Cone Volume Calculator
Calculate the volume and surface area of a cone.
How to find the volume of a cone
A cone holds exactly one-third of the cylinder that would surround it with the same base and height. So its volume is one-third of pi times the radius squared times the height. The slant height, the distance up the sloping side, comes from the Pythagorean theorem and is useful for surface area.
$$V = \frac{1}{3}\pi r^2 h \qquad l = \sqrt{r^2 + h^2}$$
How to use this calculator
- Enter the radius of the circular base.
- Enter the vertical height from the base to the tip.
- Press Calculate to get the volume.
Worked example
A cone with radius 3 and height 4 has a volume of one-third times pi times 3 squared times 4, which is one-third times pi times 36, about 37.7 cubic units. That is exactly one-third of a cylinder of the same size, which would hold about 113.1.
How to use this calculator
- Enter the cone's radius and height into the calculator.
- Choose whether you want volume, surface area, or both.
- Review the result and check the units to match your input.
The formula explained
$$ V = \frac{1}{3}\pi r^2 h, \quad A = \pi r^2 + \pi r s, \quad s = \sqrt{r^2 + h^2} $$
- \(V\) = volume of the cone
- \(A\) = surface area of the cone
- \(r\) = radius of the circular base
- \(h\) = height of the cone
- \(s\) = slant height of the cone
- \(\text{pi}\) = pi, approximately 3.14159
Step by step method
- Measure the radius of the cone's base and the vertical height from the base to the tip.
- Use the radius and height to find the slant height if surface area is needed.
- Apply the cone formulas to calculate volume and surface area.
- Read the result in cubic units for volume and square units for surface area.
Worked example
Suppose a cone has a radius of 3 cm and a height of 4 cm.
- Compute the volume with \(V = \frac{1}{3}\pi r^2 h\). Substituting the values gives \(V = \frac{1}{3}\pi(3^2)(4) = 12\pi\).
- Find the slant height with \(s = \sqrt{r^2 + h^2}\). This gives \(s = \sqrt{3^2 + 4^2} = \sqrt{25} = 5\).
- Compute the surface area with \(A = \pi r^2 + \pi r s\). Substituting gives \(A = 9\pi + 15\pi = 24\pi\).
Answer. Volume = 12 pi cubic cm, surface area = 24 pi square cm.
Tips and common mistakes
- Use the radius, not the diameter. If you have the diameter, divide it by 2 first.
- Volume uses cubic units, while surface area uses square units.
- Surface area needs the slant height, not just the vertical height.
- Keep your units consistent throughout the calculation.
Frequently asked questions
What is the difference between height and slant height?+
The height is the straight vertical distance from the base to the tip. The slant height runs up the sloping surface and is always longer. Volume uses the vertical height, while surface area uses the slant height.
Why is a cone one-third of a cylinder?+
It is a result from geometry that holds for any cone and matching cylinder. You can see it roughly by filling a cone with water and pouring it into the same-size cylinder: it takes exactly three cones to fill it.
Does this work for an ice cream cone shape?+
Yes, as long as it is a right circular cone with a flat round base and a single point. Enter the radius of the opening and the height to the tip to find how much it holds.
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