Volume tells us the amount of space there is inside a given 3D shape. It’s kind of like the 3D version of area. To find the volume of a cone or pyramid, we need to multiply the area of the base by the height, and then divide by 3.$Volume=3base area×Height $This formula is very similar to the formula for the volume of a cylinder and prism. The only difference is that there’s one additional step...dividing by 3!

We can also think about this formula in terms of cake!

In our Volume of Cylinders & Prisms Lesson 💦 we explain how the amount of cake in one layer is like the area of the base of a cylinder or prism, and the total number of layers is like the height of the cylinder or prism.

But a cone or a pyramid is smaller than a cylinder or a prism - it's like we're slicing off a piece from either side and then we're left with ⅓ of the cake. That's why we have to take the volume of the cylinder or prism and divide by 3.

Pick your cake shape:

$V=3πr_{2}×H $

What a piece of cake! 😜

Volume of Cones and Pyramids Calculator

What does your shape look like?

Step 1. Imagine & identify the base.

We can think of the cone as a cake. The base is the bottom layer, a circle.

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Step 2. Calculate the area of the base.

We start by calculating the area of the base circle.

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$0$

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$V =3πr_{2}×H =3π×(0)_{2}×H =30π×H $

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

$H=$

We can think of this step as stacking up layers of the cake base until you reach the desired height. Notice this forms a cylinder around our cone.

Step 4. Divide by 3.

Lastly, we take one piece off each side and we're left with $31 $ of the cylinder's volume.

$V =30π =0 $

Our answer is $V=0$.

Recap 🧢 How to Find the Volume of Cylinders & Prisms

Shape:

Step 1. Imagine & identify the base.

We can think of the cone 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=3πr_{2}×H $

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

We can think of this step as stacking up layers of the cake base until you reach the desired height.$V=3πr_{2}×H $

Step 4. Divide by 3.

We can think of this step as taking a piece off the left and right sides, leaving us with ⅓ of the cake.$V=3πr_{2}×H $

Continue to learn more and practice some examples of calculating the volume of cones and pyramids!

Volume of Cones & Pyramids

Like cylinders and prisms, the two main things we need to know in order to solve for the volume of a cone and pyramid are the area of the base and the height of the 3D shape.$Volume=3base area×Height $The formula for the base area depends on the shape of the base.

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$V=3πr_{2}×H $

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$V=3l×w×H $

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$V=321 bh×H $

Once we have base area, we multiply it by the height, and then divide by 3! 👌🏾👌🏽👌🏻

Great question! We divide by 3 because the volume of a cone is equal to ⅓ of the volume of a cylinder with the same base, and the volume of a pyramid is equal to ⅓ of the volume of a prism with the same base.

Think of it like we're we've divided the cylinder or prism into 3 sections, and then we take a chunk off either side, leaving us with ⅓.

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 pyramid:

$5$

$3$

$2$

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Step 1. Imagine & identify the base.

We can think of the rectangular pyramid as a cake. The base is the bottom layer, a rectangle.

$3$

$3$

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Step 2. Calculate the area of the base.

We start by calculating the area of the base rectangle.

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$V =3l×w×H =33×2×H =36×H $

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

We can think of this step as stacking up layers of the cake base until you reach the desired height. Notice this forms a prism around our rectangular pyramid.

Step 4. Divide by 3.

Lastly, we take one piece off each side and we're left with $31 $ of the prism's volume.

$V =330 =10 $

Our answer is $V=10$.

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

$6$

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$5$

$4$

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Step 1. Imagine & identify the base.

We can think of the triangular pyramid as a cake. The base is the bottom layer, a triangle.

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$4$

Step 2. Calculate the area of the base.

We start by calculating the area of the base triangle.

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$5$

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$0$

$V =321 ×b×h×H =321 ×5×4×H =3220 ×H =310×H $

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

We can think of this step as stacking up layers of the cake base until you reach the desired height. Notice this forms a prism around our triangular pyramid.

Step 4. Divide by 3.

Lastly, we take one piece off each side and we're left with $31 $ of the prism's volume.

$V =360 =20 $

Our answer is $V=20$.

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

$7$

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$2$

Step 1. Imagine & identify the base.

We can think of the cone as a cake. The base is the bottom layer, a circle.

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Step 2. Calculate the area of the base.

We start by calculating the area of the base circle.

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$V =3πr_{2}×H =3π×(4)_{2}×H =316π×H $

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

We can think of this step as stacking up layers of the cake base until you reach the desired height. Notice this forms a cylinder around our cone.

Step 4. Divide by 3.

Lastly, we take one piece off each side and we're left with $31 $ of the cylinder's volume.

$V =396π =32π=100.531 $

Our answer is $V=32π=100.531$.

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

Practice: Volume of Cones & Pyramids

Question 1 of 6: Find the volume of this cone:

$4$

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$3$

Step 1. Imagine & identify the base.

The base of a cone is a .

Nice! We can think of the cone as a piece of cake with layers shaped as circles.

Step 2. Calculate the area of the base.

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

Good work! Now plug in:

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$3$

$V$$=$$3πr_{2}×H $$=$

$π$$_{2}×H$

$3$

$=$

$π$$×H$

$3$

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 cone.

We can think of this step as stacking up layers of the cake base until you reach the desired height.

$4$

$V$$=$$39π×H $$=$

$9π×$

$3$

$=$

$π$

$3$

Nice work! Now that we've multiplied the base by the height, there's only one step left.

Step 4. Divide by 3.

We can think of this step as cutting a piece off either side, leaving us with $31 $ of the cake in the form of a cone.

$V$$=$$336π $$=$

$π$

Awesome! The volume of the cone is $12π≈37.7$.

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

Nice work, look at you go! Thanks for checking out this lesson ☺️🙏. Where to next?