Enter the base radius and the height of the cone. You get the volume (1/3 × π × radius² × height), the volume of a cylinder with the same base and height, and the volume in cm³, L and m³.
Enter the radius and the height in the same unit, as numbers 0 or greater (numbers only, no units. For example, for 3 cm enter "3").
Result and figure
Enter the base radius and the height in the fields on the left and press "Calculate". The result and a 3D shape you can turn with your mouse will appear here.
Table of Contents
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What you can do on this page
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What is this calculation used for?
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How to Use
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Formulas and figures
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Symbols and terms
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Good to know before you start
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How to calculate it in Excel
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How to calculate it in Google Sheets
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How to calculate it in Python
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How to write it in LaTeX and other math languages (copy and paste)
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How to have ChatGPT do the calculation
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DataChef Features
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Related Features
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NumberChef Calculators List
What you can do on this page
- Enter the base radius and the height, and you get the volume of the cone on the spot (volume = \(\dfrac{1}{3}\) × π × radius² × height)
- The result is also shown as a 3D shape. Turn it around with your mouse to see which length is which
- It also shows the volume of a cylinder with the same base and height (the cone is exactly 1/3 of it), so you can see what the formula says
- Volume unit conversions are shown too: gallons and ft³ when you enter inches. Switch "Units" to Metric to get liters and m³ instead
- A plain-language explanation of why it is 1/3 and copy-and-paste formulas for Excel, Google Sheets and Python are all on this page
Enter the radius and the height in the same unit (both in inches, or both in feet). This page handles a right circular cone, whose apex is directly above the center of the base circle. Other solids such as cylinders, spheres and pyramids have their own pages.
What is this calculation used for?
How much an ice cream cone holds (food)
The cone part of a soft-serve or waffle cone is a cone. For a cone with a rim radius of 1.25 in and a depth of 6 in, the amount it holds when filled level is \(\dfrac{1}{3} \times \pi \times 1.25^2 \times 6 \approx 9.8\,\mathrm{in^3}\) (about 5.4 fl oz).
The numbers show that, apart from the scoop on top, a cone holds less than you might think. Food is full of cones, from cone-shaped cups to candy and snack cones.
How much water a paper cone cup holds (office and emergency supplies)
The paper cone cups next to a water cooler are often about 2.75 in across the rim (a radius of 1.375 in) and 3.75 in deep. They hold \(\dfrac{1}{3} \times \pi \times 1.375^2 \times 3.75 \approx 7.4\,\mathrm{in^3}\) (about 4.1 fl oz).
A 5-gallon water jug holds 640 fl oz, so it fills about 155 of these cups. Estimates like "how many people can get one cup each" take just this one formula.
How much water a coffee dripper holds (home)
The inside of a cone-shaped pour-over coffee dripper is roughly a cone with a rim radius of 2 in and a depth of 3 in. Filled level, it holds \(\dfrac{1}{3} \times \pi \times 2^2 \times 3 \approx 12.6\,\mathrm{in^3}\) (about 7 fl oz).
This tells you how much hot water you can pour at one time, and whether you can brew two cups in one go.
Estimating a pile of sand or gravel (construction and landscaping)
When sand or gravel is dumped from a truck, it naturally forms a pile close to a cone (the steepest slope it can hold without sliding is called the angle of repose). If the pile is 5 ft in radius at the bottom and 3 ft high, it holds about \(\dfrac{1}{3} \times \pi \times 5^2 \times 3 \approx 78.5\,\mathrm{ft^3}\), or about 2.9 cubic yards (78.5 ÷ 27).
A real pile is not a perfect cone, so this is an estimate. Still, on job sites and in landscaping, this formula gives a quick answer to "how many truckloads do we need?" and "will it fit in the storage area?"
Estimating the size of a volcano (earth science and education)
Stratovolcanoes such as Mount Fuji in Japan are close to a cone in shape. Treat Mount Fuji as a cone with a base radius of about 12 mi and a height of about 2 mi above the surrounding land, and its volume is about \(\dfrac{1}{3} \times \pi \times 12^2 \times 2 \approx 300\,\mathrm{mi^3}\) (about 1,260 km³).
A real mountain is not a perfect cone, so this is only a rough estimate. Even in volcano research, the volume of a mountain is often estimated by treating it as a simple solid like this.
Formulas and figures
Volume of a cone
Figure
Standard notation
(the usual math form)
\(V\)
\(=\)
\(\dfrac{1}{3}\)
\(\times\)
\(\pi\)
\(\times\)
\(r\)
\(2\)
\(\times\)
\(h\)
In words
(symbols replaced with words)
⑥
\(V\): volume of the cone
\(=\)
⑤
\(\dfrac{1}{3}\): one third
\(\times\)
①
\(\pi\): pi (about 3.14)
\(\times\)
②
\(r\): base radius
③
squared (the number times itself)
\(\times\)
④
\(h\): height
The formula in words
①
Take
\(\pi\): pi (about 3.14)
②
multiply it by the
\(r\): base radius
③
squared (the number times itself)
④
and by the
\(h\): height
. This gives the volume of a cylinder with the same base and height.
⑤
Then multiply by
\(\dfrac{1}{3}\): one third
(divide by 3)
⑥
and you get the
\(V\): volume of the cone
Quick example
For a cone with a base radius of 3 in and a height of 10 in, the volume is
\(V\): volume of the cone
\(=\)
\(\dfrac{1}{3}\): one third
\(\times\)
\(\pi\): pi
\(\times\)
radius (3 in)
squared
\(\times\)
height (10 in)
\(\dfrac{1}{3} \times \pi \times 3^{2} \times 10 = \dfrac{1}{3} \times \pi \times 90 = 30\pi\)
\(30\pi \approx 30 \times 3.14 = 94.2\,\mathrm{in^3}\)
Key idea
Why 1/3? Take a cylinder-shaped container and a cone-shaped container with the same base and the same height. Fill the cone with water and pour it into the cylinder: it takes exactly 3 cones to fill it (a common classroom experiment). So the volume of a cone is exactly \(\dfrac{1}{3}\) of the cylinder's volume \(\pi r^2 h\). A full proof uses integration from calculus, so in middle school you learn it as a rule: a cone or a pyramid is 1/3 of the cylinder or prism with the same base and height.
One more thing to watch is how you measure the height. The height \(h\) is measured straight up from the base to the apex, at a right angle to the base. It is not the length along the sloping side (the slant height).
How it relates to the volume of a cylinder (an easy way to remember)
Figure
Standard notation
(the usual math form)
\(V\)
\(=\)
\(\pi r^{2} h\)
\(\div\)
\(3\)
In words
(symbols replaced with words)
③
\(V\): volume of the cone
\(=\)
①
\(\pi r^2 h\): volume of a cylinder with the same base and height
\(\div\)
②
\(3\): split into 3 equal parts
The formula in words
①
Take the
\(\pi r^2 h\): volume of a cylinder with the same base and height
②
divide it by
\(3\): split into 3 equal parts
③
and you get the
\(V\): volume of the cone
Quick example
With a base radius of 3 in and a height of 10 in, finding the cone's volume from the cylinder with the same base and height gives
\(V\): volume of the cone
\(=\)
cylinder volume (90π ≈ 282.6 in³)
\(\div\)
3
\(\pi \times 3^{2} \times 10 = 90\pi \approx 90 \times 3.14 = 282.6\,\mathrm{in^3}\)
\(90\pi \div 3 = 30\pi \approx 94.2\,\mathrm{in^3}\)
Key idea
In words, "volume of a cone = base area × height ÷ 3". The base area (the area of a circle, \(\pi r^2\)) times the height is the volume of the cylinder, and the cone is one of 3 equal parts of it. The same "base area × height ÷ 3" works in exactly the same way for pyramids, whose base is a triangle or a square (triangular pyramids and square pyramids).
Converting volume units (in³ → ft³)
Figure
Standard notation
(the usual math form)
\(V_{\mathrm{ft^3}}\)
\(=\)
\(V_{\mathrm{in^3}}\)
\(\div\)
\(1728\)
In words
(symbols replaced with words)
③
\(V_{\mathrm{ft^3}}\): volume in ft³
\(=\)
①
\(V_{\mathrm{in^3}}\): volume in in³
\(\div\)
②
\(1728\): cubic inches in 1 ft³
The formula in words
①
Take the
\(V_{\mathrm{in^3}}\): volume in in³
②
divide it by
\(1728\): cubic inches in 1 ft³
③
and you get the
\(V_{\mathrm{ft^3}}\): volume in ft³
Quick example
The cone's volume of 94.2 in³ in cubic feet is
volume in ft³
\(=\)
volume in in³ (94.2)
\(\div\)
cubic inches in 1 ft³ (1728)
\(94.2 \div 1728 \approx 0.0545\,\mathrm{ft^3}\)
Key idea
\(1\,\mathrm{ft^3}\) is the volume of a cube 1 ft (12 in) on each side, so \(1\,\mathrm{ft^3} = 12 \times 12 \times 12 = 1728\,\mathrm{in^3}\). The conversion factor for volume is the cube of the factor for length. Since \(1\,\mathrm{yd} = 3\,\mathrm{ft}\), \(1\,\mathrm{yd^3} = 3 \times 3 \times 3 = 27\,\mathrm{ft^3}\). A common mistake is to divide by 12 or 144 instead of 1728.
For liquids, the US uses gallons: \(1\,\mathrm{gal} = 231\,\mathrm{in^3}\), so divide cubic inches by 231 to get gallons (\(94.2 \div 231 \approx 0.41\,\mathrm{gal}\)).
The volume of a cone is "\(\dfrac{1}{3}\) × π × radius squared × height", which is "base area × height ÷ 3". It is exactly 1/3 of a cylinder with the same base and height. Use the same unit for the radius and the height; the answer is in the cube of that unit (in³ for inches). To go from in³ to ft³, divide by 1728; to get gallons, divide by 231.
Symbols and terms
Symbols
| \(V\) | vee | A common symbol for volume, from the first letter of "volume". On this page it is the volume of the cone. |
| \(r\) | ar | The radius of the base circle, from the first letter of "radius". It is the length from the center of the circle to its edge, half the diameter. |
| \(h\) | aitch | The height of the cone, from the first letter of "height". It is measured straight up from the base to the apex, and is not the same as the length along the side (the slant height). |
| \(r^2\) | r squared | The number \(r\) multiplied by itself (\(r \times r\)). The small 2 at the upper right is an exponent that says "use it as a factor twice". |
| \(\pi\) | pi | The number that tells how many times the circumference of a circle is its diameter. It is about 3.14159… and the decimals never end. You can use 3.14 in a calculation, or keep the symbol \(\pi\) in the answer. |
| \(\mathrm{in^3}\) | cubic inch | A unit of volume. A cube 1 in on each side has a volume of 1 in³. Do not mix it up with in² (square inch), the unit of area. |
| \(\mathrm{ft^3}\) | cubic foot | A unit of volume. A cube 1 ft on each side has a volume of 1 ft³. \(1\,\mathrm{ft^3} = 1728\,\mathrm{in^3}\), and \(1\,\mathrm{yd^3} = 27\,\mathrm{ft^3}\). |
| \(\mathrm{gal}\) | gallon | The familiar unit for liquids such as milk and gas. One US gallon is \(231\,\mathrm{in^3}\) (\(1\,\mathrm{gal} = 128\,\mathrm{fl\ oz} \approx 3.785\,\mathrm{L}\)). |
Terms
| cone | A solid with a circular base that narrows smoothly to a single point. Ice cream cones, party hats and traffic cones are shaped like cones. |
| right circular cone | A cone whose apex is directly above the center of the base circle. The cones in school textbooks are usually this kind, and this calculator handles it too. A cone whose apex is not above the center is called an oblique cone. |
| base | The circle at the bottom of the cone. Its area (\(\pi r^2\) for a cone) is called the base area. |
| apex | The pointed tip of the cone (also called the vertex). The height is the length from this point straight down to the base, at a right angle. |
| slant height | The straight length along the side of the cone, from the apex to the edge of the base circle. By the Pythagorean theorem it is \(\sqrt{r^2 + h^2}\). Be careful not to mix it up with the height. |
| volume | The amount of space a solid takes up, given as a number. It is measured by how many unit cubes (such as 1 in³ cubes) would fill it. |
| pi | The number that tells how many times the circumference is the diameter (about 3.14). It appears in every calculation for shapes that involve a circle, such as circles, cones and cylinders. |
Good to know before you start
Here is what helps you use the calculation on this page with real understanding, not just by pressing the button.
| What volume is and its units (Grade 5) |
|
| Area of a circle (Grade 7) |
|
| Expressions with letters and the number π (Grades 6–7) |
|
| Solid figures (Grade 8) |
|
| Multiplying and dividing decimals (Grades 5–6) |
|
How to calculate it in Excel
Copy the whole table below and paste it into cell A1 in Excel. It works as is.
Table to find the volume of a cone
| Base radius (in) | 3 |
| Height (in) | 10 |
| Cone volume (in³) | =PI()*B1^2*B2/3 |
Table to check the link with a cylinder's volume
| Base radius (in) | 3 |
| Height (in) | 10 |
| Cylinder with the same base and height (in³) | =PI()*B1^2*B2 |
| Cone volume, 1/3 of the cylinder (in³) | =B3/3 |
Table to convert in³ to ft³
| Volume in in³ | 94.25 |
| Volume in ft³ | =B1/1728 |
After pasting, column A holds the labels and column B holds the numbers. The upper rows are your inputs, and the formula in the last row calculates from them automatically.
In a formula, "B1" and "B2" tell the formula to use the number in that cell. "PI()" is π (3.14159…), "^" raises to a power (how many times to multiply), "*" is multiplication and "/" is division.
In the first table, for example, B3 shows about 94.25 (= 1/3 × π × 3² × 10), in in³ because the inputs are in inches. In the second table, B3 shows about 282.74 and B4 shows one third of it, about 94.25, so you can check that the cone is 1/3 of the cylinder. The third table shows about 0.05454 in B2. Just replace the input numbers with the size of your own cone.
In a formula, "B1" and "B2" tell the formula to use the number in that cell. "PI()" is π (3.14159…), "^" raises to a power (how many times to multiply), "*" is multiplication and "/" is division.
In the first table, for example, B3 shows about 94.25 (= 1/3 × π × 3² × 10), in in³ because the inputs are in inches. In the second table, B3 shows about 282.74 and B4 shows one third of it, about 94.25, so you can check that the cone is 1/3 of the cylinder. The third table shows about 0.05454 in B2. Just replace the input numbers with the size of your own cone.
How to calculate it in Google Sheets
Copy the whole table below and paste it into cell A1 in Google Sheets. It works as is.
Table to find the volume of a cone
| Base radius (in) | 3 |
| Height (in) | 10 |
| Cone volume (in³) | =PI()*B1^2*B2/3 |
Table to check the link with a cylinder's volume
| Base radius (in) | 3 |
| Height (in) | 10 |
| Cylinder with the same base and height (in³) | =PI()*B1^2*B2 |
| Cone volume, 1/3 of the cylinder (in³) | =B3/3 |
Table to convert in³ to ft³
| Volume in in³ | 94.25 |
| Volume in ft³ | =B1/1728 |
The same formulas as in Excel (including the PI() function) work as is. Copy the whole table, paste it into cell A1, and replace the input numbers with the size of your own cone.
How to calculate it in Python
import math
radius = 3 # base radius (inches in this example)
height = 10 # height (same unit as the radius)
volume = math.pi * radius ** 2 * height / 3 # cone volume (cube of the input unit; in3 in this example)
volume_gal = volume / 231 # volume in US gallons, if you entered inches (1 gal = 231 in3)
print(f"Cone volume: {volume} in3")
print(f"In gallons: {volume_gal} gal")
Runs with the standard library only. "math.pi" is π (3.14159…), "**" raises to a power (radius ** 2 is the radius squared), "*" is multiplication and "/" is division. Change the radius and height at the top and run it. (This example uses inches. If you enter feet, the answer is in ft³, so multiply by 1728 and divide by 231 to get gallons.)
How to write it in LaTeX and other math languages (copy and paste)
Volume of a cone
V = (1/3)πr²h
V = \frac{1}{3} \pi r^{2} h
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
<mrow>
<mi>V</mi>
<mo>=</mo>
<mfrac><mn>1</mn><mn>3</mn></mfrac>
<mi>π</mi>
<msup><mi>r</mi><mn>2</mn></msup>
<mi>h</mi>
</mrow>
</math>
V = 1/3 pi r^2 h
Pi*r^2*h/3
V := (1/3)*Pi*r^2*h;
V = (1/3)*pi*r^2*h;
V = (1/3)πr^2 h
How it relates to the volume of a cylinder (an easy way to remember)
V = πr²h ÷ 3
V = \frac{\pi r^{2} h}{3}
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
<mrow>
<mi>V</mi>
<mo>=</mo>
<mfrac>
<mrow><mi>π</mi><msup><mi>r</mi><mn>2</mn></msup><mi>h</mi></mrow>
<mn>3</mn>
</mfrac>
</mrow>
</math>
V = (pi r^2 h)/3
Pi*r^2*h/3
V := Pi*r^2*h/3;
V = pi*r^2*h/3;
V = πr^2 h/3
Converting volume units (in³ → ft³)
V[ft³] = V[in³] ÷ 1728
V_{\mathrm{ft^3}} = V_{\mathrm{in^3}} \div 1728
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
<mrow>
<msub><mi>V</mi><mrow><msup><mi mathvariant="normal">ft</mi><mn>3</mn></msup></mrow></msub>
<mo>=</mo>
<msub><mi>V</mi><mrow><msup><mi mathvariant="normal">in</mi><mn>3</mn></msup></mrow></msub>
<mo>÷</mo>
<mn>1728</mn>
</mrow>
</math>
V_(ft^3) = V_(in^3) -: 1728
vIn3/1728
vFt3 := vIn3/1728;
v_ft3 = v_in3/1728;
V(ft³) = V(in³)/1728
How to have ChatGPT do the calculation
You are a math calculation assistant. Do the following calculation by actually running Python code, and base your answer only on the numbers from the execution result (do not answer by mental math or guessing). A cone has a base radius of 3 in and a height of 10 in. Find each of the following: 1. The volume of this cone in in³ (V = 1/3 × π × radius² × height) 2. That volume in US gallons (1 gal = 231 in³) Show the formulas you used and the numbers from the execution result.
How to Use
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1Enter your numbersType the numbers you want to calculate with into the input fields
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2CalculatePress the "Calculate" button
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3Check the resultThe result appears on the spot. The same page also explains the idea behind the calculation and the formula
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Related Features
Circle Calculator (Radius, Diameter, Circumference and Area)
The base of a cone is a circle. Find the base area (πr²) from the radius and convert between the basic measures of a circle
Pythagorean Theorem Calculator (Hypotenuse and Legs of a Right Triangle)
Calculates the Pythagorean theorem, which you use to find the slant height of a cone (√(radius² + height²))
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