Enter the base side length and the height of a square pyramid (a pyramid with a square base). You get the volume (side × side × height ÷ 3), converted to cm³, m³ and L as well.
Enter the base side and the height in the same unit, as numbers only (no units. For example, for 5 cm enter "5"). The height is the straight up-and-down distance from the base to the apex.
Result and figure
Enter the base side length and the height in the fields on the left and press "Calculate". The result and a 3D figure 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
- Just enter the side length of the base and the height, and you get the volume of a square pyramid (side × side × height ÷ 3) on the spot
- The pyramid you entered is drawn in 3D (drag to spin it around)
- The base area and the steps are shown too, so you can check how the answer was reached, not just the answer
- The volume is also converted to other units (gal and ft³ if you enter inches, gal if you enter feet)
- A plain-language explanation of why we divide by 3 and copy-and-paste formulas for Excel, Google Sheets and Python are all on this page
Enter the base side and the height in the same unit (both in inches, or both in feet). The height is the straight up-and-down distance from the base to the apex (not the length of a slanted edge). Cones, triangular pyramids and other solids are covered on their own pages.
What is this calculation used for?
Getting a feel for the size of the pyramids of Egypt (history and travel)
The Great Pyramid of Giza (the Pyramid of Khufu) is almost a right square pyramid, with a base side of about 756 ft and an original height of about 481 ft. Its volume is about \(756 \times 756 \times 481 \div 3 \approx 91{,}600{,}000\,\mathrm{ft^3}\), or about 92 million cubic feet (about 2.6 million m³). That is enough stone to fill roughly 1,000 Olympic-size swimming pools.
The real pyramid has passages and rooms inside, and its faces are not perfectly flat, so this is an estimate. Still, one formula lets you put the scale of a World Heritage Site into numbers.
Estimating the air inside a pyramid tent (camping)
A tent held up by a single center pole (a pyramid tent, also called a teepee-style tent) is almost a square pyramid if its floor is square. For a 10 ft × 10 ft floor and a 7 ft pole, the space inside is about \(10 \times 10 \times 7 \div 3 \approx 233\,\mathrm{ft^3}\).
Knowing the amount of air in the tent helps you judge how well a heater can warm it on a winter camping trip, or how likely condensation is (many real tents have short vertical walls at the bottom, so this is an estimate).
Finding the attic volume under a pyramid roof (construction)
A pyramid hip roof on a square building is exactly a square pyramid. For a 20 ft × 20 ft roof that rises 6 ft, the attic space under it is about \(20 \times 20 \times 6 \div 3 = 800\,\mathrm{ft^3}\).
The attic volume is a number used when planning attic ventilation and insulation. Many pointed roofs you see around you, such as gazebo and tower roofs, can be estimated with this formula.
Calculating the capacity of an upside-down pyramid hopper (manufacturing and farming)
A hopper is a funnel-shaped bin that stores grain, plastic pellets or powder and lets it out little by little from the bottom. It is often an upside-down square pyramid. For a hopper 6 ft across the top and 4.5 ft deep, the capacity is \(6 \times 6 \times 4.5 \div 3 = 54\,\mathrm{ft^3}\), which is 2 yd³, or about 43 bushels of grain (1 bushel ≈ 1.24 ft³).
Questions like "how many more tons will fit?" or "can it hold a day's worth of material?" start from this volume calculation (real hoppers are often cut off at the narrow bottom, forming a frustum; then you subtract the small pyramid from the large one).
Solving a classic geometry test question (school)
The volume of pyramids and cones is a staple of middle school and high school geometry, and the pyramid formula appears on the formula sheet of tests such as the SAT. For example, "What is the volume of a right square pyramid with a base side of 6 in and a height of 8 in?" The answer is \(6 \times 6 \times 8 \div 3 = 96\,\mathrm{in^3}\).
Forgetting to divide by 3 is the most common mistake. Remember it together with "a pointed solid is 1/3 of the prism", and you will get these points on the test.
Formulas and figures
Square pyramid volume formula
Figure
Standard notation
(the usual math form)
\(V\)
\(=\)
\(a\)
\(2\)
\(\times\)
\(h\)
\(\div\)
\(3\)
In words
(symbols replaced with words)
⑤
\(V\): pyramid volume
\(=\)
①
\(a\): base side
②
squared (the length times itself)
\(\times\)
③
\(h\): height
\(\div\)
④
\(3\): split into 3 equal parts
The formula in words
①
Take the
\(a\): base side
,
②
find its
square (the length times itself)
to get the base area,
③
multiply by the
\(h\): height
,
④
and divide by
\(3\): split into 3 equal parts
to get the
⑤
\(V\): pyramid volume
Quick example
The volume of a square pyramid with a base side of 5 in and a height of 12 in is
\(V\): pyramid volume
\(=\)
base side (5 in)
squared
\(\times\)
height (12 in)
\(\div\)
split into 3 equal parts (3)
\(5 \times 5 = 25\)
\(25 \times 12 \div 3 = 300 \div 3 = 100\,\mathrm{in^3}\)
Key idea
Why divide by 3? A cube (the shape of a die) can be cut from one corner into exactly 3 identical "slanted" square pyramids. So the volume of a pyramid is exactly \(\dfrac{1}{3}\) of the volume of a prism (a box) with the same base and the same height. This formula writes "\(\times \dfrac{1}{3}\)" as "\(\div 3\)".
Note that the height \(h\) is the straight up-and-down distance from the base to the apex, not the length of a slanted edge (an edge running to the apex). Also, even for a "slanted" pyramid whose apex is not directly above the center of the base, the volume is the same as long as the base area and the height are the same.
Volume formula for all pyramids and cones (from the base area)
Figure
Standard notation
(the usual math form)
\(V\)
\(=\)
\(S\)
\(\times\)
\(h\)
\(\div\)
\(3\)
In words
(symbols replaced with words)
④
\(V\): volume of the pyramid or cone
\(=\)
①
\(S\): base area
\(\times\)
②
\(h\): height
\(\div\)
③
\(3\): split into 3 equal parts
The formula in words
①
Multiply the
\(S\): base area
②
by the
\(h\): height
,
③
and divide by
\(3\): split into 3 equal parts
to get the
④
\(V\): volume of the pyramid or cone
Quick example
The volume of a pyramid with a base area of 25 in² and a height of 12 in is
\(V\): volume of the pyramid or cone
\(=\)
base area (25 in²)
\(\times\)
height (12 in)
\(\div\)
split into 3 equal parts (3)
\(25 \times 12 \div 3 = 300 \div 3 = 100\,\mathrm{in^3}\)
Key idea
This formula is not just for square pyramids. It works for every pyramid, such as triangular and pentagonal pyramids, and for cones too (every solid that narrows to a point, an apex, has volume "base area × height ÷ 3"). In US textbooks it is often written \(V = \dfrac{1}{3}Bh\), with \(B\) for the base area.
In a square pyramid the base is a square, so the base area is \(S = a \times a\). Substituting this gives the first formula, \(V = \dfrac{1}{3} a^{2} h\). For a cone, the base area is the area of a circle, \(S = \pi r^{2}\).
Converting volume units (in³ to 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
Divide the
\(V_{\mathrm{in^3}}\): volume in in³
②
by
\(1728\): cubic inches in 1 ft³
to get the
③
\(V_{\mathrm{ft^3}}\): volume in ft³
Quick example
The volume of the square pyramid with a base side of 5 in and a height of 12 in, 100 in³, converted to ft³ is
volume in ft³
\(=\)
volume in in³ (100)
\(\div\)
cubic inches in 1 ft³ (1728)
\(100 \div 1728 \approx 0.0579\,\mathrm{ft^3}\)
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 same idea works for cubic yards: \(1\,\mathrm{yd} = 3\,\mathrm{ft}\), so \(1\,\mathrm{yd^3} = 3 \times 3 \times 3 = 27\,\mathrm{ft^3}\). The conversion factor for volume is the conversion factor for length cubed (squared for area, cubed for volume). 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 (\(100 \div 231 \approx 0.433\,\mathrm{gal}\)).
The volume of a square pyramid is "base area (side × side) × height ÷ 3". We divide by 3 because a pyramid or cone has exactly 1/3 the volume of a prism (a box) with the same base and the same height. Use the straight up-and-down height from the base to the apex; the answer is in the length unit cubed (in³ for inches).
Symbols and terms
Symbols
| \(V\) | vee | A common symbol for volume, from the first letter of "volume". On this page it stands for the volume of the square pyramid. |
| \(a\) | a | The side length of the square base. The length and width of the base are the same, so one symbol is enough. |
| \(a^2\) | a squared | The number \(a\) multiplied by itself (\(a \times a\)). The small 2 at the upper right is an exponent. On this page it is the base area (the area of the square). |
| \(h\) | aitch | The height, from the first letter of "height". It is measured straight up (at a right angle) from the base to the apex, not along a slanted edge. |
| \(S\) | ess | A symbol used here for the base area (the area of the base). US textbooks often use \(B\) instead. For a square pyramid, \(S = a \times a\). |
| \(\mathrm{in^3}\) | cubic inches | A unit of volume. A cube with 1-inch sides has a volume of 1 in³. Do not mix it up with in² (square inches), the unit of area. |
| \(\mathrm{ft^3}\) | cubic feet | A unit of volume. A cube with 1-foot sides has a volume of 1 ft³, and \(1\,\mathrm{ft^3} = 1728\,\mathrm{in^3}\). |
| \(\mathrm{gal}\) | gallons | A unit of volume, mostly used for liquids. One US gallon is \(1\,\mathrm{gal} = 231\,\mathrm{in^3}\). It is the familiar unit on milk jugs and at the gas pump. |
Terms
| square pyramid | A solid with a four-sided base that narrows to a single point (the apex). This page covers the ones with a square base (the classic pyramid shape). |
| right square pyramid | A square pyramid whose apex is directly above the center of the square base. The pyramids of Egypt have this shape. Even if the apex is not directly above the center, the volume does not change as long as the base area and the height stay the same. |
| pyramid | The general name for a solid with a polygon base (a triangle, a four-sided shape, a pentagon and so on) that narrows to a single point. With a triangle base it is a triangular pyramid; with a four-sided base, a square or rectangular pyramid. |
| base area | The area of the bottom face (the base) of a solid. For a square pyramid, it is the area of the square base (side × side). |
| volume | The amount of space a solid takes up, given as a number. It is measured by how many cubes with 1-inch sides (1 in³) would fill it. |
| apex | The single point at the top of a pyramid or a cone, where the faces meet. Every solid that narrows from its base to one apex, pyramids and cones alike, has volume "base area × height ÷ 3". |
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.
| Multiplication and division (Grades 3–4) |
|
| Area of a square (Grade 3) |
|
| What volume is and its units (Grade 5) |
|
| Pyramids as solids (Grade 7 and geometry) |
|
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 square pyramid
| Base side length | 5 |
| Height | 12 |
| Pyramid volume | =B1^2*B2/3 |
Table to find the volume of any pyramid or cone (from the base area)
| Base area | 25 |
| Height | 12 |
| Volume | =B1*B2/3 |
Table to convert in³ to ft³
| Volume in in³ | 100 |
| Volume in ft³ | =B1/1728 |
After pasting, column A holds the item names and column B holds the numbers. The upper rows are your inputs, and the formula in the last row calculates automatically from them.
In a formula, "B1" and "B2" tell Excel to use the number in that cell. "*" is multiplication, "/" is division and "^" is a power (how many times to multiply).
In the first table, for example, B3 shows 5 × 5 × 12 ÷ 3 = 100 (in³ if you entered inches). The second table shows 100 in B3, and the third table shows about 0.0579 in B2. Just replace the input numbers with your own lengths.
In a formula, "B1" and "B2" tell Excel to use the number in that cell. "*" is multiplication, "/" is division and "^" is a power (how many times to multiply).
In the first table, for example, B3 shows 5 × 5 × 12 ÷ 3 = 100 (in³ if you entered inches). The second table shows 100 in B3, and the third table shows about 0.0579 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 square pyramid
| Base side length | 5 |
| Height | 12 |
| Pyramid volume | =B1^2*B2/3 |
Table to find the volume of any pyramid or cone (from the base area)
| Base area | 25 |
| Height | 12 |
| Volume | =B1*B2/3 |
Table to convert in³ to ft³
| Volume in in³ | 100 |
| Volume in ft³ | =B1/1728 |
The same formulas as in Excel 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
a = 5 # base side length (in inches in this example)
h = 12 # height (straight up from the base to the apex; same unit as a)
base_area = a ** 2 # base area (area of the square)
volume = base_area * h / 3 # pyramid volume (input unit cubed; in3 in this example)
volume_gal = volume / 231 # volume in US gallons, if you entered inches
print(f"Base area: {base_area} in2")
print(f"Pyramid volume: {volume} in3")
print(f"In gallons: {volume_gal} gal")
Runs with the standard library only. "**" is a power (squared), "*" is multiplication and "/" is division. Change the side and the height at the top and run it (this example uses inches; to get ft³ instead, divide the volume by 1728).
How to write it in LaTeX and other math languages (copy and paste)
Square pyramid volume formula
V = a² × h ÷ 3
V = \frac{1}{3} a^{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>
<msup><mi>a</mi><mn>2</mn></msup>
<mi>h</mi>
</mrow>
</math>
V = 1/3 a^2 h
a^2*h/3
V := a^2*h/3;
V = a^2*h/3;
V = (a^2 × h)/3
Volume formula for all pyramids and cones (from the base area)
V = S × h ÷ 3
V = \frac{1}{3} S 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>S</mi>
<mi>h</mi>
</mrow>
</math>
V = 1/3 S h
s*h/3
V := S*h/3;
V = s*h/3;
V = (S × h)/3
Converting volume units (in³ to 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 square pyramid has a square base with 5 in sides and a height of 12 in. Find each of the following: 1. The base area of this pyramid in in² 2. The volume of this pyramid in in³ (volume = base area × height ÷ 3) 3. 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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Finds the area of a square or rectangle (length × width), which is the base area of a pyramid
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