Volume tells us the amount of space there is inside a 3D shape. It’s kind of like the 3D version of area 🤯. To find the volume of a cylinder or prism, we multiply the area of the base by the height.$Volume=base area×Height$To help us understand why this is the formula for volume, let’s imagine we have a cake with lots of layers.

Pick your cake shape:

If we wanted to determine the total amount of cake, we could do so by first finding the amount of cake in one layer...And then multiplying it by the number of layers.

The amount of cake in one layer is like the area of the base of the cylinder, and the total number of layers is like the height of the cylinder.

Multiplying the two together gives us:

$Volume =base area×Height=πr_{2}×H $

We can use this logic for any 3D shape where we stack up the exact same base shape over and over again:

Shape

Volume

Cylinder

$H$

$undefined$

$undefined$

$undefined$

$undefined$

$r$

$V=πr_{2}×H$

Rectangular Prism

$H$

$l$

$w$

$undefined$

$w$

$undefined$

$V=lw×H$

Triangular Prism

$H$

$undefined$

$h$

$b$

$h$

$undefined$

$V=21 bh×H$

Volume of Cylinders & Prisms Calculator

What is your shape?

Step 1. Imagine & identify the base.

We can think of the cylinder as a cake with layers shaped as circles. The base is the bottom layer, a circle.

$l$

$l$

$w$

$w$

$l$

$l$

$w$

$w$

$r$

$r$

$r$

$b$

$b$

$h$

$h$

$b$

$b$

$h$

$h$

Step 2. Calculate the area of the base.

We start by calculating the area of the base circle.

$l$

$w$

$b$

$h$

$r=$

$l=$

$w=$

$b=$

$h=$

$0$

$0$

$0$

$0$

$0$

$V =(πr_{2})×H=(π×(0)_{2})×H=(0π)×H $

Step 3. Multiply the area of the base by the height (H) of the cylinder.

Next, we multiply the base area by the height, or the number of layers.

$H=$

Recap 🧢 How to Find the Volume of Cylinders & Prisms

Shape:

Step 1. Imagine & identify the base.

We can think of the cylinder as a cake with layers shaped as circles. The base is the bottom layer, a circle.

Step 2. Calculate the area of the base.

We start by calculating the area of the base circle.$V=(πr_{2})×H$

Step 3. Multiply the area of the base by the height (H) of the cylinder.

Then, we multiply the base area by the height to get the volume.$V=(πr_{2})×H$

Continue to learn more and practice some examples of calculating the volume of cylinders and prisms!

Volume of Cylinders & Prisms

The two main things we need to know in order to solve for the volume of a cylinder and prism are the area of the base and the height of the 3D shape.$Volume=base area×Height$Once we have those two values all that’s left is to multiply the base area by the height, and then we're set! 👌🏿👌🏽👌🏻

In cake terms, the base area is like the amount of cake in one layer, and the height is like the total number of layers in the cake.

When we multiply the base area by the height, we can think of this step as stacking the layers of the cake until we reach the desired height.

Pick your cake shape:

The formula for the base area depends on the shape of the base.

$l$

$l$

$w$

$w$

$l$

$l$

$w$

$w$

$r$

$r$

$r$

$b$

$b$

$h$

$h$

$b$

$b$

$h$

$h$

$V=(πr_{2})×H$

$l$

$l$

$w$

$w$

$l$

$l$

$w$

$w$

$r$

$r$

$r$

$b$

$b$

$h$

$h$

$b$

$b$

$h$

$h$

$V=(l×w)×H$

$l$

$l$

$w$

$w$

$l$

$l$

$w$

$w$

$r$

$r$

$r$

$b$

$b$

$h$

$h$

$b$

$b$

$h$

$h$

$V=(21 bh)×H$

Let’s take a look at some examples of finding the volume of each of these 3D shapes!

Let's start by finding the volume of this rectangular prism:

$5$

$10$

$4$

$undefined$

$4$

$undefined$

Step 1. Imagine & identify the base.

We can think of the rectangular prism as a cake with layers shaped as rectangles. The base is the bottom layer, a rectangle.

$10$

$10$

$4$

$4$

$10$

$10$

$4$

$4$

$0$

$0$

$0$

$0$

$0$

$0$

$0$

$0$

$0$

$0$

$0$

Step 2. Calculate the area of the base.

We start by calculating the area of the base rectangle.

$10$

$4$

$0$

$0$

$0$

$V =(l×w)×H=(10×4)×H=(40)×H $

Step 3. Multiply the area of the base by the height (H) of the rectangular prism.

Next, we multiply the base area by the height, or the number of layers.

Nice work! Now let's try finding the volume of this triangular prism:

$4$

$undefined$

$3$

$6$

$3$

$undefined$

Step 1. Imagine & identify the base.

We can think of the triangular prism as a cake with layers shaped as triangles. The base is the bottom layer, a triangle.

$0$

$0$

$0$

$0$

$0$

$0$

$0$

$0$

$0$

$0$

$0$

$6$

$6$

$3$

$3$

$6$

$6$

$3$

$3$

Step 2. Calculate the area of the base.

We start by calculating the area of the base triangle.

$0$

$0$

$6$

$3$

$0$

$V =(21 bh)×H=(21 ×6×3)×H=(218 )×H $

Step 3. Multiply the area of the base by the height (H) of the triangular prism.

Next, we multiply the base area by the height, or the number of layers.

Amazing! Finally, let's find the volume of this cylinder:

$5$

$undefined$

$undefined$

$undefined$

$undefined$

$2$

Step 1. Imagine & identify the base.

We can think of the cylinder as a cake with layers shaped as circles. The base is the bottom layer, a circle.

$0$

$0$

$0$

$0$

$0$

$0$

$0$

$0$

$2$

$2$

$2$

$0$

$0$

$0$

$0$

$0$

$0$

$0$

$0$

Step 2. Calculate the area of the base.

We start by calculating the area of the base circle.

$0$

$0$

$0$

$0$

$2$

$V =(πr_{2})×H=(π×(2)_{2})×H=(4π)×H $

Step 3. Multiply the area of the base by the height (H) of the cylinder.

Next, we multiply the base area by the height, or the number of layers.

You're on fire 🔥 Try a couple problems on your own now or keep scrolling to wrap up this lesson.

Practice: Volume of Cylinders & Prisms

Question 1 of 3: Find the volume of this cylinder ⬇️

$4$

$undefined$

$undefined$

$undefined$

$undefined$

$3$

Step 1. Imagine & identify the base.

The base of a cylinder is a.

Step 2. Calculate the area of the base.

$undefined$

$undefined$

$undefined$

$undefined$

$undefined$

$undefined$

$undefined$

$undefined$

$3$

$3$

$3$

$undefined$

$undefined$

$undefined$

$undefined$

$undefined$

$undefined$

$undefined$

$undefined$

Which formula should we use to solve for the area of this circle base?

That's right! Plug in the right values to calculate the area of the base:

$V$$=$$(πr_{2})$$×$$H$$=$

$(π$$_{2})$

$×$$H$

$=$

$($$π)$

$×$

$H$

Nice work! Now that we have the base area, we can go on to the next step.

Step 3. Multiply the area of the base by the height (H) of the cylinder.

$4$

$V$$=$$9π×$$H$$=$$9π×$

$=$

$π$

Nice work! Our answer is $V=36π$.

When you feel like you've mastered this lesson, click for a celebration ⬇️!

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