Enter the base side length and the height of a square pyramid (a pyramid with a square base). You get the surface area (base area + lateral area), plus the slant height, base area and lateral area used along the way.
Enter the base side and the height in the same unit, as numbers only (no units. For example, for 6 cm enter "6"). The height is the straight up-and-down distance from the base to the apex (it is different from the slant height, the height of a triangular side).
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 surface area of a square pyramid (base area + lateral area) on the spot
- The pyramid you entered is drawn in 3D (drag to spin it around)
- The slant height (the height of each triangular side), the base area, the lateral area and the steps are shown too, so you can check how the answer was reached, not just the answer
- A plain-language explanation of why the Pythagorean theorem comes in and how the height differs from the slant height, plus 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 or a distance along a side face). This page assumes a right square pyramid, with the apex directly above the center of the base (unlike the volume, the surface area changes if the apex is off center). Cones, rectangular prisms and other solids are covered on their own pages.
What is this calculation used for?
Estimating the area of the casing stones that once covered a pyramid (history)
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 outside was once covered with white limestone "casing stones". Their area (the lateral area) is about \(2 \times 756 \times 612 \approx 925{,}000\,\mathrm{ft^2}\), since the slant height is \(\sqrt{378^2 + 481^2} \approx 612\,\mathrm{ft}\). That is about 21 acres, or about 16 American football fields including the end zones.
The real surface is stepped stonework, and almost all the casing stones are gone today, so this is an estimate. Still, one lateral area formula lets you put the scale of an ancient mega-project into numbers (the base sits on the ground, so for buildings you normally look at the lateral area only).
Estimating the fabric of a pyramid tent (camping)
A tent held up by a single center pole (a pyramid tent) is almost a square pyramid if its floor is square. For a 12 ft × 12 ft floor and an 8 ft pole, the slant height is \(\sqrt{6^2 + 8^2} = 10\,\mathrm{ft}\), and the fabric for the sides is \(2 \times 12 \times 10 = 240\,\mathrm{ft^2}\). Including the floor, it is \(240 + 144 = 384\,\mathrm{ft^2}\).
Knowing the fabric area helps you estimate how many cans of waterproofing spray you need, how much seam tape to prepare, and about how much the fabric weighs (many real tents have short vertical walls at the bottom, so this is an estimate).
Estimating roofing material for a pyramid roof (construction)
A pyramid hip roof on a square building is exactly the 4 sides of a square pyramid. For a 24 ft × 24 ft roof that rises 5 ft, the slant height is \(\sqrt{12^2 + 5^2} = 13\,\mathrm{ft}\), and the roof area is \(2 \times 24 \times 13 = 624\,\mathrm{ft^2}\), or about 6.2 "squares" (roofers measure in squares of 100 ft²).
The roof area is the basis for how many shingles, how much metal roofing or underlayment you need, and for repainting estimates (a real roof is larger by the overhang of the eaves, so this is only a guide). A roof always has more area than its footprint measured flat, and this formula counts that extra area correctly.
Solving a classic geometry test question (school)
The surface area of a right square pyramid is a staple of middle school math, where it is found from a net, and problems with the slant height go together with the Pythagorean theorem (Grade 8) in geometry class and on tests. For example, "What is the surface area of a right square pyramid with a base side of 6 in and a height of 4 in?" The slant height is \(\sqrt{3^2 + 4^2} = 5\,\mathrm{in}\), so the answer is \(36 + 6 \times 5 \div 2 \times 4 = 96\,\mathrm{in^2}\).
The most common mistake is to use the height of the pyramid instead of the slant height for the triangle area. Remember "the height of a triangular side = the slant height", and you will get these points on the test.
Formulas and figures
Surface area of a square pyramid (add the base area and the lateral area)
Figure
Standard notation
(the usual math form)
\(S\)
\(=\)
\(S_1\)
\(+\)
\(S_2\)
In words
(symbols replaced with words)
③
\(S\): surface area of the pyramid
\(=\)
①
\(S_1\): base area
\(+\)
②
\(S_2\): lateral area
The formula in words
①
Add the
\(S_1\): base area
②
and the
\(S_2\): lateral area
to get the
③
\(S\): surface area of the pyramid
Quick example
The surface area of a square pyramid with a base side of 6 in and a height of 4 in (base area 36 in², lateral area 60 in²; see Formulas 2 to 4 below for how to find them) is
\(S\): surface area of the pyramid
\(=\)
base area (36 in²)
\(+\)
lateral area (60 in²)
\(36 + 60 = 96\,\mathrm{in^2}\)
Key idea
Surface area is the area of the whole outside of a solid, which is the total area of its net (the flat shape you get by cutting the solid open and laying it flat). The net of a square pyramid is "1 square (the base) + 4 congruent isosceles triangles (the sides)", so the surface area is the base area plus the lateral area (the total of the 4 triangles).
When you only want the area of the 4 triangles (the lateral area), use \(S_2\) before adding. This is the one you need for things with no bottom, such as the fabric of a tent or the area of a roof.
Base area (area of the square)
Figure
Standard notation
(the usual math form)
\(S_1\)
\(=\)
\(a\)
\(2\)
In words
(symbols replaced with words)
③
\(S_1\): base area
\(=\)
①
\(a\): base side
②
squared (the length times itself)
The formula in words
①
Take the
\(a\): base side
and
②
find its
square (the length times itself)
to get the
③
\(S_1\): base area
Quick example
For a square pyramid with a base side of 6 in, the base area is
\(S_1\): base area
\(=\)
base side (6 in)
squared
\(6 \times 6 = 36\,\mathrm{in^2}\)
Key idea
The base is a square, so its area is "side × side". It is the same number multiplied by itself, so it can be written \(a^2\) (\(a\) squared). For a rectangular base (length and width different), do not use this page's formula: find the base area as "length × width", and treat the sides as two different kinds of triangles.
Finding the slant height (height of a triangular side) with the Pythagorean theorem
Figure
Standard notation
(the usual math form)
\(\left(\dfrac{a}{2}\right)^{\!2}\)
\(+\)
\(h^2\)
\(=\)
\(l^2\)
In words
(symbols replaced with words)
①
half the side, \(\dfrac{a}{2}\), squared
\(+\)
②
height \(h\) squared
\(=\)
③
slant height \(l\) squared
The formula in words
①
Add
half the side, \(\dfrac{a}{2}\), squared
②
and
height \(h\) squared
to get
③
slant height \(l\) squared
(take the square root at the end to get the slant height \(l\) itself)
Quick example
For a square pyramid with a base side of 6 in and a height of 4 in, the slant height is
half the side (3 in) squared
\(+\)
height (4 in) squared
\(=\)
slant height squared
\(3^2 + 4^2 = 9 + 16 = 25\)
\(l = \sqrt{25} = 5\,\mathrm{in}\)
Key idea
The slant height \(l\) is the length of the line drawn from the top of a triangular side straight down to its base edge, at a right angle. In other words, it is the height of a triangular side. It is different from the height \(h\) of the pyramid (the straight up-and-down distance from the base to the apex), so be careful not to mix them up.
Hidden inside the pyramid is a right triangle whose three sides are the height \(h\), the distance \(\dfrac{a}{2}\) from the center of the base to the middle of a base edge, and the slant height \(l\). The rule for the three sides of a right triangle is the Pythagorean theorem, and this formula applies it. The copy-and-paste formulas below are solved for \(l\), with the square root taken: \(l = \sqrt{\left(\dfrac{a}{2}\right)^2 + h^2}\).
Lateral area (4 congruent triangles)
Figure
Standard notation
(the usual math form)
\(S_2\)
\(=\)
\(a\)
\(\times\)
\(l\)
\(\div\)
\(2\)
\(\times\)
\(4\)
In words
(symbols replaced with words)
⑤
\(S_2\): lateral area
\(=\)
①
\(a\): base side
\(\times\)
②
\(l\): slant height
\(\div\)
③
\(2\): split in half
\(\times\)
④
\(4\): number of side faces
The formula in words
①
Multiply the
\(a\): base side
②
by the
\(l\): slant height
,
③
divide by
\(2\): split in half
to get the area of one triangular side,
④
and multiply by
\(4\): number of side faces
to get the
⑤
\(S_2\): lateral area
Quick example
For a base side of 6 in and a slant height of 5 in (the pyramid with a height of 4 in; see Formula 3 for how to find the slant height), the lateral area is
\(S_2\): lateral area
\(=\)
base side (6 in)
\(\times\)
slant height (5 in)
\(\div\)
split in half (2)
\(\times\)
number of side faces (4)
\(6 \times 5 \div 2 = 15\,\mathrm{in^2}\)
\(15 \times 4 = 60\,\mathrm{in^2}\)
Key idea
The sides are 4 congruent isosceles triangles, each with base \(a\) and height \(l\) (the slant height). Find the area of one with the triangle area formula "base × height ÷ 2", then multiply by 4. The biggest stumbling block is that the "height" of the triangle here is the slant height \(l\), not the height \(h\) of the pyramid.
"\(\div 2 \times 4\)" simplifies to "\(\times 2\)", so the copy-and-paste formulas use the neater form \(S_2 = 2al\).
All in one formula (from just the side and the height)
Standard notation
(the usual math form)
\(S\)
\(=\)
\(a^2\)
\(+\)
\(2a\sqrt{\left(\dfrac{a}{2}\right)^{\!2} + h^2}\)
In words
(symbols replaced with words)
③
\(S\): surface area of the pyramid
\(=\)
①
base area (side \(a\) squared)
\(+\)
②
lateral area \(2al\)
The formula in words
①
Add the
base area (side \(a\) squared)
②
and the
lateral area \(2al\)
(the slant height \(l\) is replaced by the square root from Formula 3, \(\sqrt{\left(\dfrac{a}{2}\right)^2 + h^2}\)) to get the
③
\(S\): surface area of the pyramid
Quick example
The surface area of a square pyramid with a base side of 6 in and a height of 4 in, calculated with one formula, is
\(S\): surface area of the pyramid
\(=\)
base area (6 in squared)
\(+\)
lateral area (2 × 6 in × slant height)
\(l = \sqrt{3^2 + 4^2} = \sqrt{25} = 5\)
\(S = 6^2 + 2 \times 6 \times 5 = 36 + 60 = 96\,\mathrm{in^2}\)
Key idea
This formula puts Formulas 2 to 4 into Formula 1 to make a single formula, and many math references show it in this form. It is handy when you want to type everything into a calculator or a program at once.
By hand, it is better to go step by step, as in Formulas 2 to 4 (base area → slant height → lateral area → total), because that makes it easier to check your work along the way.
The surface area of a square pyramid is "base area (side × side) + lateral area (side × slant height ÷ 2 × 4 triangles)". The slant height (the height of a triangular side) comes from half the side and the height, using the Pythagorean theorem. The height of the pyramid and the slant height are different, so always use the slant height for the triangle area. The answer is in the length unit squared (in² for inches).
Symbols and terms
Symbols
| \(S\) | ess | A common symbol for area, from the first letter of "surface". On this page it stands for the surface area of the square pyramid. |
| \(S_1\) | S sub 1 | The symbol for the base area (the area of the square base) on this page. The small 1 at the lower right is a subscript for "the first \(S\)". \(S_1 = a^2\). |
| \(S_2\) | S sub 2 | The symbol for the lateral area (the area of the 4 triangular sides) on this page. \(S_2 = 2al\) (side × slant height ÷ 2 × 4 triangles). |
| \(a\) | 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 of the pyramid, from the first letter of "height". It is measured straight up (at a right angle) from the base to the apex, and is different from a slanted edge or the slant height. |
| \(l\) | el | The symbol for the slant height (the height of a triangular side) on this page, as in US textbooks. It is found with \(l = \sqrt{\left(\dfrac{a}{2}\right)^2 + h^2}\). |
| \(\sqrt{\ }\) | square root (radical sign) | The symbol for a square root (a number that gives the number under the root when squared). For example, \(\sqrt{25} = 5\), because 5 squared is 25. It is used in the last step of finding the slant height. |
| \(\mathrm{in^2}\) | square inches | A unit of area. A square with 1-inch sides has an area of 1 in². Do not mix it up with in³ (cubic inches), the unit of volume. |
| \(\mathrm{ft^2}\) | square feet | A unit of area. A square with 1-foot sides has an area of 1 ft², and \(1\,\mathrm{ft^2} = 144\,\mathrm{in^2}\) (the length factor 12, squared). |
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. The surface area formulas on this page assume a right square pyramid (if the apex is off center, the 4 triangles are no longer congruent, and the surface area changes). |
| surface area | The area of the whole outside of a solid. It equals the total area of the net, and for a square pyramid it is "base area + lateral area". It is a different quantity from volume, which measures the space inside. |
| 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). |
| lateral area | The total area of the side faces of a solid (all faces except the base). For a square pyramid it is 4 congruent isosceles triangles, found with "side × slant height ÷ 2 × 4". |
| slant height | The length of the line from the top of a triangular side straight down to its base edge, at a right angle (the height of a triangular side). It is different from the height of the pyramid, and it is found with the Pythagorean theorem \(\left(\dfrac{a}{2}\right)^2 + h^2 = l^2\). |
| net | The flat shape you get by cutting a solid along its edges and laying it out flat. The net of a square pyramid is "1 square and 4 isosceles triangles", and their total area is the surface area. |
| Pythagorean theorem | The theorem that in a right triangle, "the sum of the squares of the two legs = the square of the hypotenuse". On this page it is applied to the right triangle formed by the height, half the side and the slant height, to find the slant height. |
| congruent | Exactly the same shape and size. The 4 side faces of a right square pyramid are congruent isosceles triangles, so the area of one times 4 gives the lateral area. |
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) |
|
| Area of a triangle (Grade 6) |
|
| Pyramids and nets (Grade 6) |
|
| The Pythagorean theorem (Grade 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 surface area of a square pyramid (base area + lateral area)
| Base area | 36 |
| Lateral area | 60 |
| Surface area | =B1+B2 |
Table to find the base area (area of the square)
| Base side length | 6 |
| Base area | =B1^2 |
Table to find the slant height (height of a triangular side)
| Base side length | 6 |
| Height | 4 |
| Slant height | =SQRT((B1/2)^2+B2^2) |
Table to find the lateral area (all 4 triangles)
| Base side length | 6 |
| Slant height | 5 |
| Lateral area | =B1*B2/2*4 |
Table for the all-in-one formula (from the side and the height)
| Base side length | 6 |
| Height | 4 |
| Surface area | =B1^2+2*B1*SQRT((B1/2)^2+B2^2) |
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, "^" is a power (how many times to multiply) and "SQRT" is the function for a square root.
In the first table, for example, B3 shows 36 + 60 = 96 (in² if you entered inches). The third table shows 5 in B3, the fourth shows 60 in B3, and the fifth shows 96 in B3. 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, "^" is a power (how many times to multiply) and "SQRT" is the function for a square root.
In the first table, for example, B3 shows 36 + 60 = 96 (in² if you entered inches). The third table shows 5 in B3, the fourth shows 60 in B3, and the fifth shows 96 in B3. 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 surface area of a square pyramid (base area + lateral area)
| Base area | 36 |
| Lateral area | 60 |
| Surface area | =B1+B2 |
Table to find the base area (area of the square)
| Base side length | 6 |
| Base area | =B1^2 |
Table to find the slant height (height of a triangular side)
| Base side length | 6 |
| Height | 4 |
| Slant height | =SQRT((B1/2)^2+B2^2) |
Table to find the lateral area (all 4 triangles)
| Base side length | 6 |
| Slant height | 5 |
| Lateral area | =B1*B2/2*4 |
Table for the all-in-one formula (from the side and the height)
| Base side length | 6 |
| Height | 4 |
| Surface area | =B1^2+2*B1*SQRT((B1/2)^2+B2^2) |
The same formulas as in Excel (the SQRT function has the same name) 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
a = 6 # base side length (in inches in this example)
h = 4 # height (straight up from the base to the apex; same unit as a)
base_area = a ** 2 # base area (area of the square)
slant_height = math.sqrt((a / 2) ** 2 + h ** 2) # slant height (Pythagorean theorem)
lateral_area = a * slant_height / 2 * 4 # lateral area (1 triangle x 4)
total_area = base_area + lateral_area # surface area of the pyramid
print(f"Base area: {base_area} in2")
print(f"Slant height: {slant_height} in")
print(f"Lateral area: {lateral_area} in2")
print(f"Surface area: {total_area} in2")
Runs with the standard library only. "**" is a power (squared), "math.sqrt" is the square root, "*" is multiplication and "/" is division. Change the side and the height at the top and run it (this example prints a base area of 36, a slant height of 5.0, a lateral area of 60.0 and a surface area of 96.0).
How to write it in LaTeX and other math languages (copy and paste)
Surface area of a square pyramid (add the base area and the lateral area)
S = S₁ + S₂
S = S_{1} + S_{2}
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
<mrow>
<mi>S</mi>
<mo>=</mo>
<msub><mi>S</mi><mn>1</mn></msub>
<mo>+</mo>
<msub><mi>S</mi><mn>2</mn></msub>
</mrow>
</math>
S = S_1 + S_2
s1 + s2
S := S1 + S2;
S = S1 + S2;
S = S_1 + S_2
Base area (area of the square)
S₁ = a²
S_{1} = a^{2}
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
<mrow>
<msub><mi>S</mi><mn>1</mn></msub>
<mo>=</mo>
<msup><mi>a</mi><mn>2</mn></msup>
</mrow>
</math>
S_1 = a^2
a^2
S1 := a^2;
S1 = a^2;
S_1 = a^2
Finding the slant height (height of a triangular side) with the Pythagorean theorem
l = √((a/2)² + h²)
l = \sqrt{\left(\frac{a}{2}\right)^{2} + h^{2}}
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
<mrow>
<mi>l</mi>
<mo>=</mo>
<msqrt>
<mrow>
<msup>
<mrow><mo>(</mo><mfrac><mi>a</mi><mn>2</mn></mfrac><mo>)</mo></mrow>
<mn>2</mn>
</msup>
<mo>+</mo>
<msup><mi>h</mi><mn>2</mn></msup>
</mrow>
</msqrt>
</mrow>
</math>
l = sqrt((a/2)^2 + h^2)
Sqrt[(a/2)^2 + h^2]
l := sqrt((a/2)^2 + h^2);
l = sqrt((a/2)^2 + h^2);
l = √((a/2)^2 + h^2)
Lateral area (4 congruent triangles)
S₂ = 2al
S_{2} = 2al
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
<mrow>
<msub><mi>S</mi><mn>2</mn></msub>
<mo>=</mo>
<mn>2</mn>
<mi>a</mi>
<mi>l</mi>
</mrow>
</math>
S_2 = 2 a l
2*a*l
S2 := 2*a*l;
S2 = 2*a*l;
S_2 = 2al
All in one formula (from just the side and the height)
S = a² + 2a√((a/2)² + h²)
S = a^{2} + 2a\sqrt{\left(\frac{a}{2}\right)^{2} + h^{2}}
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
<mrow>
<mi>S</mi>
<mo>=</mo>
<msup><mi>a</mi><mn>2</mn></msup>
<mo>+</mo>
<mn>2</mn>
<mi>a</mi>
<msqrt>
<mrow>
<msup>
<mrow><mo>(</mo><mfrac><mi>a</mi><mn>2</mn></mfrac><mo>)</mo></mrow>
<mn>2</mn>
</msup>
<mo>+</mo>
<msup><mi>h</mi><mn>2</mn></msup>
</mrow>
</msqrt>
</mrow>
</math>
S = a^2 + 2 a sqrt((a/2)^2 + h^2)
a^2 + 2*a*Sqrt[(a/2)^2 + h^2]
S := a^2 + 2*a*sqrt((a/2)^2 + h^2);
S = a^2 + 2*a*sqrt((a/2)^2 + h^2);
S = a^2 + 2a√((a/2)^2 + h^2)
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 right square pyramid has a square base with 6 in sides and a height of 4 in. Find each of the following: 1. The base area of this pyramid in in² 2. The slant height of this pyramid (the height of a triangular side) in inches (slant height = √((half the side)² + height²)) 3. The lateral area of this pyramid in in² (lateral area = side × slant height ÷ 2 × 4) 4. The surface area of this pyramid in in² (surface area = base area + lateral area) Show the formulas you used and the numbers from the execution result.
How to Use
-
1Enter your numbersType the numbers you want to calculate with into the input fields
-
2CalculatePress the "Calculate" button
-
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
Square Pyramid Volume Calculator (Base Area × Height ÷ 3)
Finds the volume (how much space is inside) of the same pyramid. A good match for the surface area
Pythagorean Theorem Calculator (Hypotenuse and Legs of a Right Triangle)
Works with the relation between the three sides of a right triangle, the idea used to find the slant height
Triangle Area Calculator (from Three Sides with Heron's Formula)
Finds the area of a triangle, the shape of each side face of the pyramid
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