On This Page
- How to use the Cone Volume Calculator
- Cone volume formula
- How to calculate the volume of a cone
- Diameter, radius, and slant height
- Reverse cone volume calculations
- Worked examples
- Truncated cone and frustum volume
- Composite cone problems
- Frequently asked questions
How to Use the Cone Volume Calculator
Enter the base radius and vertical height. The calculator shows volume, slant height, surface area, and liquid conversions. Before you type anything in, check which height your problem gives you, cone volume uses the perpendicular distance from base center to apex (vertical height), not the diagonal distance along the side (slant height). If your problem gives slant height l and radius r, find vertical height first with h = √(l² − r²).
For composite shapes such as an ice cream cone with a hemisphere scoop, or a silo that is a cylinder topped with a cone, calculate each part separately and add the volumes.
Cone Volume Formula
The standard formula for calculating cone volume is:
- V = volume
- π ≈ 3.14159
- r = radius of the circular base
- h = vertical height from the base center to the apex
A cone has one-third the volume of a cylinder with the same base radius and vertical height. That is why the formula includes the factor 1/3. See Why a Cone Is Exactly One-Third of a Cylinder for the full proof and intuition, including why the same 1/3 factor shows up in the Pyramid Volume Calculator for an identical geometric reason.
How to Calculate the Volume of a Cone
To calculate the volume of a cone by hand, identify the radius and vertical height first. Make sure both measurements use the same unit before multiplying.
Step 1: Find the Radius and Vertical Height
The radius is half the diameter of the circular base. The height is the straight vertical distance from the base center to the apex.
Step 2: Square the Radius
Multiply the radius by itself. For example, if r = 4, then r² = 16.
Step 3: Multiply by Height and Pi
Multiply r² by the height, then multiply by π. This gives the matching cylinder volume.
Step 4: Divide by 3
A cone is one-third of the matching cylinder, so divide the result by 3.
- Find radius: r
- Find vertical height: h
- Square the radius: r²
- Multiply by height and π
- Divide by 3
- Write the answer in cubic units
Diameter, Radius, and Slant Height
Cone volume problems often give different measurements. Radius and vertical height go directly into the formula. Diameter and slant height need extra care.
Cone Volume From Diameter
If a cone has diameter instead of radius, divide the diameter by 2 first.
Slant Height Is Not the Same as Height
Slant height is the diagonal distance along the side of the cone. Cone volume uses vertical height. If you know radius and slant height, use the Pythagorean theorem to find vertical height.
Oblique Cone Volume
An oblique cone leans to one side, but the volume formula is still V = (1/3)πr²h. The key is using the perpendicular height, not the slanted side.
Reverse Cone Volume Calculations
Sometimes you know the cone volume and need to find a missing dimension. Rearranging the formula lets you solve for height or radius.
Find Height From Volume and Radius
Find Radius From Volume and Height
Worked Cone Volume Examples
Cone problems often come down to small choices: radius instead of diameter, vertical height instead of slant height, and the final answer in cubic units.
Example 1: Traffic Cone Volume
Problem: A traffic cone has a circular base with radius 4 inches and a vertical height of 18 inches. Estimate the volume of the cone. Use 3.14 for π.
- Radius = 4 in, height = 18 in
- Use V = (1/3)πr²h
- r² = 4² = 16
- V = (1/3) × 3.14 × 16 × 18
- V = 301.44 in³
Answer: The traffic cone volume is about 301.4 cubic inches.
Example 2: Party Hat Volume From Diameter
Problem: A party hat is shaped like a cone. The base diameter is 10 cm and the vertical height is 24 cm. Find the volume of the hat in cubic centimeters.
- Diameter = 10 cm, so radius = 5 cm
- Height = 24 cm
- V = (1/3)π(5²)(24)
- V = 200π cm³
- Decimal approximation: 200 × 3.14159 ≈ 628.3 cm³
Answer: The exact volume is 200π cm³, or about 628.3 cm³.
Example 3: Cone Volume From Slant Height
Problem: A cone has a radius of 6 cm and a slant height of 10 cm. Find its volume. First calculate the vertical height.
- Use h = √(l² - r²)
- h = √(10² - 6²) = √(100 - 36) = √64 = 8 cm
- Use V = (1/3)πr²h
- V = (1/3)π(6²)(8) = 96π cm³
- Decimal approximation: 96 × 3.14159 ≈ 301.6 cm³
Answer: The cone volume is 96π cm³, or about 301.6 cm³.
Example 4: Find Missing Radius From Volume
Problem: A cone has a volume of 192π cubic inches and a height of 16 inches. Find the radius of the cone.
- Use r = √(3V ÷ πh)
- Substitute V = 192π and h = 16
- r = √((3 × 192π) ÷ (π × 16))
- r = √(576 ÷ 16) = √36
- r = 6 inches
Answer: The radius is 6 inches.
Example 5: Ice Cream Cone With a Hemisphere Scoop
Problem: An ice cream dessert has a cone with radius 3 cm and height 9 cm, plus a hemispherical scoop with the same radius. Find the total volume of the cone and scoop.
- Cone volume = (1/3)π(3²)(9) = 27π cm³
- Hemisphere volume = (2/3)π(3³) = 18π cm³
- Total volume = 27π + 18π = 45π cm³
- Decimal approximation: 45 × 3.14159 ≈ 141.4 cm³
Answer: The total volume is 45π cm³, or about 141.4 cm³.
Truncated Cone Volume (Frustum)
A truncated cone, also called a conical frustum, is a cone with the top cut off parallel to the base. It has two circular faces with different radii, so the simple cone formula does not apply. The formula for truncated cone volume is:
Where R is the larger radius, r is the smaller radius, and h is the vertical height between the two faces. Buckets, plant pots, and paper cups are common truncated cone shapes. If your shape has both a top radius and a bottom radius, use the Frustum Volume Calculator instead of this page.
Composite Cone Problems
Some textbook problems combine a cone with another shape, such as a hemisphere scoop on an ice cream cone. In those cases, calculate each part separately and add the volumes.
Ice Cream Cone Made From a Cone and a Hemisphere
Problem: An ice cream cone has a cone-shaped base with radius 3 inches and height 8 inches. A hemispherical scoop with radius 3 inches sits on top. Find the total volume.
- Cone volume = (1/3)π × 3² × 8 = 24π in³
- Hemisphere volume = (2/3)π × 3³ = 18π in³
- Total volume = 24π + 18π = 42π in³
- Decimal approximation: 42 × 3.14159 ≈ 131.9 in³
Answer: The combined volume is 42π cubic inches, or about 131.9 cubic inches.
Filling a Cone With a Hose
Problem: A hose fills a cone at 1,413 cm³ per minute. The cone has radius 60 cm and height 150 cm. Estimate how long it takes to fill the cone.
- Cone volume = (1/3) × 3.14 × 60² × 150
- Volume = 565,200 cm³
- Time = 565,200 ÷ 1,413 = 400 minutes
Answer: It takes about 400 minutes to fill the cone at that rate.
References
- Cone, Wikipedia: geometric definition, volume formula derivation, oblique cone, and frustum relationship.
- Cone, Wolfram MathWorld: formal mathematical treatment including slant height, lateral surface area, and volume proof.
- Cone, Math is Fun: approachable explanation of the cone volume formula with diagrams and examples.
Cone Volume Units and Capacity
Cone volume is written in cubic units such as cm³, m³, in³, or ft³. If the cone is a funnel or container, cubic volume can be converted to liters or gallons.
| Input Unit | Volume Unit | Useful Conversion |
|---|---|---|
| Centimeters | cm³ | 1,000 cm³ = 1 liter |
| Meters | m³ | 1 m³ = 1,000 liters |
| Inches | in³ | 231 in³ = 1 US gallon |
| Feet | ft³ | 1 ft³ ≈ 7.48052 US gallons |
Cone Volume Mistakes to Avoid
- Use vertical height: Cone volume uses straight vertical height, not slant height.
- Do not forget one-third: A cone is one-third of the same-base cylinder.
- Radius vs. diameter: If you are given the full width of the base, divide by 2 to get radius.
- Use matching units: Convert all dimensions before calculating.
- Do not use frustum formula for a full cone: A full cone has one base and one apex. A frustum has two circular faces.
Frequently Asked Questions
How do you calculate the volume of a cone?
Use V = (1/3)πr²h. Square the radius, multiply by the height and pi, then divide by 3. For a cone with radius 4 in and height 9 in: V = (1/3) × 3.14159 × 16 × 9 ≈ 150.8 in³.
Do I use slant height or vertical height for cone volume?
Always use vertical height, the straight perpendicular distance from the center of the base to the apex. If your problem only gives slant height l and radius r, find vertical height first with h = √(l² − r²). Slant height appears in surface area calculations, not volume.
Can I calculate cone volume from diameter?
Yes. Divide the base diameter by 2 to get the radius, then apply V = (1/3)πr²h. A cone with base diameter 10 cm has radius 5 cm.
Why does a cone hold exactly one-third of the same-sized cylinder?
A cone tapers from a full circular base to a single point, so each horizontal cross-section becomes smaller toward the apex. When you sum all those shrinking circles, the total is exactly one-third of the cylinder. You can demonstrate it physically: fill a cone-shaped cup with water and pour it into a matching cylinder, you need exactly three pours to fill it.
How do I calculate the volume of a truncated cone?
Use the frustum formula: V = (1/3)πh(R² + Rr + r²), where R is the larger radius, r is the smaller radius, and h is the vertical height. The Frustum Volume Calculator handles this with both a top radius and bottom radius input.
What common objects are cone-shaped?
Traffic cones, party hats, funnels, waffle cones, and ice cream cones are close enough to true cones for volume estimates. Sand, salt, or grain piled into a conical mound in a bin or stockpile is another practical case, volume gives you how much material is there before moving it.