters to the 2nd powerorcentimeters 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 powerorcentimeters 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 partThe 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 weve 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 calledsurface area”.

• Surface area is a lot like it soundsit’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,

• were 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”, “widthandheightfor example.

• If I have a rectangle, I could name its two dimensionslengthandwidthlike we did in the Area video.

• But I could also name themwidthandheightif 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 theyre called.

• Okayback 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 thebaseof 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.

• Nowto 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.

• Were 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

• Mannow 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 axislike 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 well 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,