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In this video, we’ll learn how to calculate volumes of rectangular prisms and cubes, given their dimensions, and solve problems, including real-life situations.
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We’ll consider what we actually mean by the word “volume” and how the properties of rectangular prisms and cubes can help us to derive and use a formula for their volume.
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So, our first question is, what is a prism?
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A prism is a three-dimensional shape with a constant cross section.
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In other words, the cross section has the same shape and size throughout its length.
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A triangular prism, for example, has a triangular cross section.
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I could slice down here or down here, and the size and shape of that triangle would stay the same.
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Similarly, a cylinder has a constant cross section; it’s a circle.
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Now, we’re interested in rectangular prisms, like this, and cubes.
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A cube is simply a rectangular prism whose dimensions are all the same.
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We notice its faces — that’s the flat surfaces of the shape — are all squares.
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Now, in this video, we’re learning how to calculate the volume of these shapes.
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The volume of a shape is a measure of its total three-dimensional space.
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And the easiest way to measure the three-dimensional space is to consider how many cubic units a shape contains.
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A cubic unit is simply a cube that measures one unit by one unit by one unit.
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Let’s have a look at an example of this.
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Find the volume of the cuboid.
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The volume of a cuboid or a rectangular prism is a measure of its three-dimensional space.
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We can work out the volume by considering how many unit cubes the shape contains.
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In this case, we see that our cuboid is split into cubes which measure one centimeter by one centimeter by one centimeter.
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That’s one cubic centimeter.
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And one method we have is to simply count them, beginning with the front of our shape.
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Here, we have one, two, three, four, five, six, seven, eight, nine, 10, 11, 12, 13, 14, 15 cubes.
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But is there a quicker way to calculate this?
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Well, yes!
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We have five rows, each containing three cubes.
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So, we could have simply multiplied five and three to see that there are 15 cubes on the front of our shape.
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We now look at the depth of our shape.
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It’s two centimeters.
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This means we’ve essentially got two identical slices, each containing 15 cubes.
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The volume is, therefore, 15 multiplied by two, which is 30.
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Now, each cube is one centimeter cubed.
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So, the total volume of our cuboid is 30 cubic centimeters.
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But is there a quicker way we could have done this?
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Well, yes.
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Rather than counting cubes, we saw that we can multiply five by three, and then we multiply this by two.
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In other words, we can multiply the width by the height by the length, and that’s the formula for the volume of a rectangular prism or a cuboid.
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It’s the product of its length, its width, and its height.
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In this case, that was five times three times two.
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But remember, multiplication is commutative.
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So, we can actually do this in any order and get the same answer of 30 cubic centimeters.
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Let’s now have a look at how we can use this formula to solve problems involving the volume of rectangular prisms and cubes.
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Which of the following describes how the volume of a rectangular prism is affected after doubling all three dimensions?
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Is it (A) 𝑉 sub new is equal to six times 𝑉 sub old?
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(B) 𝑉 new is equal to 𝑉 old squared.
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(C) 𝑉 new is equal to two times 𝑉 old.
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(D) 𝑉 new is equal to four times 𝑉 old.
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Or (E) 𝑉 new is equal to eight times 𝑉 old.
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To answer this question, we’ll begin by recalling the formula for the volume of a rectangular prism.
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The volume 𝑉 of a rectangular prism is the product of its length, its width, and its height.
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So, let’s call the volume of our original shape 𝑉 old.
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It’s 𝑤𝑙ℎ.
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Now, of course, we could do this in any order.
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So, we could write 𝑙𝑤ℎ or any other combination.
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We’re now going to take our original rectangular prism and double all of its dimensions.
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The height of our new shape is two times ℎ, which is two ℎ.
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The width is now two times 𝑤.
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That’s two 𝑤.
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And the length is two times 𝑙.
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That’s two 𝑙.
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And so, we can now calculate the volume of the new shape.
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It’s still the product of all of its dimensions, but this time that’s two 𝑤 times two 𝑙 times two ℎ.
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When we multiply algebraic expressions, such as this, we begin by multiplying the numbers.
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And so, two times two times two is eight.
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And the new volume is eight 𝑤𝑙ℎ.
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We now compare the original volume to the new volume.
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And since the original volume is 𝑤𝑙ℎ and the new volume is eight times this, this must mean that the new volume is eight times the old volume.
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The correct answer, therefore, is (E) 𝑉 sub new is equal to eight times 𝑉 sub old.
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We’ll now consider how we can use the formula for the volume of a rectangular prism within a real-world context.
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A man needs to store 16,170 cubic centimeters of rice in a container.
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He has one box which is a cuboid with dimensions of 35 centimeters, 22 centimeters, and 21 centimeters and another box which is a cube with length 22 centimeters.
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Which box should he use?
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Remember a measure of capacity or the amount of space a three-dimensional shape holds is volume.
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And the formula for the volume of a rectangular prism or a cuboid is width times height times length.
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Essentially, it’s the product of its three dimensions.
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Now, we can calculate this product in any order.
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We also have a cube in this question, but a cube is simply a cuboid whose dimensions are all equal.
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And so, the volume of a cube is simply the length of one of its edges cubed.
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So, we’ll begin by calculating the volume of each shape.
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The dimensions of the cuboid are 35 centimeters, 22 centimeters, and 21 centimeters.
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So, the volume is 35 times 22 times 21, which is 16,170.
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Now, we’re multiplying centimeters by centimeters by centimeters.
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And so, the units here are cubic centimeters or centimeters cubed.
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Now, in fact, this is the exact same value as the amount of rice that the man needs to store.
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But let’s double-check with the volume of the cube.
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This time, the length of each edge is 22 centimeters.
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And so, the volume of our cube is 22 times 22 times 22 or 22 cubed.
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That gives us a volume of 10,648 cubic centimeters.
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This is indeed going to be too small, so he should use the cuboid to store the rice.
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In our next question, we’ll consider how to calculate the volume of a rectangular prism given the area of one of its faces.
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Cuboid 𝐴 has dimensions of 56 centimeters, 40 centimeters, and 34 centimeters.
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Cuboid 𝐵 has a base area of 2,904 square centimeters and a height of 36 centimeters.
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Which cuboid is greater in volume?
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We begin by recalling the formula for the volume of a rectangular prism or a cuboid.
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It’s the product of its three dimensions.
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We could write that as width times height times length in any order.
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And so, we can calculate the volume of cuboid 𝐴 fairly easily.
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It’s 56 multiplied by 40 multiplied by 34, which is 76,160.
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Our units are centimeters, so the units for the volume are cubic centimeters or centimeters cubed.
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But what about the volume of cuboid 𝐵?
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Well, we’ll go back to our formula for the volume of a cuboid.
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And since multiplication is commutative, we know we can do it in any order.
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So, we can rewrite this as width times length multiplied by height.
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But, of course, width times length gives us the area of a rectangle.
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In this case, that’s the area of the base of our cuboid.
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And so, we can alternatively say that the volume of a cuboid is equal to the area of its base multiplied by its height, where the height is the side that’s perpendicular to the base.
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We also sometimes say that the volume of a cuboid is equal to the area of its cross section multiplied by its length or its height.
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We can, therefore, calculate the volume of cuboid 𝐵 by multiplying 2,904 — that’s the area of its base — by its height; that’s 36.
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That gives us a value of 104,544 cubic centimeters.
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We can quite clearly see that 76,160 is less than 104,544, meaning that cuboid 𝐵 is greater in volume.
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We’ll now have a look at how we can use information about the volume to solve problems in a real-world context.
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Given that 405 cubic centimeters of water is poured into a rectangular-prism-shaped vessel with a square base whose side length is nine centimeters, find the height of water in the vessel.
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In his question, we’ve been given some information about the volume of water being poured into a rectangular-prism-shaped vessel.
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This vessel has a square base with side length of nine centimeters.
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So let’s sketch this out.
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Here is this vessel.
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Now, we don’t know what the height of the water is in the vessel when it’s poured in.
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So, let’s call that ℎ centimeters.
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We do know that the amount of space this takes up in three dimensions is 405 cubic centimeters.
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And we also know that this is the volume.
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And the volume of a cuboid is equal to the area of its base multiplied by its perpendicular height.
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Now, we can calculate the area of the base of our vessel.
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It’s simply a square.
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So, its area is nine multiplied by nine, which is 81.
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We’re working in centimeters.
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So, the area of the base of our vessel is 81 square centimeters.
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We, in fact, also know that the volume of our water is 405 cubic centimeters.
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And we’ve said that its height is equal to ℎ.
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We can, therefore, form an equation in ℎ.
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We can say that 405, remember, that’s the volume, is equal to the area of the base, that’s 81, times ℎ or 405 equals 81ℎ.
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We want to solve for ℎ.
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So, we’re going to divide both sides of our equation by 81.
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That gives us ℎ is equal to five.
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And we can, therefore, say that the height of water in the vessel is five centimeters.
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In our very final example, we’ll look at how we need to be extra careful when dealing with different units in our question.
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Find the volume of the cuboid.
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We recall that the volume of a cuboid or a rectangular prism is the product of its three dimensions.
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So, it’s its width multiplied by its length multiplied by its height.
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Now, we need to be extra careful here.
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We are given that the width of our cuboid is 0.5 meters and its length is four meters.
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Its height, however, is 85 centimeters.
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The units are different to the other dimensions.
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And so, before we calculate the volume, we’re going to ensure that all our units are the same.
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Now, we could do this in one of two ways.
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We could convert all of our measurements to centimeters and give our volume in cubic centimeters or convert our measurements to meters and give our volume in cubic meters.
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We’ll have a look at both examples.
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We know that there are 100 centimeters in a meter.
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And so, to convert from meters to centimeters, we multiply by 100.
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0.5 meters is 0.5 times 100, which is 50 centimeters.
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Similarly, four meters is four times 100, which is 400 centimeters.
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The volume, therefore, of our cuboid in cubic centimeters is 50 times 400 times 85, which is 1,700,000 cubic centimeters.
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And now, let’s see what happens when we calculate the volume in cubic meters.
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This time, to change from centimeters into meters, we’re going to divide by 100.
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So, 85 centimeters is 85 divided by 100, which is 0.85 meters.
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In cubic meters then, our volume is 0.5 times four times 0.85, which gives us a volume of 1.7 cubic meters.
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And so, we’ve calculated the volume of our cuboid in both cubic centimeters and cubic meters.
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Now, a common mistake is to think that to convert between cubic centimeters and cubic meters, we multiply or divide by 100.
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We can see quite clearly that that’s not the case here.
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In fact, to convert from cubic centimeters to cubic meters, we divide by 100 cubed.
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And to convert the other way, we multiply by 100 cubed.
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In this video, we’ve learned that a rectangular prism is a solid shape that looks a little bit like a box.
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It has six rectangular faces, and we call its dimensions length, width, and height.
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We learned that the volume of this prism is the product of these dimensions.
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It’s width times length times height.
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And we can calculate this in any order.
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We also saw that we can alternatively say that the volume of a rectangular prism is equal to the area of its base or its cross section multiplied by its height, where the height is the dimension perpendicular to the base.
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And remember, we should always check that dimensions are given in the same unit before working out volume of a three-dimensional shape.