Calculate cone volume instantly with our free calculator. Enter radius and height for full cones or truncated cones (frustums). Get accurate results in seconds for engineering, architecture, and education projects.
A cone volume calculator is an essential mathematical tool that instantly computes the volume of both full cones and truncated cones (frustums) with precision. Whether you're working in engineering, architecture, education, or need to calculate the volume of a cone for manufacturing, this free online tool provides accurate results in seconds for any cone dimensions you input.
A cone is a three-dimensional geometric shape featuring a circular base that tapers smoothly to a single point called the apex. A truncated cone (or frustum) is created when the top portion of a cone is removed by cutting parallel to the base, leaving a shape with two circular faces of different sizes.
Follow these simple steps to calculate cone volume:
The volume (V) of a full cone is given by the formula:
Where:
The volume (V) of a truncated cone is calculated using the formula:
Where:
The calculator performs the following steps to compute the volume:
For a full cone: a. Square the radius (r^2) b. Multiply by pi (π) c. Multiply by the height (h) d. Divide the result by 3
For a truncated cone: a. Square both radii (R^2 and r^2) b. Calculate the product of the radii (Rr) c. Sum the results of steps a and b d. Multiply by pi (π) e. Multiply by the height (h) f. Divide the result by 3
The calculator uses double-precision floating-point arithmetic to ensure accuracy.
Cone volume calculations have numerous practical applications across various industries:
While cone volume is crucial for conical shapes, there are other related measurements that might be more appropriate in certain situations:
Cylinder Volume: For cylindrical objects without tapering.
Pyramid Volume: For objects with a polygonal base that tapers to a point.
Sphere Volume: For perfectly round objects.
Surface Area: When the outer surface of the cone is more relevant than its volume.
The concept of cone volume calculation dates back to ancient civilizations. The ancient Egyptians and Babylonians had some understanding of conical volumes, but it was the ancient Greeks who made significant advancements in this area.
Democritus (c. 460-370 BCE) is credited with first determining that the volume of a cone is one-third the volume of a cylinder with the same base and height. However, it was Eudoxus of Cnidus (c. 408-355 BCE) who provided the first rigorous proof of this relationship using the method of exhaustion.
Archimedes (c. 287-212 BCE) later refined and extended these concepts in his work "On Conoids and Spheroids," where he also addressed the volumes of truncated cones.
In the modern era, the development of calculus by Newton and Leibniz in the 17th century provided new tools for understanding and calculating cone volumes, leading to the formulas we use today.
Here are some code examples to calculate the volume of cones:
1import math
2
3def cone_volume(radius, height):
4 return (1/3) * math.pi * radius**2 * height
5
6def truncated_cone_volume(radius1, radius2, height):
7 return (1/3) * math.pi * height * (radius1**2 + radius2**2 + radius1*radius2)
8
9## Example usage:
10full_cone_volume = cone_volume(3, 4)
11truncated_cone_volume = truncated_cone_volume(3, 2, 4)
12
13print(f"Full Cone Volume: {full_cone_volume:.2f} cubic units")
14print(f"Truncated Cone Volume: {truncated_cone_volume:.2f} cubic units")
151function coneVolume(radius, height) {
2 return (1/3) * Math.PI * Math.pow(radius, 2) * height;
3}
4
5function truncatedConeVolume(radius1, radius2, height) {
6 return (1/3) * Math.PI * height * (Math.pow(radius1, 2) + Math.pow(radius2, 2) + radius1 * radius2);
7}
8
9// Example usage:
10const fullConeVolume = coneVolume(3, 4);
11const truncatedConeVolume = truncatedConeVolume(3, 2, 4);
12
13console.log(`Full Cone Volume: ${fullConeVolume.toFixed(2)} cubic units`);
14console.log(`Truncated Cone Volume: ${truncatedConeVolume.toFixed(2)} cubic units`);
151public class ConeVolumeCalculator {
2 public static double coneVolume(double radius, double height) {
3 return (1.0/3.0) * Math.PI * Math.pow(radius, 2) * height;
4 }
5
6 public static double truncatedConeVolume(double radius1, double radius2, double height) {
7 return (1.0/3.0) * Math.PI * height * (Math.pow(radius1, 2) + Math.pow(radius2, 2) + radius1 * radius2);
8 }
9
10 public static void main(String[] args) {
11 double fullConeVolume = coneVolume(3, 4);
12 double truncatedConeVolume = truncatedConeVolume(3, 2, 4);
13
14 System.out.printf("Full Cone Volume: %.2f cubic units%n", fullConeVolume);
15 System.out.printf("Truncated Cone Volume: %.2f cubic units%n", truncatedConeVolume);
16 }
17}
18Full Cone:
Truncated Cone:
Edge Case: Zero Radius
Edge Case: Truncated Height Equals Full Height
To calculate cone volume, use the formula V = (1/3)πr²h, where r is the base radius and h is the height. Simply multiply π (3.14159) by the square of the radius, then by the height, and divide by 3. For example, a cone with radius 5cm and height 10cm has volume: (1/3) × π × 5² × 10 = 261.8 cubic centimeters.
The cone volume formula is V = (1/3)πr²h for a full cone, where V is volume, r is the radius of the circular base, and h is the perpendicular height from base to apex. For a truncated cone (frustum), the formula is V = (1/3)πh(R² + r² + Rr), where R and r are the two base radii.
A full cone has one circular base and tapers to a point, while a truncated cone (frustum) has two parallel circular bases of different sizes. The truncated cone formula accounts for both radii: V = (1/3)πh(R² + r² + Rr), making it larger than a full cone with the same height.
If you have the diameter instead of radius, simply divide the diameter by 2 to get the radius, then use the standard formula V = (1/3)πr²h. For example, if diameter = 10cm, then radius = 5cm.
Yes, the cone volume calculator accepts decimal values for radius and height measurements, providing precise calculations for any real-world application requiring fractional dimensions.
The calculator works with any unit of measurement (inches, centimeters, meters, feet, etc.). The resulting volume will be in cubic units matching your input measurements (e.g., cm input gives cm³ output).
To calculate truncated cone volume, use the formula V = (1/3)πh(R² + r² + Rr), where R is the lower base radius, r is the upper base radius, and h is the height. Enter both radii and the height in our calculator for instant results.
Our cone volume calculator uses double-precision floating-point arithmetic, ensuring high accuracy (up to 15 decimal places) for both small and large dimensional values.
A right circular cone (where the apex is directly above the center of the base) uses the standard formula V = (1/3)πr²h. This is the most common type of cone and what our calculator computes by default.
Absolutely! The cone volume calculator is perfect for determining ice cream cone volumes, helping food manufacturers and consumers understand serving sizes for waffle cones, sugar cones, and cake cones.
If you input zero for either radius or height, the cone volume calculator will correctly return a volume of zero cubic units, since a cone with no dimension has no volume.
After calculating cone volume in cubic units, convert to liters or gallons: 1 cubic meter = 1000 liters = 264.17 gallons; 1 cubic foot = 28.32 liters = 7.48 gallons; 1 cubic inch = 0.0164 liters.
Ready to use our cone volume calculator? Simply input your cone dimensions above and get instant, accurate results for any cone volume calculation. Whether you're working on engineering projects, educational assignments, or everyday calculations, our tool provides the precision you need.
Meta Title: Cone Volume Calculator - Free Cone & Frustum Volume Tool Meta Description: Calculate cone volume instantly with our free calculator. Enter radius and height for full cones or truncated cones (frustums). Get accurate results in seconds for any project.
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