Cone Volume Calculator
This calculator finds the volume, slant height, base area, lateral surface area, and total surface area of a right circular cone based on its radius and height.
Relation to "Volume of a Circle": A cone has a circular base. Its volume calculation uses the area of this circle (π * r²). While a 2D circle doesn't have volume, understanding its area is key to finding the volume of cone shapes.
Enter the radius (r) of the base and the perpendicular height (h) of the cone below to calculate its properties. The correct cone volume formula and other relevant equations are used.
Enter Cone Dimensions
Understanding Cone Volume & Formulas
What is a Cone?
A cone is a three-dimensional shape that tapers smoothly from its typically circular base to a common point called the apex (or vertex). Mathematically, a cone is formed similarly to a circle, by a set of line segments connected to a common center point, except that the center point is not included in the plane that contains the circle. This calculator considers a finite right circular cone, where the apex is directly above the center of the base.
The Cone Volume Formula
The formula for cone volume is one-third of the product of its base area and its height. Since the base is a circle (Area = π * r²), the formula is:
V = (1/3) * Base Area * Height = (1/3) * π * r² * h
Where:
- V is the Volume
- π (Pi) is approximately 3.14159...
- r is the radius of the circular base
- h is the perpendicular height from the base to the apex
This cone volume equation calculates the 3D space inside the cone. It's exactly one-third the volume of a cylinder with the same radius and height.
Other Cone Formulas
- Slant Height (l): The distance from the apex to any point on the edge of the circular base. Found using the Pythagorean theorem:
l = √(r² + h²) - Base Area (BA): The area of the circular base.
BA = π * r² - Lateral Surface Area (LSA): The area of the curved side surface only.
LSA = π * r * l = π * r * √(r² + h²) - Total Surface Area (TSA): The sum of the base area and the lateral surface area.
TSA = BA + LSA = π * r² + π * r * l
Example Calculation (Provided in Original Text)
EX: An ice cream waffle cone has a circular base radius (r) of 1.5 inches and a height (h) of 5 inches. Calculate the volume:
V = (1/3) * π * r² * h = (1/3) * π * (1.5)² * 5 ≈ (1/3) * π * 2.25 * 5 ≈ 11.78 cubic inches (in³).
Real-Life Cone Volume Examples
Click on an example to see the step-by-step calculation:
Example 1: Ice Cream Sugar Cone
Scenario: Calculate the volume of a typical sugar cone.
1. Known Values: Approx. Radius (r) = 1 inch, Height (h) = 4 inches.
2. Formula: V = (1/3) * π * r² * h
3. Calculation: V = (1/3) * π * (1)² * 4 ≈ (1/3) * π * 1 * 4 ≈ 1.333π
4. Result: V ≈ 4.19 cubic inches.
Conclusion: A typical sugar cone can hold about 4.2 cubic inches of ice cream (if filled level).
Example 2: Traffic Cone
Scenario: Find the volume occupied by a standard orange traffic cone (ignoring the square base it often sits on).
1. Known Values: Approx. Base Radius (r) = 7 inches, Height (h) = 28 inches.
2. Formula: V = (1/3) * π * r² * h
3. Calculation: V = (1/3) * π * (7)² * 28 ≈ (1/3) * π * 49 * 28 ≈ 457.33π
4. Result: V ≈ 1436.76 cubic inches.
Conclusion: The conical part of a large traffic cone has a volume of roughly 1437 cubic inches.
Example 3: Party Hat
Scenario: Calculate the volume inside a conical party hat.
1. Known Values: Approx. Radius (r) = 3 inches, Height (h) = 7 inches.
2. Formula: V = (1/3) * π * r² * h
3. Calculation: V = (1/3) * π * (3)² * 7 ≈ (1/3) * π * 9 * 7 = 21π
4. Result: V ≈ 65.97 cubic inches.
Conclusion: A standard party hat encloses about 66 cubic inches of space.
Example 4: Funnel Volume (Conical Part)
Scenario: Estimate the volume of the main conical section of a kitchen funnel.
1. Known Values: Approx. Base (top) Radius (r) = 5 cm, Height (h) = 10 cm (of the cone part).
2. Formula: V = (1/3) * π * r² * h
3. Calculation: V = (1/3) * π * (5)² * 10 ≈ (1/3) * π * 25 * 10 ≈ 83.33π
4. Result: V ≈ 261.8 cubic cm (or 261.8 mL).
Conclusion: The conical part of the funnel holds about 262 mL.
Example 5: Tepee Volume (Approximate)
Scenario: Estimate the internal volume of a tepee, approximating it as a cone.
1. Known Values: Approx. Base Radius (r) = 6 feet, Height (h) = 12 feet.
2. Formula: V = (1/3) * π * r² * h
3. Calculation: V = (1/3) * π * (6)² * 12 ≈ (1/3) * π * 36 * 12 = 144π
4. Result: V ≈ 452.39 cubic feet.
Conclusion: The tepee has an approximate internal volume of 452 cubic feet.
Example 6: Volcano Shape (Approximate)
Scenario: Estimate the volume of Mount Fuji, approximating its shape as a cone.
1. Known Values: Approx. Base Radius (r) ≈ 18 km, Height (h) ≈ 3.8 km.
2. Formula: V = (1/3) * π * r² * h
3. Calculation: V = (1/3) * π * (18)² * 3.8 ≈ (1/3) * π * 324 * 3.8 ≈ 410.4π
4. Result: V ≈ 1289.3 cubic kilometers.
Conclusion: Approximating as a cone, Mount Fuji's volume is roughly 1290 cubic kilometers.
Example 7: Conical Roof Section
Scenario: Find the volume under a conical roof section of a building.
1. Known Values: Radius (r) = 10 feet, Vertical Height (h) = 5 feet.
2. Formula: V = (1/3) * π * r² * h
3. Calculation: V = (1/3) * π * (10)² * 5 ≈ (1/3) * π * 100 * 5 ≈ 166.67π
4. Result: V ≈ 523.6 cubic feet.
Conclusion: The space under the conical roof is about 524 cubic feet.
Example 8: Disposable Paper Water Cup
Scenario: Estimate the volume of a standard conical paper water cup.
1. Known Values: Approx. Top Radius (r) ≈ 4 cm, Height (h) ≈ 9 cm.
2. Formula: V = (1/3) * π * r² * h
3. Calculation: V = (1/3) * π * (4)² * 9 ≈ (1/3) * π * 16 * 9 = 48π
4. Result: V ≈ 150.8 cubic cm (or 150.8 mL).
Conclusion: The paper cup holds approximately 150 mL.
Example 9: Sharpened Pencil Tip (Approximate)
Scenario: Estimate the volume of the conical tip of a sharpened pencil.
1. Known Values: Approx. Radius (r) ≈ 0.3 cm (base of cone), Height (h) ≈ 1 cm.
2. Formula: V = (1/3) * π * r² * h
3. Calculation: V = (1/3) * π * (0.3)² * 1 ≈ (1/3) * π * 0.09 * 1 = 0.03π
4. Result: V ≈ 0.094 cubic cm.
Conclusion: The pencil tip has a very small volume, less than 0.1 cubic cm.
Example 10: Pile of Sand/Gravel
Scenario: A pile of gravel forms a natural cone shape.
1. Known Values: Measured Base Radius (r) = 2 meters, Height (h) = 1 meter.
2. Formula: V = (1/3) * π * r² * h
3. Calculation: V = (1/3) * π * (2)² * 1 ≈ (1/3) * π * 4 * 1 ≈ 1.333π
4. Result: V ≈ 4.19 cubic meters.
Conclusion: The pile contains about 4.2 cubic meters of gravel.
Understanding Volume Measurement
Volume is the quantification of the three-dimensional space...
Common Volume Units Reference
Ensure your input radius and height use a consistent unit...
Frequently Asked Questions about Cone Volume
1. What is the formula for volume of a cone?
The volume (V) is V = (1/3) * π * r² * h, where 'π' is Pi, 'r' is the base radius, and 'h' is the perpendicular height.
2. How is cone volume related to cylinder volume?
A cone's volume is exactly one-third (1/3) the volume of a cylinder that has the same base radius and the same height.
3. What is Slant Height (l) and how is it calculated?
Slant height is the distance from the tip (apex) of the cone down the side to the edge of the base. It forms the hypotenuse of a right triangle with the radius (r) and height (h). It's calculated using the Pythagorean theorem: l = √(r² + h²).
4. What's the difference between Lateral Surface Area and Total Surface Area?
Lateral Surface Area (LSA = πrl) is the area of the sloping side surface only. Total Surface Area (TSA = πr² + πrl) includes the area of the circular base plus the lateral surface area.
5. Does this calculator work for oblique cones (where the tip isn't centered)?
No, this calculator assumes a 'right circular cone' where the apex is directly above the center of the base. The volume formula (V = 1/3 * Base Area * Height) still works for oblique cones if 'h' is the *perpendicular* height, but the surface area formulas change.
6. How do I calculate the volume if I only know the slant height (l) and radius (r)?
You first need to find the perpendicular height (h) using the Pythagorean theorem rearranged: h = √(l² - r²). Then use this 'h' in the volume formula V = (1/3) * π * r² * h.
7. Why is the cone volume equation V = (1/3)πr²h? Where does the 1/3 come from?
It arises from calculus (integration) when summing the volumes of infinitesimally thin circular disks stacked to form the cone. Empirically, it was also found by comparing cone and cylinder volumes (it takes 3 cones to fill a cylinder of the same base/height).
8. What units will the results be in?
If radius and height are in inches, slant height will be in inches, areas (Base, Lateral, Total) in square inches (in²), and volume in cubic inches (in³). Maintain consistent input units.
9. What if the top of my cone is cut off (like a bucket)?
That shape is called a 'conical frustum'. It requires a different formula involving both the top and bottom radii and the height of the frustum section.
10. Can the radius or height be zero?
For a valid cone, both radius and height must be positive numbers (greater than zero). If either is zero, the volume is zero.
11. How do I find the volume of just the curved part, not the whole cone?
Volume refers to the space *inside*. If you mean the area of the curved surface, that's the Lateral Surface Area (LSA = πrl), which this calculator provides.
