Enter the top radius, the bottom radius and the height of a frustum of a cone (a solid like a bucket, with parallel circles at the top and bottom and a tapered side). You get the volume and the volume in liters (L).
Enter all three lengths in the same unit, as numbers only (for example, for 15 cm enter "15"). Either the top or the bottom radius can be the larger one. Enter 0 for one radius to get a cone, or the same value for both to get a cylinder.
Result and figure
Enter the top and bottom radii and the height in the fields on the left and press "Calculate". The result 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 top radius, the bottom radius and the height, and you get the volume of a frustum of a cone (a shape like a bucket or a pudding cup) on the spot
- The volume is also converted to US gallons (when you enter inches or feet). Switch "Units" to Metric to get liters instead
- You can also see the frustum as a 3D shape that you can turn by dragging with your mouse
- Enter 0 for one radius to get the volume of a cone, or the same radius for both to get the volume of a cylinder
- A plain-language explanation of the formulas and copy-and-paste formulas for Excel, Google Sheets and Python are all on this page
Enter all three lengths in the same unit (all in inches, for example). The "height" here is the straight-up distance from the top circle to the bottom circle, not the length of the slanted side.
What is this calculation used for?
How many gallons a bucket holds (home)
Most buckets are frustums, wide at the top and narrow at the bottom. For a bucket with a top radius of 6 in, a bottom radius of 5 in and a depth of 14 in, the volume is about 1334 in³, or about 5.8 gallons.
Measure a typical 5-gallon bucket and you get about these numbers. It is a handy calculation for working out how many trips it takes to move a certain amount of water, or the rough capacity of a container with no label. (You never fill a container to the very brim, so plan on using a bit less than the calculated amount.)
How much potting soil a pot needs (gardening)
A flower pot is a typical frustum. For a pot with a top radius of 6 in, a bottom radius of 4.5 in and a height of 10 in, the volume is about 872 in³, which is about 0.5 ft³ (872 ÷ 1728). Potting mix is sold in bags such as 1 ft³ and 2 ft³, so with the number of pots you can work out how many bags to buy before you go to the store.
You leave some space below the rim for watering, so plan on using about 80 to 90% of the calculated amount.
Designing the capacity of cups and food containers (food and manufacturing)
A paper cup is a frustum that is wider at the top. For a cup with a top radius of 1.5 in, a bottom radius of 1 in and a height of 3.5 in, the volume is about 17.4 in³, or about 9.6 fl oz filled to the brim. Drinks are poured below the rim, so the working capacity is a bit less. This formula is the basis for designing cup sizes and the capacity printed on them.
Pudding cups, yogurt containers and measuring cups: the capacity of every frustum-shaped food container comes from this calculation.
Estimating a pile of sand or soil (construction and landscaping)
A pile of sand or gravel delivered by truck is roughly a cone (a frustum with a top radius of 0). For a pile with a bottom radius of 6 ft and a height of 4 ft, the volume is about 151 ft³, or about 5.6 cubic yards (151 ÷ 27).
On job sites and in landscaping, this estimate tells you how many truckloads it is and whether you ordered enough. A real pile is not a perfect cone, so it is used only as a rough estimate.
Capacity of a hopper, a funnel-shaped bin (industry)
A hopper stores powder or grain in a factory and lets it out little by little from the bottom. Its lower part is an upside-down frustum. For a section with a top radius of 3 ft, a bottom radius of 0.5 ft and a height of 4 ft, the capacity is about 45 ft³ (about 337 gallons).
Equipment designers use this formula to work out how much a hopper can hold, and then decide the tank size and how often to refill it. Silos, funnels and drain basins: frustums show up everywhere in industrial equipment.
Formulas and figures
Formula for the volume of a frustum
Figure
Standard notation
(the usual math form)
\(V\)
\(=\)
\((\)
\(r\)
\(2\)
\(+\)
\(r\)
\(\times\)
\(R\)
\(+\)
\(R\)
\(2\)
\()\)
\(\times\)
\(\pi\)
\(\times\)
\(h\)
\(\div 3\)
In words
(symbols replaced with words)
⑦
\(V\): volume of the frustum
\(=\)
\((\)
①
\(r\): top radius
③
squared (the number times itself)
\(+\)
\(r\): top radius
\(\times\)
②
\(R\): bottom radius
\(+\)
\(R\): bottom radius
squared
\()\)
\(\times\)
④
\(\pi\): pi
\(\times\)
⑤
\(h\): height
⑥
\(\div 3\)
The formula in words
①
Using the
\(r\): top radius
②
and the
\(R\): bottom radius
③
make three products, the
squared (the number times itself)
values \(r^2\) and \(R^2\), and \(rR\) (the two radii multiplied together), and add them all up
④
multiply by
\(\pi\): pi (about 3.14)
⑤
multiply by the
\(h\): height
⑥
and finally divide by 3 (split into 3 equal parts)
⑦
to get the
\(V\): volume of the frustum
Quick example
For a frustum with a top radius of 1 in, a bottom radius of 2 in and a height of 3 in (wider at the bottom, like a small pudding cup), the volume is
\(V\): volume of the frustum
\(=\)
\((\)
top radius (1 in)
squared
\(+\)
top radius (1 in)
\(\times\)
bottom radius (2 in)
\(+\)
bottom radius (2 in)
squared
\()\)
\(\times\)
\(\pi\): pi
\(\times\)
height (3 in)
\(\div 3\)
\(1 \times 1 + 1 \times 2 + 2 \times 2 = 1 + 2 + 4 = 7\)
\(7 \times \pi \times 3 \div 3 = 7\pi\)
\(7\pi = 7 \times 3.14159\ldots \approx 21.99\,\mathrm{in^3}\)
Key idea
An easy way to remember it: add the three products \(r^2\), \(rR\) and \(R^2\), multiply by π as you do for the area of a circle, and since this is a relative of the cone (a pointed solid), divide by 3 at the end.
This formula also covers cones and cylinders. Set \(r = 0\) and it becomes the cone formula \(V = \dfrac{1}{3}\pi R^2 h\); set \(r = R\) and it becomes the cylinder formula \(V = \pi R^2 h\). Swapping the top and bottom radii does not change the answer (the formula keeps the same form when you swap \(r\) and \(R\)).
Big cone minus small cone (where the formula comes from)
Figure
Standard notation
(the usual math form)
\(V\)
\(=\)
\(V_{1}\)
\(-\)
\(V_{2}\)
In words
(symbols replaced with words)
③
\(V\): volume of the frustum
\(=\)
①
\(V_1\): volume of the original big cone
\(-\)
②
\(V_2\): volume of the small cone cut off
The formula in words
①
From the
\(V_1\): volume of the original big cone
②
subtract the
\(V_2\): volume of the small cone cut off
③
and you get the
\(V\): volume of the frustum
Quick example
Check it with the same frustum as in formula 1 (top radius 1 in, bottom radius 2 in, height 3 in). Extend the side upward and you get a big cone 6 in tall; the small cone that was cut off is 3 in tall
\(V\): volume of the frustum
\(=\)
big cone (radius 2 in, height 6 in)
\(-\)
small cone (radius 1 in, height 3 in)
\(V_1 = 2 \times 2 \times \pi \times 6 \div 3 = 8\pi\)
\(V_2 = 1 \times 1 \times \pi \times 3 \div 3 = \pi\)
\(V = 8\pi - \pi = 7\pi \approx 21.99\,\mathrm{in^3}\)
Key idea
A frustum is what you get when you cut a cone with a plane parallel to its base and remove the part with the apex. So you can find its volume by extending the tapered side to form the big cone, and then subtracting the small cone that was cut off.
How far you have to extend the side to reach the apex comes from the idea of similar figures. In this example, the top and bottom radii are in the ratio \(1 : 2\), so the heights of the small cone and the big cone are also in the ratio \(1 : 2\). The frustum's height of 3 in is the difference between them, so the small cone is 3 in tall and the big cone is 6 in tall. Formula 1 is this subtraction worked out with letters and simplified, so the answer is exactly the same \(7\pi\). Even if you forget the formula, you can still find the volume of a frustum with this idea.
Converting volume units (in³ → gallons)
Figure
Standard notation
(the usual math form)
\(V_{\mathrm{gal}}\)
\(=\)
\(V_{\mathrm{in^3}}\)
\(\div\)
\(231\)
In words
(symbols replaced with words)
③
\(V_{\mathrm{gal}}\): volume in gallons
\(=\)
①
\(V_{\mathrm{in^3}}\): volume in in³
\(\div\)
②
\(231\): cubic inches in 1 gallon
The formula in words
①
Take the
\(V_{\mathrm{in^3}}\): volume in in³
②
divide it by
\(231\): cubic inches in 1 gallon
③
and you get the
\(V_{\mathrm{gal}}\): volume in gallons
Quick example
A 5-gallon bucket (a frustum with a top radius of 6 in, a bottom radius of 5 in and a height of 14 in) has a volume of about 1334.13 in³. In gallons, that is
volume in gallons
\(=\)
volume in in³ (about 1334.13)
\(\div\)
cubic inches in 1 gallon (231)
\(1334.13 \div 231 = 5.775\ldots \approx 5.8\,\mathrm{gal}\)
Key idea
One US gallon is defined as exactly 231 in³. That is the volume of a cube about 6.14 in on each side (\(6.14^3 \approx 231\)). Unlike a liter, which is a 10 cm cube, a gallon is not a cube of a round length, so just remember the number 231. To turn cubic inches into gallons, divide by 231.
If you entered feet, the volume is in ft³. First multiply by 1728 to turn it into in³ (a cube 1 ft = 12 in on each side is \(12 \times 12 \times 12 = 1728\,\mathrm{in^3}\)), then divide by 231. That makes \(1\,\mathrm{ft^3} \approx 7.48\,\mathrm{gal}\). The bucket above holds about 5.8 gal when filled to the very brim, which is why it is sold as a 5-gallon bucket.
The volume of a frustum of a cone is "(top radius squared + top radius × bottom radius + bottom radius squared) × π × height ÷ 3". If you forget the formula, go back to the subtraction "big cone − small cone". To get gallons from cubic inches, divide by 231.
Symbols and terms
Symbols
| \(r\) | lowercase r | The radius of the top circle, from the first letter of "radius". On this page it is always the top radius, whether or not the top is the smaller circle. |
| \(R\) | capital R | The radius of the bottom circle (the base). It is written as a capital letter to tell it apart from the top radius \(r\). |
| \(h\) | aitch | The height of the frustum, from the first letter of "height". It is the straight-up distance from the bottom circle to the top circle, not the length of the slanted side. |
| \(\pi\) | pi | The number that tells how many times the circumference of a circle is its diameter. It is \(3.14159265\ldots\) and the decimals never end. In simple calculations, 3.14 is often used. |
| \(r^2\) | 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". |
| \(V\) | vee | A common symbol for volume, from the first letter of "volume". |
| \(\mathrm{in^3}\) | cubic inch | A unit of volume. A cube 1 in on each side has a volume of 1 in³. |
| \(\mathrm{gal}\) | gallon | The familiar unit for amounts of liquid such as water. One US gallon is exactly \(231\,\mathrm{in^3}\), and \(1\,\mathrm{ft^3} \approx 7.48\,\mathrm{gal}\) (\(1\,\mathrm{gal} \approx 3.785\,\mathrm{L}\)). |
Terms
| frustum | The solid you get when you cut a cone with a plane parallel to its base and remove the part with the apex (a frustum of a cone, also called a truncated cone). Its top and bottom are parallel circles of different sizes, and its side is tapered. Buckets, flower pots, paper cups and pudding cups all have this very common shape. |
| cone | A solid with a circular base that narrows to a single point, the apex, like an ice cream cone. A frustum is made from a cone. |
| base | The face at the bottom of a solid. For a frustum it is the bottom circle (the circle at the top is called the top face). |
| 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. |
| similar | Having the same shape but a different size. In formula 2, finding the height of the big cone uses the fact that, for similar cones, the ratio of the radii equals the ratio of the heights. |
| pyramid | A solid whose base is a polygon and whose sides are triangles that meet at one point. Cones and pyramids both narrow to a point, and their volume is always "base area × height ÷ 3". The "÷ 3" in the frustum formula comes from this. |
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.
| Area of a circle and pi (Grade 7) |
|
| What volume is and its units (Grade 5) |
|
| Exponents (Grade 6) |
|
| Volume of cylinders and cones (Grade 8) |
|
| Similar figures (Grades 7–8) |
|
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 frustum
| Top radius r (in) | 6 |
| Bottom radius R (in) | 5 |
| Height h (in) | 14 |
| Frustum volume (in³) | =(B1^2+B1*B2+B2^2)*PI()*B3/3 |
Table to find it as big cone − small cone
| Base radius of the big cone | 2 |
| Height of the big cone | 6 |
| Base radius of the small cone | 1 |
| Height of the small cone | 3 |
| Volume of the big cone | =B1^2*PI()*B2/3 |
| Volume of the small cone | =B3^2*PI()*B4/3 |
| Frustum volume | =B5-B6 |
Table to convert in³ to gallons
| Volume in in³ | 1334.13 |
| Volume in US gallons | =B1/231 |
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. "*" is multiplication, "/" is division, "^2" squares a number, and "PI()" is the Excel function for π (3.14159…).
In the first table, for example, B4 shows about 1334.13 (in in³, because the inputs are in inches). The second table shows about 21.99 in B7, and the third table shows about 5.78 in B2. Just replace the input numbers with your own lengths.
In a formula, "B1" and "B2" tell the formula to use the number in that cell. "*" is multiplication, "/" is division, "^2" squares a number, and "PI()" is the Excel function for π (3.14159…).
In the first table, for example, B4 shows about 1334.13 (in in³, because the inputs are in inches). The second table shows about 21.99 in B7, and the third table shows about 5.78 in B2. Just replace the input numbers with your own lengths.
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 frustum
| Top radius r (in) | 6 |
| Bottom radius R (in) | 5 |
| Height h (in) | 14 |
| Frustum volume (in³) | =(B1^2+B1*B2+B2^2)*PI()*B3/3 |
Table to find it as big cone − small cone
| Base radius of the big cone | 2 |
| Height of the big cone | 6 |
| Base radius of the small cone | 1 |
| Height of the small cone | 3 |
| Volume of the big cone | =B1^2*PI()*B2/3 |
| Volume of the small cone | =B3^2*PI()*B4/3 |
| Frustum volume | =B5-B6 |
Table to convert in³ to gallons
| Volume in in³ | 1334.13 |
| Volume in US gallons | =B1/231 |
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 your own lengths.
How to calculate it in Python
import math
radius_top = 6 # radius of the top circle
radius_bottom = 5 # radius of the bottom circle (the base)
height = 14 # height (same unit as the radii)
volume = (radius_top**2 + radius_top*radius_bottom + radius_bottom**2) * math.pi * height / 3
gallons = volume / 231 # volume in US gallons, if you entered inches (1 gal = 231 in3)
print(f"Frustum volume: {volume} in3")
print(f"In gallons: {gallons} gal")
Runs with the standard library only. "math.pi" is π, "**2" squares a number, "*" is multiplication and "/" is division. Change the three values at the top and run it. (This example uses inches. If you enter feet, multiply by 1728 and then divide by 231 to get gallons.)
How to write it in LaTeX and other math languages (copy and paste)
Formula for the volume of a frustum
V = (1/3)πh(r² + rR + R²)
V = \frac{1}{3}\pi h \left( r^{2} + rR + R^{2} \right)
<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>
<mi>h</mi>
<mo>(</mo>
<msup><mi>r</mi><mn>2</mn></msup>
<mo>+</mo>
<mi>r</mi>
<mi>R</mi>
<mo>+</mo>
<msup><mi>R</mi><mn>2</mn></msup>
<mo>)</mo>
</mrow>
</math>
V = (1/3) pi h (r^2 + r R + R^2)
Pi*h*(r^2 + r*R + R^2)/3
V := Pi*h*(r^2 + r*R + R^2)/3;
V = pi*h*(r^2 + r*R + R^2)/3;
V = (1/3)πh(r^2 + rR + R^2)
Big cone minus small cone (where the formula comes from)
V = V₁ − V₂
V = V_{1} - V_{2}
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
<mrow>
<mi>V</mi>
<mo>=</mo>
<msub><mi>V</mi><mn>1</mn></msub>
<mo>−</mo>
<msub><mi>V</mi><mn>2</mn></msub>
</mrow>
</math>
V = V_1 - V_2
v1 - v2
V := V1 - V2;
V = V1 - V2;
V = V_1 - V_2
Converting volume units (in³ → gallons)
V[gal] = V[in³] ÷ 231
V_{\mathrm{gal}} = V_{\mathrm{in^3}} \div 231
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
<mrow>
<msub><mi>V</mi><mi>gal</mi></msub>
<mo>=</mo>
<msub><mi>V</mi><mrow><msup><mi mathvariant="normal">in</mi><mn>3</mn></msup></mrow></msub>
<mo>÷</mo>
<mn>231</mn>
</mrow>
</math>
V_(gal) = V_(in^3) -: 231
vIn3/231
vGal := vIn3/231;
v_gal = v_in3/231;
V(gal) = V(in³)/231
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 frustum of a cone (shaped like a bucket) has a top radius of 6 in, a bottom radius of 5 in and a height of 14 in. Find each of the following: 1. The volume of this frustum in in³ (use the formula V = (1/3)πh(r² + rR + R²)) 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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Calculates the area of a trapezoid, the shape of a frustum seen straight from the side
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