Volume of Cylinders, Cones & Spheres

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Grades 8–9GeometryGuided journey: Discover, Learn, Practice, Apply

Apply CCSS 8.G.C.9 formulas for cylinders, cones, and spheres. Connect Grade 7 cylinder work to cones and spheres, use consistent radius and height labels, and model real containers such as rain barrels, funnel pours, and round storage tanks.

What you’ll learn

Cylinder volume is $V=\pi r^2 h$ where $r$ is the base radius and $h$ is the perpendicular height between circular faces.
Cone volume is $V=\frac{1}{3}\pi r^2 h$ , exactly one-third of a cylinder with the same base radius and height.
Sphere volume is $V=\frac{4}{3}\pi r^3$ ; convert diameter to radius with $r=\frac{d}{2}$ before substituting.
Always express answers in cubic units and label whether you leave $\pi$ in the answer or use a decimal approximation.
Sketching the solid and marking $r$ and $h$ prevents confusing slant height with vertical height on cones.

Aligned to 8.G.C.9

Studying for a final? Geometry Final Exam Review or 8th Grade Math Final Exam Review

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Volume of Cylinders, Cones & Spheres

Free

Grades 8–9GeometryGuided journey: Discover, Learn, Practice, Apply

What you’ll learn

Cylinder volume is $V=\pi r^2 h$ where $r$ is the base radius and $h$ is the perpendicular height between circular faces.
Cone volume is $V=\frac{1}{3}\pi r^2 h$ , exactly one-third of a cylinder with the same base radius and height.
Sphere volume is $V=\frac{4}{3}\pi r^3$ ; convert diameter to radius with $r=\frac{d}{2}$ before substituting.
Always express answers in cubic units and label whether you leave $\pi$ in the answer or use a decimal approximation.
Sketching the solid and marking $r$ and $h$ prevents confusing slant height with vertical height on cones.

Aligned to 8.G.C.9

Studying for a final? Geometry Final Exam Review or 8th Grade Math Final Exam Review

For teachers

Share to Google Classroom

Not started

Reference

Volume of Cylinders, Cones & Spheres

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Share this journey with a parent or teacher.

For parents & teachers

Get the free 7-Day Study Reset

A research-backed week of short routines that help a student study less and remember more: spacing, self-quizzing, and a calmer week before tests.

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Reference

Volume of Cylinders, Cones & Spheres

Discover, learn, practice, and apply what you know

Guide

Explore

Tools

Where you are

Units

Share this journey with a parent or teacher.

Discover

Think about this

Why is cone volume exactly one-third of the cylinder that shares its base and height, and how could you justify that with water-pour experiments?

A round metal rainwater tank in a garden setting, shaped like a vertical cylinder. — A rain barrel is modeled as a cylinder: measure radius and height, then apply $V=\pi r^2 h$ in cubic meters or gallons after unit conversion.

Cylinder volume and surface area | Perimeter, area, and volume | Geometry | Khan Academy (8:07)

Learn

Read each section like a formula card: why it matters, the steps, one worked example, then the check. Try the steps yourself before moving to Practice.

Cylinder: $V=\pi r^2 h$

How to do it:

1. $r$ is the radius of the circular base (half the diameter).

2. $h$ is the perpendicular height between the two circular faces, not the slant length along the side.

3. Compute base area $\pi r^2\text{,}$ then multiply by height.

Worked example: A rain barrel has $r=0.4\text{ m}$ and $h=1.2\text{ m}\text{.}$

V=\pi(0.4)^2(1.2)=0.192\pi\approx 0.60\text{ m}^3.

Check: diameter is not radius. A $0.8$ m diameter barrel uses $r=0.4\text{,}$ not $0.8\text{.}$ Replay arithmetic in Volume of a Cylinder.

Round metal rainwater tank beside a garden, same cylinder model as a rain barrel

Cone: $V=\frac{1}{3}\pi r^2 h$

A right cone has a circular base and a single apex directly above the center. Its volume is one-third of a cylinder with the same $r$ and $h\text{.}$

How to do it:

1. Confirm $r$ is the base radius and $h$ is the vertical height to the tip.

2. Compute $\pi r^2 h\text{,}$ then multiply by $\tfrac{1}{3}\text{.}$

3. Compare with the matching cylinder to see the $\tfrac{1}{3}$ factor.

Worked example: A funnel-shaped pour spout is modeled as a cone with $r=3\text{ cm}$ and $h=12\text{ cm}\text{.}$

V=\tfrac{1}{3}\pi(3)^2(12)=36\pi\approx 113\text{ cm}^3.

Check: $h$ is not the slant edge along the funnel wall. Forgetting $\tfrac{1}{3}$ gives a cylinder, three times too large.

Ice cream in a cone, a cone volume model with the same $r$ and vertical $h$

Sphere: $V=\frac{4}{3}\pi r^3$

A sphere is the set of points a fixed radius $r$ from a center: think basketball, marble, or a round propane tank.

How to do it:

1. If a problem gives diameter $d\text{,}$ use $r=\tfrac{d}{2}$ first.

2. Cube the radius, then multiply by $\tfrac{4}{3}\pi\text{.}$

3. Report cubic units.

Worked example: A ball-shaped tank has radius $r=5\text{ ft}\text{.}$

V=\tfrac{4}{3}\pi(5)^3=\tfrac{500}{3}\pi\approx 523.6\text{ ft}^3.

Check: plugging diameter into $r^3$ inflates the volume. Doubling radius multiplies volume by $8\text{,}$ not by $2\text{.}$

Basketball on court, a sphere whose volume depends only on radius $r$

Compare the three

How to do it:

1. Match the solid to its formula before you compute.

2. Cylinder and cone share $r$ and perpendicular $h\text{;}$ the cone is one-third of that cylinder.

3. A sphere uses only $r\text{.}$

Solid	Formula	Key label
Cylinder	$\pi r^2 h$	perpendicular $h$
Cone	$\tfrac{1}{3}\pi r^2 h$	same $r\text{,}$ $h$ as matching cylinder
Sphere	$\tfrac{4}{3}\pi r^3$	only $r$

Worked example: Same $r$ and $h\text{:}$ cone volume is one-third of the cylinder. If you double the radius of a sphere, volume multiplies by $2^3=8\text{.}$

Check: reporting square units instead of cubic units mixes area with volume.

Word problems (rain, pour, tank)

How to do it:

1. Sketch the solid and mark $r$ and $h$ on the picture.

2. Convert diameter to radius when needed.

3. Keep units consistent; leave answers in terms of $\pi$ when asked for exact form. Round at the end if the prompt requests a decimal.

Worked example: A cylindrical silo has diameter $12\text{ m}$ so $r=6\text{ m}\text{,}$ and height $20\text{ m}\text{.}$ Volume $V=\pi(36)(20)=720\pi\text{ m}^3\text{.}$ A conical pile with the same footprint and height $6\text{ m}$ has $V=\tfrac{1}{3}\pi(36)(6)=72\pi\text{ m}^3\text{.}$ The silo is larger.

Check: mixing slant height with vertical height on cones, or using diameter as radius, will blow the answer.

Key points to remember

Cylinder volume is $V=\pi r^2 h$ where $r$ is the base radius and $h$ is the perpendicular height between circular faces.
Cone volume is $V=\frac{1}{3}\pi r^2 h\text{,}$ exactly one-third of a cylinder with the same base radius and height.
Sphere volume is $V=\frac{4}{3}\pi r^3\text{;}$ convert diameter to radius with $r=\frac{d}{2}$ before substituting.
Always express answers in cubic units and label whether you leave $\pi$ in the answer or use a decimal approximation.
Sketching the solid and marking $r$ and $h$ prevents confusing slant height with vertical height on cones.

Practice

Practice what you learned

Apply what you read with the activities below. Use calculators and formula sheets as needed; scored quizzes are in Apply.

Practice for you

8 questions picked from what you need most

Flashcards

12 cards

Practice sets

1 matching · 1 fill-in

Games

Number Cross

Calculator Study guide

Your place saves as you go. Use Back to journey when you are ready to continue the path.

Apply

Apply what you learned

The best check is explaining Volume of Cylinders, Cones & Spheres in your own words. The tutor listens like a curious friend and asks clarifying questions. No score, no grade.

Explain it in your own words

Explain Volume of Cylinders, Cones & Spheres as if to someone who has never heard of it. The tutor responds like a curious learner with clarifying questions, not a score.

Reflection prompts (optional)

Why is cone volume exactly one-third of the cylinder that shares its base and height, and how could you justify that with water-pour experiments?
If two spheres have radii in a $2:3$ ratio, how do their volumes compare? Why does the exponent $3$ matter?
When would a real container be modeled better as a cylinder than as a sphere, even if it looks rounded?

Optional: scored check

Want a scored check? Open it here. The teach-back above does not require finishing it.

The scored check opens up once you have spent a little time in Learn and Practice. Recall sticks best after you have engaged with the material first.

For parents & teachers

Get the free 7-Day Study Reset

A research-backed week of short routines that help a student study less and remember more: spacing, self-quizzing, and a calmer week before tests.

Discover

Think about this

Why is cone volume exactly one-third of the cylinder that shares its base and height, and how could you justify that with water-pour experiments?

Cylinder volume and surface area | Perimeter, area, and volume | Geometry | Khan Academy (8:07)

Learn

Read each section like a formula card: why it matters, the steps, one worked example, then the check. Try the steps yourself before moving to Practice.

Cylinder: $V=\pi r^2 h$

How to do it:

1. $r$ is the radius of the circular base (half the diameter).

2. $h$ is the perpendicular height between the two circular faces, not the slant length along the side.

3. Compute base area $\pi r^2\text{,}$ then multiply by height.

Worked example: A rain barrel has $r=0.4\text{ m}$ and $h=1.2\text{ m}\text{.}$

V=\pi(0.4)^2(1.2)=0.192\pi\approx 0.60\text{ m}^3.

Check: diameter is not radius. A $0.8$ m diameter barrel uses $r=0.4\text{,}$ not $0.8\text{.}$ Replay arithmetic in Volume of a Cylinder.

Cone: $V=\frac{1}{3}\pi r^2 h$

A right cone has a circular base and a single apex directly above the center. Its volume is one-third of a cylinder with the same $r$ and $h\text{.}$

How to do it:

1. Confirm $r$ is the base radius and $h$ is the vertical height to the tip.

2. Compute $\pi r^2 h\text{,}$ then multiply by $\tfrac{1}{3}\text{.}$

3. Compare with the matching cylinder to see the $\tfrac{1}{3}$ factor.

Worked example: A funnel-shaped pour spout is modeled as a cone with $r=3\text{ cm}$ and $h=12\text{ cm}\text{.}$

V=\tfrac{1}{3}\pi(3)^2(12)=36\pi\approx 113\text{ cm}^3.

Check: $h$ is not the slant edge along the funnel wall. Forgetting $\tfrac{1}{3}$ gives a cylinder, three times too large.

Sphere: $V=\frac{4}{3}\pi r^3$

A sphere is the set of points a fixed radius $r$ from a center: think basketball, marble, or a round propane tank.

How to do it:

1. If a problem gives diameter $d\text{,}$ use $r=\tfrac{d}{2}$ first.

2. Cube the radius, then multiply by $\tfrac{4}{3}\pi\text{.}$

3. Report cubic units.

Worked example: A ball-shaped tank has radius $r=5\text{ ft}\text{.}$

V=\tfrac{4}{3}\pi(5)^3=\tfrac{500}{3}\pi\approx 523.6\text{ ft}^3.

Check: plugging diameter into $r^3$ inflates the volume. Doubling radius multiplies volume by $8\text{,}$ not by $2\text{.}$

Compare the three

How to do it:

1. Match the solid to its formula before you compute.

2. Cylinder and cone share $r$ and perpendicular $h\text{;}$ the cone is one-third of that cylinder.

3. A sphere uses only $r\text{.}$

Solid	Formula	Key label
Cylinder	$\pi r^2 h$	perpendicular $h$
Cone	$\tfrac{1}{3}\pi r^2 h$	same $r\text{,}$ $h$ as matching cylinder
Sphere	$\tfrac{4}{3}\pi r^3$	only $r$

Worked example: Same $r$ and $h\text{:}$ cone volume is one-third of the cylinder. If you double the radius of a sphere, volume multiplies by $2^3=8\text{.}$

Check: reporting square units instead of cubic units mixes area with volume.

Word problems (rain, pour, tank)

How to do it:

1. Sketch the solid and mark $r$ and $h$ on the picture.

2. Convert diameter to radius when needed.

3. Keep units consistent; leave answers in terms of $\pi$ when asked for exact form. Round at the end if the prompt requests a decimal.

Check: mixing slant height with vertical height on cones, or using diameter as radius, will blow the answer.

Key points to remember

Cylinder volume is $V=\pi r^2 h$ where $r$ is the base radius and $h$ is the perpendicular height between circular faces.
Cone volume is $V=\frac{1}{3}\pi r^2 h\text{,}$ exactly one-third of a cylinder with the same base radius and height.
Sphere volume is $V=\frac{4}{3}\pi r^3\text{;}$ convert diameter to radius with $r=\frac{d}{2}$ before substituting.
Always express answers in cubic units and label whether you leave $\pi$ in the answer or use a decimal approximation.
Sketching the solid and marking $r$ and $h$ prevents confusing slant height with vertical height on cones.

Practice

Practice what you learned

Apply what you read with the activities below. Use calculators and formula sheets as needed; scored quizzes are in Apply.

Practice for you

8 questions picked from what you need most

Flashcards

12 cards

Practice sets

1 matching · 1 fill-in

Games

Number Cross

Calculator Study guide

Your place saves as you go. Use Back to journey when you are ready to continue the path.

Apply

Apply what you learned

The best check is explaining Volume of Cylinders, Cones & Spheres in your own words. The tutor listens like a curious friend and asks clarifying questions. No score, no grade.

Explain it in your own words

Explain Volume of Cylinders, Cones & Spheres as if to someone who has never heard of it. The tutor responds like a curious learner with clarifying questions, not a score.

Reflection prompts (optional)

Why is cone volume exactly one-third of the cylinder that shares its base and height, and how could you justify that with water-pour experiments?
If two spheres have radii in a $2:3$ ratio, how do their volumes compare? Why does the exponent $3$ matter?
When would a real container be modeled better as a cylinder than as a sphere, even if it looks rounded?

Optional: scored check

Want a scored check? Open it here. The teach-back above does not require finishing it.

The scored check opens up once you have spent a little time in Learn and Practice. Recall sticks best after you have engaged with the material first.

Volume of Cylinders, Cones & Spheres

What you’ll learn

Volume of Cylinders, Cones & Spheres

What you’ll learn

Where you are

Units

Get the free 7-Day Study Reset

Where you are

Units

Learning acts

Cylinder: V=πr2hV=\pi r^2 hV=πr2h

Cone: V=13πr2hV=\frac{1}{3}\pi r^2 hV=31​πr2h

Sphere: V=43πr3V=\frac{4}{3}\pi r^3V=34​πr3

Compare the three

Word problems (rain, pour, tank)

Key points to remember

Practice what you learned

Practice for you

Flashcards

Practice sets

Games

Apply what you learned

Explain it in your own words

Get the free 7-Day Study Reset

Learning acts

Cylinder: V=πr2hV=\pi r^2 hV=πr2h

Cone: V=13πr2hV=\frac{1}{3}\pi r^2 hV=31​πr2h

Sphere: V=43πr3V=\frac{4}{3}\pi r^3V=34​πr3

Compare the three

Word problems (rain, pour, tank)

Key points to remember

Practice what you learned

Practice for you

Flashcards

Practice sets

Games

Apply what you learned

Explain it in your own words

Cylinder: $V=\pi r^2 h$

Cone: $V=\frac{1}{3}\pi r^2 h$

Sphere: $V=\frac{4}{3}\pi r^3$

Cylinder: $V=\pi r^2 h$

Cone: $V=\frac{1}{3}\pi r^2 h$

Sphere: $V=\frac{4}{3}\pi r^3$