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
Use this cone volume calculator for traffic cones, funnels, party hats, cone-shaped piles, tapered containers, and geometry homework problems. Enter the base radius and vertical height, choose the unit, and the page shows volume, base area, diameter, slant height, surface area, liters, gallons, and calculation steps.
If the problem gives the full width of the base, divide it by 2 to get the radius. If it gives slant height, pause there: cone volume uses vertical height, not the diagonal side length.
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.
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 and Frustum Volume
A truncated cone, also called a conical frustum, is a cone with the top cut off. It does not use the simple cone formula because it has two circular faces with different radii.
If your shape has a top radius and bottom radius, use the Frustum Volume Calculator instead.
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.
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 height and pi, then divide by 3.
Can I calculate cone volume from diameter?
Yes. Divide the diameter by 2 to get radius, then use V = (1/3)πr²h.
Do I use slant height for cone volume?
No. Cone volume uses vertical height. If only slant height is given, calculate vertical height first using h = √(l² - r²).
What is the difference between cone volume and cone surface area?
Volume measures the space inside the cone. Surface area measures the outside covering of the cone, including the base and curved side.
How do I calculate the volume of a truncated cone?
Use the frustum formula V = (1/3)πh(R² + Rr + r²), where R and r are the two radii.