Volume of a Cylinder, Cone, and Sphere, With Examples

Cylinders, cones, and spheres show up together in grade 8 because their volume formulas are closely related. The Common Core grade 8 geometry standards ask students to "know the formulas for the volumes of cones, cylinders, and spheres and use them to solve real-world and mathematical problems." All three are worked below with the same radius for comparison, then converted to liters.

1. The three formulas

Shape	Formula	What you need
Cylinder	V = πr²h	radius r and height h
Cone	V = (1/3)πr²h	radius r and height h
Sphere	V = (4/3)πr³	radius r only

Two details matter in every problem:

• r is the radius, half of the diameter. Many problems give the diameter on purpose.
• Volume is in cubic units, such as cm³ or m³, because three lengths are multiplied together.

2. Example 1: a cylinder

A cylinder has radius 4 cm and height 10 cm.

1. Square the radius: 4² = 16
2. Multiply by the height: 16 · 10 = 160
3. Multiply by π: V = 160π cm³

As a decimal, 160π is about 502.65 cm³.

Many teachers want the exact answer, 160π, unless the problem says to round. If you use 3.14 for π, you get 502.4 cm³. That is close but not identical, so use whatever value of π the problem tells you to use.

3. Example 2: a cone with the same radius and height

A cone has radius 4 cm and height 10 cm.

V = (1/3) · π · 4² · 10 = 160π / 3, which is about 167.55 cm³.

That is exactly one third of the cylinder in Example 1. A cone always holds one third of what a cylinder with the same base and height holds, which is an easy way to remember the 1/3 in the formula.

Height, not slant height. The h in the cone formula is the straight-up height from the tip to the center of the base. If a problem gives the slant height (the length along the side), use the Pythagorean theorem to find the true height first.

4. Example 3: a sphere with the same radius

A sphere has radius 4 cm.

1. Cube the radius: 4³ = 64
2. Multiply by 4/3: 64 · 4/3 = 256/3
3. Multiply by π: V = 256π / 3 cm³

As a decimal, that is about 268.08 cm³.

How the sphere compares to a cylinder. Picture a cylinder that fits the sphere snugly: radius 4 cm and height 8 cm (the sphere's diameter). That cylinder has volume π · 16 · 8 = 128π cm³. The sphere's 256π/3 is exactly two thirds of 128π. So a sphere fills two thirds of the cylinder that wraps around it, and a cone with that same base and height fills one third.

5. Summary for radius 4 cm

Shape	Exact volume	Approximate volume
Cylinder, h = 10 cm	160π cm³	502.65 cm³
Cone, h = 10 cm	160π/3 cm³	167.55 cm³
Sphere	256π/3 cm³	268.08 cm³

6. Common mistakes

Using the diameter as the radius. If the cylinder in Example 1 were described as "8 cm across" and you used 8 as r, you would get π · 64 · 10 = 640π, about 2,010.62 cm³. That is four times too large, because the radius is squared.

Squaring instead of cubing for a sphere. The sphere formula uses r³. Because the radius is cubed, doubling the radius makes a sphere 8 times larger in volume, not 2 times.

Forgetting the 1/3 for a cone. If your cone answer equals the cylinder answer for the same radius and height, the 1/3 is missing.

Square units on a volume. Area answers use square units. Volume answers always use cubic units.

7. Turning cubic units into liters

Volume problems about tanks, cans, or bottles often ask for liters. NIST, the US agency that maintains measurement standards, gives the key equivalences:

• 1 cm³ = 1 milliliter (mL)
• 1,000 cm³ = 1 liter (L)
• 1,000 L = 1 m³

So the cylinder in Example 1 holds about 502.65 mL, or about 0.50 L.

Example 4: a water tank. A cylindrical tank has radius 0.5 m and height 1.2 m.

1. V = π · 0.5² · 1.2 = π · 0.25 · 1.2 = 0.3π m³
2. 0.3π is about 0.942 m³
3. Convert using 1 m³ = 1,000 L: about 942 L

NIST also points out a convenient fact about the metric system: volume is defined the same way for solids, liquids, and gases, so cubic meters and liters work for all of them. In US customary units, by contrast, dry and liquid volumes use different units, such as bushels and gallons.

8. A quick checklist

1. Is the given length a radius or a diameter? Halve the diameter.
2. Is the cone's height the vertical height, not the slant height?
3. Did you square r for a cylinder or cone, and cube r for a sphere?
4. Did you include the 1/3 or 4/3?
5. Are the units cubic, and does the problem want liters?

9. Key takeaways

• Cylinder: V = πr²h. Cone: V = (1/3)πr²h. Sphere: V = (4/3)πr³.
• A cone holds one third of a cylinder with the same base and height, and a sphere fills two thirds of the cylinder that just contains it.
• Always use the radius, and remember the radius is squared or cubed, so an error there grows fast.
• Volume uses cubic units; 1 cm³ is 1 mL, and 1 m³ is 1,000 L.