Enter the base radius and the height of the cone. You get the surface area (base area + lateral area), its breakdown into base area and lateral area, and the slant height.
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"). The height is measured straight up from the base to the apex, not along the sloping side (that is the slant height).
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 surface area of the cone on the spot (surface area = base area \(\pi r^2\) + lateral area \(\pi r l\))
- It also shows the breakdown into base area and lateral area, and the slant height \(l = \sqrt{r^2 + h^2}\)
- The result is also shown as a 3D shape. Turn it around with your mouse to see which length is which
- A plain-language explanation of why the lateral area is \(\pi r l\) 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, and calculates the total surface area including the base circle. For how much a cone holds (its volume), see the Volume of a Cone Calculator.
What is this calculation used for?
How much paper a party hat or craft cone needs (school and parties)
A party hat or a paper cone for a craft project has only the side, with no base, so the paper you need is the lateral area \(\pi r l\). For a hat with a rim radius of 3 in and a height of 8 in, the slant height is \(\sqrt{3^2 + 8^2} \approx 8.54\,\mathrm{in}\) and the lateral area is \(\pi \times 3 \times 8.54 \approx 80\,\mathrm{in^2}\).
You also need extra for the glue flap and trimming, so prepare a little more paper than this. Being able to estimate materials before you start is what makes this formula handy.
Roofing for a cone-shaped roof (construction)
For cone-shaped roofs on turrets, gazebos and silos, the amount of roofing (metal or shingles) is estimated from the lateral area. For a roof with a base radius of 10 ft and a height of 6 ft, the slant height is \(\sqrt{10^2 + 6^2} \approx 11.66\,\mathrm{ft}\) and the lateral area is \(\pi \times 10 \times 11.66 \approx 366\,\mathrm{ft^2}\), or about 3.7 roofing squares (1 square = 100 ft²).
Real jobs need overlaps between pieces and produce offcuts, so roofers add extra to this number when they order. This formula is the starting point of the estimate.
Fabric for a single-pole tent (outdoors)
A cone-shaped tent with a single pole in the middle, such as a tipi or a bell tent, uses about the lateral area for its walls and the base area for its floor (the groundsheet). With a radius of 6.5 ft at the bottom and a height of 8 ft, the walls are about \(210\,\mathrm{ft^2}\), the floor is \(\pi \times 6.5^2 \approx 133\,\mathrm{ft^2}\), and the total is about \(343\,\mathrm{ft^2}\).
A real tent has seam allowances and a door, so this is an estimate. Still, it gives you a feel for how heavy the fabric may be, or whether one can of waterproofing spray is enough.
Estimating the cost of plating or painting a part (manufacturing)
For plating or painting metal parts, the amount of chemicals or paint depends mostly on the surface area, so a cost estimate starts with it. For a cone-shaped part with a radius of 0.75 in and a height of 2 in, the surface area is \(\pi \times 0.75 \times (0.75 + \sqrt{0.75^2 + 2^2}) \approx 6.8\,\mathrm{in^2}\).
For parts made of cylinders and cones, you add up the surface area of each piece with formulas like this one to work out the processing cost and the amount of paint.
Batter for ice cream cones and cone sleeves (food)
An ice cream cone is the classic cone shape. For a cone with a rim radius of 1.25 in and a depth of 6 in, the slant height is \(\sqrt{1.25^2 + 6^2} \approx 6.13\,\mathrm{in}\) and the lateral area is \(\pi \times 1.25 \times 6.13 \approx 24\,\mathrm{in^2}\).
This lateral area is a guide to how much batter to spread for baking a cone, or how large a paper sleeve around the cone needs to be. If you also know that it unrolls into a sector, you can even make a paper pattern.
Formulas and figures
Surface area of a cone (base area + lateral area)
Figure
Standard notation
(the usual math form)
\(S\)
\(=\)
\(\pi r^{2}\)
\(+\)
\(\pi r l\)
In words
(symbols replaced with words)
③
\(S\): surface area of the cone
\(=\)
①
\(\pi r^2\): base area
\(+\)
②
\(\pi r l\): lateral area
The formula in words
①
Add the
\(\pi r^2\): base area
(the area of the base circle)
②
and the
\(\pi r l\): lateral area
(the area of the sloping side)
③
and you get the
\(S\): surface area of the cone
Quick example
For a cone with a base radius of 3 in and a height of 4 in (the slant height is 5 in; the next formula shows how to find it), the surface area is
\(S\): surface area of the cone
\(=\)
base area (9π in²)
\(+\)
lateral area (15π in²)
\(\pi \times 3^{2} + \pi \times 3 \times 5 = 9\pi + 15\pi = 24\pi\)
\(24\pi \approx 24 \times 3.14 = 75.36\,\mathrm{in^2}\)
Key idea
Surface area is the area of the whole outside of a solid. Cut a cone open with scissors and lay it flat (this flat pattern is called a net), and it splits into two pieces: the base circle and the side, which is a sector. The total area of these two pieces is the surface area.
The \(l\) in the formula is the slant height, the straight length along the sloping side from the apex to the edge of the base. It is not the same as the height \(h\), which is measured straight up. Also, the answer is an area, so its unit is in² (square inches). Do not mix it up with in³, the unit of volume.
Slant height (from the Pythagorean theorem)
Figure
Standard notation
(the usual math form)
\(l\)
\(=\)
\(\sqrt{r^{2} + h^{2}}\)
In words
(symbols replaced with words)
②
\(l\): slant height
\(=\)
①
\(\sqrt{r^2 + h^2}\): square root
The formula in words
①
Work out the
\(\sqrt{r^2 + h^2}\): square root
(square the radius and the height, add them, and take the square root of the sum)
②
and you get the
\(l\): slant height
Quick example
For a cone with a base radius of 3 in and a height of 4 in, the slant height is
\(l\): slant height
\(=\)
square root \(\sqrt{3^2 + 4^2}\)
\(l = \sqrt{3^{2} + 4^{2}} = \sqrt{9 + 16} = \sqrt{25} = 5\,\mathrm{in}\)
Key idea
The slant height is the straight segment along the sloping side from the apex to the edge of the base. Slice the cone straight down through the middle, and the height \(h\) (vertical), the base radius \(r\) (horizontal) and the slant height \(l\) (slanted) form the three sides of a right triangle. So the slant height, which is the hypotenuse, comes from the Pythagorean theorem.
This calculator asks for the radius and the height, so it first finds the slant height with this formula. When a problem gives you the slant height from the start (common in textbook problems), you can skip this step and go straight to the lateral area formula.
Lateral area (the side unrolls into a sector)
Figure
Standard notation
(the usual math form)
\(\pi r l\)
\(=\)
\(\dfrac{1}{2}\)
\(\times\)
\(2\pi r\)
\(\times\)
\(l\)
In words
(symbols replaced with words)
④
\(\pi r l\): lateral area
\(=\)
③
\(\dfrac{1}{2}\): one half
\(\times\)
①
\(2\pi r\): arc length
\(\times\)
②
\(l\): radius of the sector = slant height
The formula in words
①
Take the
\(2\pi r\): arc length
(= the circumference of the base)
②
multiply it by the
\(l\): radius of the sector = slant height
③
then multiply by
\(\dfrac{1}{2}\): one half
(divide by 2)
④
and you get the
\(\pi r l\): lateral area
Quick example
For a cone with a base radius of 3 in and a slant height of 5 in, the lateral area is
lateral area
\(=\)
\(\dfrac{1}{2}\): one half
\(\times\)
arc length (2π × 3 = 6π in)
\(\times\)
slant height (5 in)
\(\dfrac{1}{2} \times 2\pi \times 3 \times 5 = \dfrac{1}{2} \times 30\pi = 15\pi\)
\(15\pi \approx 15 \times 3.14 = 47.1\,\mathrm{in^2}\)
Key idea
Cut the side of a cone open and it becomes a sector whose radius is the slant height \(l\). The length of the sector's arc (the curved edge) is exactly the circumference of the base, \(2\pi r\), because before you cut it, the bottom edge of the side wrapped once around the edge of the base.
The area of a sector is "\(\dfrac{1}{2}\) × arc length × radius", so the lateral area is \(\dfrac{1}{2} \times 2\pi r \times l = \pi r l\). This form is very handy because you do not need the central angle. Remember it as "lateral area = π × radius × slant height".
One formula from the radius and height (the one this calculator uses)
Standard notation
(the usual math form)
\(S\)
\(=\)
\(\pi\)
\(\times\)
\(r\)
\(\times\)
\((\)
\(r\)
\(+\)
\(\sqrt{r^{2} + h^{2}}\)
\()\)
In words
(symbols replaced with words)
⑤
\(S\): surface area of the cone
\(=\)
①
\(\pi\): pi (about 3.14)
\(\times\)
②
\(r\): base radius
\(\times\)
\((\)
③
\(r\): base radius
\(+\)
④
\(l = \sqrt{r^2 + h^2}\): slant height
\()\)
The formula in words
①
Take
\(\pi\): pi (about 3.14)
②
multiply it by the
\(r\): base radius
③
then multiply by the sum of the
\(r\): base radius
④
and the
\(l = \sqrt{r^2 + h^2}\): slant height
⑤
and you get the
\(S\): surface area of the cone
Quick example
Finding the surface area of a cone with a base radius of 3 in and a height of 4 in in one go with this formula gives
\(S\): surface area of the cone
\(=\)
\(\pi\): pi
\(\times\)
radius (3 in)
\(\times\)
\((\)
radius (3 in)
\(+\)
slant height (√(3² + 4²) = 5 in)
\()\)
\(S = \pi \times 3 \times \left( 3 + \sqrt{3^{2} + 4^{2}} \right)\)
\(= \pi \times 3 \times (3 + 5) = 24\pi\)
\(24\pi \approx 24 \times 3.14 = 75.36\,\mathrm{in^2}\)
Key idea
This is formula 1 (base area + lateral area), \(\pi r^2 + \pi r l\), with the common factor \(\pi r\) factored out (\(\pi r^2 + \pi r l = \pi r (r + l)\)). The \(\sqrt{r^2 + h^2}\) from formula 2 is put in place of the slant height \(l\), so you can get the surface area in one step from just the radius and the height, without finding the slant height first. This calculator uses this formula too.
The surface area of a cone is "base area \(\pi r^2\) + lateral area \(\pi r l\)". If you do not know the slant height \(l\), the Pythagorean theorem gives \(l = \sqrt{r^2 + h^2}\). Put together, \(S = \pi r (r + \sqrt{r^2 + h^2})\), which you can calculate from just the radius and the height. The answer is in the square of the length unit (in² if you enter inches).
Symbols and terms
Symbols
| \(S\) | ess | A common symbol for area, from "surface". On this page it is the surface area of the cone (some textbooks write SA). |
| \(r\) | 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). |
| \(l\) | ell | The slant height of the cone (some textbooks use \(s\) or \(\ell\)). It is the length along the side from the apex to the edge of the base, and \(l = \sqrt{r^2 + h^2}\). |
| \(r^2\) | 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. |
| \(\sqrt{\phantom{x}}\) | square root (radical sign) | The symbol for a square root. \(\sqrt{25}\) is the positive number whose square is 25, which is 5. It can also be a number whose decimals never end, like \(\sqrt{2}\) (about 1.414). |
| \(\mathrm{in^2}\) | square inch | A unit of area. A square 1 in on each side has an area of 1 in². Do not mix it up with in³ (cubic inch), the unit of volume. |
| \(\mathrm{ft^2}\) | square foot | A unit of area. A square 1 ft on each side has an area of 1 ft². \(1\,\mathrm{ft^2} = 12 \times 12 = 144\,\mathrm{in^2}\) (note that it is not 12). |
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. |
| surface area | The area of the whole outside of a solid. For a cone, it is the area of the base (a circle) plus the area of the side (a sector). You use it to work out how much is needed to cover a surface, such as the amount of paint or the size of wrapping paper. |
| base area | The area of the face at the bottom of a solid. The base of a cone is a circle, so its base area is \(\pi r^2\) from the area of a circle. |
| lateral area | The area of the side of a solid (also called the lateral surface area). The side of a cone unrolls into a sector, and its area is \(\pi r l\) (π × radius × slant height). |
| 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. |
| net | The flat pattern you get by cutting a solid open and laying it flat. The net of a cone has two pieces, the base circle and a sector for the side. The surface area equals the total area of the net. |
| sector | A slice of a circle cut out by two radii, shaped like a folding fan or a slice of pizza. The side of a cone unrolls into a sector whose radius is the slant height \(l\). |
| arc | The curved edge of a sector (part of a circle). In the sector you get by unrolling the side of a cone, the arc length is exactly the circumference of the base, \(2\pi r\). |
| Pythagorean theorem | The rule that in a right triangle, the square of the hypotenuse equals the sum of the squares of the other two sides. In a cone, the radius, the height and the slant height form a right triangle, so the slant height is \(\sqrt{r^2 + h^2}\). |
| 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.
| Circumference and area of a circle (Grade 7) |
|
| Solid figures and nets (Grades 6–8) |
|
| Expressions with letters and the number π (Grades 6–7) |
|
| Square roots and the Pythagorean theorem (Grade 8) |
|
| Multiplying 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 surface area of a cone (base area + lateral area)
| Base radius (in) | 3 |
| Height (in) | 4 |
| Base area (in²) | =PI()*B1^2 |
| Lateral area (in²) | =PI()*B1*SQRT(B1^2+B2^2) |
| Surface area, base + lateral (in²) | =B3+B4 |
Table to find the slant height
| Base radius (in) | 3 |
| Height (in) | 4 |
| Slant height (in) | =SQRT(B1^2+B2^2) |
Table to find the lateral area (when you know the radius and slant height)
| Base radius (in) | 3 |
| Slant height (in) | 5 |
| Lateral area (in²) | =PI()*B1*B2 |
Table to find the surface area from the radius and height in one step
| Base radius (in) | 3 |
| Height (in) | 4 |
| Surface area (in²) | =PI()*B1*(B1+SQRT(B1^2+B2^2)) |
After pasting, column A holds the labels and column B holds the numbers. The upper rows are your inputs, and the formulas in the lower rows calculate from them automatically.
In a formula, "B1" and "B2" tell the formula to use the number in that cell. "PI()" is π (3.14159…), "SQRT()" is the square root, "^" raises to a power (how many times to multiply) and "*" is multiplication.
In the first table, for example, B3 shows about 28.27 (base area), B4 about 47.12 (lateral area) and B5 about 75.40 (surface area), in in² because the inputs are in inches. The second table shows 5 in B3, the third about 47.12 in B3 and the fourth about 75.40 in B3. 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…), "SQRT()" is the square root, "^" raises to a power (how many times to multiply) and "*" is multiplication.
In the first table, for example, B3 shows about 28.27 (base area), B4 about 47.12 (lateral area) and B5 about 75.40 (surface area), in in² because the inputs are in inches. The second table shows 5 in B3, the third about 47.12 in B3 and the fourth about 75.40 in B3. 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 surface area of a cone (base area + lateral area)
| Base radius (in) | 3 |
| Height (in) | 4 |
| Base area (in²) | =PI()*B1^2 |
| Lateral area (in²) | =PI()*B1*SQRT(B1^2+B2^2) |
| Surface area, base + lateral (in²) | =B3+B4 |
Table to find the slant height
| Base radius (in) | 3 |
| Height (in) | 4 |
| Slant height (in) | =SQRT(B1^2+B2^2) |
Table to find the lateral area (when you know the radius and slant height)
| Base radius (in) | 3 |
| Slant height (in) | 5 |
| Lateral area (in²) | =PI()*B1*B2 |
Table to find the surface area from the radius and height in one step
| Base radius (in) | 3 |
| Height (in) | 4 |
| Surface area (in²) | =PI()*B1*(B1+SQRT(B1^2+B2^2)) |
The same formulas as in Excel (including the PI() and SQRT() functions) 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 = 4 # height (same unit as the radius)
slant = math.sqrt(radius ** 2 + height ** 2) # slant height (Pythagorean theorem)
base_area = math.pi * radius ** 2 # base area (area of the base circle)
lateral_area = math.pi * radius * slant # lateral area (pi x radius x slant height)
total_area = base_area + lateral_area # surface area (base area + lateral area)
print(f"Slant height: {slant} in")
print(f"Base area: {base_area} in2")
print(f"Lateral area: {lateral_area} in2")
print(f"Surface area of the cone: {total_area} in2")
Runs with the standard library only. "math.sqrt()" is the square root, "math.pi" is π (3.14159…), "**" raises to a power (radius ** 2 is the radius squared) and "*" is multiplication. Change the radius and height at the top and run it (enter inches and the answer is in in²; enter feet and it is in ft²).
How to write it in LaTeX and other math languages (copy and paste)
Surface area of a cone (base area + lateral area)
S = πr² + πrl
S = \pi r^{2} + \pi r l
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
<mrow>
<mi>S</mi>
<mo>=</mo>
<mi>π</mi>
<msup><mi>r</mi><mn>2</mn></msup>
<mo>+</mo>
<mi>π</mi>
<mi>r</mi>
<mi>l</mi>
</mrow>
</math>
S = pi r^2 + pi r l
Pi*r^2 + Pi*r*l
S := Pi*r^2 + Pi*r*l;
S = pi*r^2 + pi*r*l;
S = πr^2 + πrl
Slant height (from the Pythagorean theorem)
l = √(r² + h²)
l = \sqrt{r^{2} + h^{2}}
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
<mrow>
<mi>l</mi>
<mo>=</mo>
<msqrt>
<mrow>
<msup><mi>r</mi><mn>2</mn></msup>
<mo>+</mo>
<msup><mi>h</mi><mn>2</mn></msup>
</mrow>
</msqrt>
</mrow>
</math>
l = sqrt(r^2 + h^2)
Sqrt[r^2 + h^2]
l := sqrt(r^2 + h^2);
l = sqrt(r^2 + h^2);
l = √(r^2 + h^2)
Lateral area (the side unrolls into a sector)
πrl = 1/2 × 2πr × l
\pi r l = \frac{1}{2} \times 2\pi r \times l
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
<mrow>
<mi>π</mi><mi>r</mi><mi>l</mi>
<mo>=</mo>
<mfrac><mn>1</mn><mn>2</mn></mfrac>
<mo>×</mo>
<mn>2</mn><mi>π</mi><mi>r</mi>
<mo>×</mo>
<mi>l</mi>
</mrow>
</math>
pi r l = 1/2 xx 2 pi r xx l
Pi*r*l == (1/2)*(2*Pi*r)*l
Pi*r*l = (1/2)*(2*Pi*r)*l;
pi*r*l == (1/2)*(2*pi*r)*l
πrl = (1/2)(2πr)l
One formula from the radius and height (the one this calculator uses)
S = πr(r + √(r² + h²))
S = \pi r \left( r + \sqrt{r^{2} + h^{2}} \right)
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
<mrow>
<mi>S</mi>
<mo>=</mo>
<mi>π</mi>
<mi>r</mi>
<mo>(</mo>
<mi>r</mi>
<mo>+</mo>
<msqrt>
<mrow>
<msup><mi>r</mi><mn>2</mn></msup>
<mo>+</mo>
<msup><mi>h</mi><mn>2</mn></msup>
</mrow>
</msqrt>
<mo>)</mo>
</mrow>
</math>
S = pi r (r + sqrt(r^2 + h^2))
Pi*r*(r + Sqrt[r^2 + h^2])
S := Pi*r*(r + sqrt(r^2 + h^2));
S = pi*r*(r + sqrt(r^2 + h^2));
S = πr(r + √(r^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 cone has a base radius of 3 in and a height of 4 in (a right circular cone, with its apex directly above the center of the base). Find each of the following: 1. The slant height (l = √(radius² + height²)) 2. The base area (πr²) 3. The lateral area (π × radius × slant height) 4. The surface area of the cone (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
Volume of a Cone Calculator (from Radius and Height)
Calculates how much the same cone holds, its volume (1/3 × π × radius² × height)
Area of a Sector Calculator (from Radius and Central Angle)
The side of a cone unrolls into a sector. Calculates the area, arc length and central angle of a sector
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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