Geometry desk
Cone Volume Calculator
Calculate a cone's volume from radius and perpendicular height.
Cone Volume Calculator: Calculate the space enclosed by a complete circular cone using its base radius and perpendicular height. This gives the tapered shape's volume, with guidance for distinguishing vertical height from the slant measurement along its side. Runs 100% locally in your browser with zero server file uploads.
- Category
- Calculators
- Runs
- In your browser
- Cost
- Free · no sign-up
- Availability
- Ready to use
Volume
94.25 cm³
Volume = π × radius² × height ÷ 3- Height is measured perpendicular to the circular base.
- Dimensions are measured in the same unit.
- Values must be finite and greater than zero.
Formula and worked example
Volume = π × radius² × height ÷ 3
A worked example with sample values:
| Inputs | |
|---|---|
| Radius | 3 cm |
| Height | 10 cm |
| Results | |
|---|---|
| Volume | 94.25 cm³ |
Where the one-third factor comes from
Let V be volume, r base radius and h perpendicular height. V = π × r² × h ÷ 3. A cylinder with the same base and height has volume πr²h, but a cone narrows continuously towards its tip. At distance x from the tip, a parallel slice has radius rx ÷ h and area πr²(x ÷ h)². Summing these slices gives one third of the matching cylinder's volume.
For radius 3 cm and height 10 cm, square the radius to get 9 cm², multiply by π and 10 to get 90π cm³, then divide by 3. Thus V = 30π = 94.247779… cm³ and the result reads Volume = 94.25 cm³.
The cone slicing exercise in OpenStax (https://openstax.org/books/calculus-volume-2/pages/2-2-determining-volumes-by-slicing) derives this factor. Slant height appears in surface calculations, not directly in this volume formula.
Right-cone height and dimensional checks
The slant-height relationship assumes the apex lies directly above the centre of the circular base. It is not a general shortcut for an oblique cone. With r = 3 cm and l = 5 cm, h = 4 cm and volume is 12π, about 37.70 cm³; using 5 as the height would overestimate it. Volume carries cubic units because radius is squared and then multiplied by height. Convert a cm³ answer to m³ by dividing by 1,000,000. Keep all dimensions on the same scale and round only after the calculation.
How to use it
- Enter positive dimensions in the same unit.
- Choose the unit used for every dimension.
- Calculate, then review the result and formula.
Privacy & limitations
Every value and calculation stays in your browser. No dimensions are uploaded or sent to an external service.
Related tools
Frequently asked questions
Can I enter a cone's slant height in the height field?
No. For a right circular cone, slant height l, radius r and perpendicular height h satisfy l² = r² + h². Convert using h = √(l² − r²). A slant height of 5 cm and radius 3 cm give height 4 cm.
How do I solve for cone radius when volume is known?
If perpendicular height is known, r = √(3V ÷ (πh)). With V = 30π cm³ and h = 10 cm, r = √9 = 3 cm. Alternatively, a known radius gives h = 3V ÷ (πr²). These are separate rearrangements, not alternative input modes.
Does this work for a cone with its tip cut off?
The form describes a complete cone ending at a point. A truncated cone needs both end radii; calculate its volume separately or subtract the removed small cone from the original full cone when their dimensions are known.
Embed this tool
Free to use on any website. Paste the code where the tool should appear; the link under it lets readers open the full page.
<iframe src="https://toea.com/embed/cone-volume-calculator" title="Cone Volume Calculator" width="100%" height="750" style="border:0;max-width:680px" loading="lazy"></iframe>
<p style="margin:4px 0 16px;font-size:13px"><a href="https://toea.com/cone-volume-calculator">Cone Volume Calculator</a> · TOEA</p>
<script>addEventListener("message",function(e){if(e.origin!=="https://toea.com"||!e.data||e.data.type!=="toea-embed-height")return;document.querySelectorAll("iframe").forEach(function(f){if(f.contentWindow===e.source)f.style.height=e.data.height+"px"})})</script>Free tool · runs in your browser · no account required