Geometry
How To Calculate The Surface Area Of A Cone
The surface area of a cone is the flat circular base plus the curved side that tapers to the point. The only extra quantity you need beyond radius and height is the slant height, the distance down the outside of the cone.
Quick Answer
total surface area = pi x r x l + pi x r^2, where l = sqrt(r^2 + h^2)
- r
- The radius of the circular base
- h
- The perpendicular height from base to point
- l
- The slant height, measured down the side
- total
- Base area plus lateral area
Find the slant height first with the Pythagorean theorem, then add the curved lateral area to the flat base. A radius of 5 with a height of 12 gives a slant height of 13, a lateral area of 204.20352248333654, a base of 78.53981633974483, and a total of 282.7433388230814.
What Is The Surface Area Of A Cone?
A cone has two distinct surfaces. The base is a flat circle, and the side is a curved surface that tapers from the rim of that circle to a single point above the centre. Any surface area calculation has to handle both, and they are different shapes.
The base is straightforward: pi times the radius squared. For a radius of 5 that is 78.53981633974483, the same formula as any circle.
The side is where the work is. Its area is pi times the radius times the slant height, and the slant height is not the vertical height. It is the distance measured along the outside of the cone from the rim to the point.
The slant height comes from the Pythagorean theorem. The radius, the vertical height and the slant height form a right-angled triangle inside the cone, so the slant height is the square root of the radius squared plus the height squared. For a radius of 5 and a height of 12 that is the square root of 169, which is exactly 13.
Adding the two pieces gives the total. The side contributes 204.20352248333654 and the base contributes 78.53981633974483, for a total surface area of 282.7433388230814. The side is the larger part of the two, which is true of any cone taller than it is wide.
If you only need the curved surface, skip the base. Lateral area of pi times radius times slant height gives 204.20352248333654 on its own. Ask whether the question wants the base included, because the two answers differ by the whole circle area.
A right cone has its point directly above the centre of the base, which is what these formulas assume. An oblique cone, where the point is off to one side, has a larger surface area for the same base and height and needs a different treatment.
Doubling the radius does not double the area, it roughly quadruples it. Both terms grow with the square of the radius when the height is held in proportion, and the lateral term additionally grows with the slant height. Scaling is a common source of error when people reason about cones intuitively.
Net of a cone is a useful check. Cut the side along a straight line and unroll it, and you get a circular sector whose radius is the slant height and whose arc is the circumference of the base. That sector area works out to pi times radius times slant height, which is exactly the lateral formula.
Practical uses include funnels, hoppers, lamp shades and filters. In each case the material needed is the surface area, and the seam or overlap allowance is added on top, which is why the exact number matters before you cut anything.
Formula
l = sqrt(r^2 + h^2)
Formula page →The Pythagorean theorem applied to the triangle inside the cone.
| Symbol | Meaning | Unit | Notes |
|---|---|---|---|
| l | Slant height | same length unit | The distance along the side from rim to point. |
| r | Radius | same length unit | The radius of the circular base. |
| h | Height | same length unit | The perpendicular height of the cone. |
lateral area = pi x r x l
Formula page →The curved side only, which unrolls into a circular sector.
| Symbol | Meaning | Unit | Notes |
|---|---|---|---|
| A_lateral | Lateral area | square length units | The area of the curved side. |
| r | Radius | same length unit | The radius of the base, which is the arc of the sector. |
| l | Slant height | same length unit | The radius of the unrolled sector. |
total surface area = pi x r x l + pi x r^2
Formula page →The curved side plus the flat circular base.
| Symbol | Meaning | Unit | Notes |
|---|---|---|---|
| A_total | Total surface area | square length units | The whole outside of the cone. |
| A_lateral | Lateral area | square length units | The curved side alone. |
| A_base | Base area | square length units | The flat circle, pi times radius squared. |
How To Calculate The Surface Area Of A Cone
- 1
Find the slant height
Square the radius, square the height, add them and take the square root. For a radius of 5 and a height of 12 the slant height is sqrt(25 + 144), which is exactly 13.
- 2
Compute the base area
Pi times the radius squared. For a radius of 5 that is 78.53981633974483. Skip this if you only need the lateral area.
- 3
Compute the lateral area
Pi times the radius times the slant height. For a radius of 5 and a slant height of 13 that is 204.20352248333654.
- 4
Add the two together
204.20352248333654 plus 78.53981633974483 gives a total surface area of 282.7433388230814.
- 5
Sanity check the shape
The lateral area must be larger than the base for any cone taller than it is wide, and the total must exceed both parts. For these numbers the side is the dominant term.
Examples
Example 1: A cone with radius 5 and height 12
- Radius
- 5
- Height
- 12
| Step | Calculation | Result |
|---|---|---|
| Radius squared | 5 x 5 | 25 |
| Height squared | 12 x 12 | 144 |
| Slant height | sqrt(25 + 144) | 13 |
| Base area | pi x 5 x 5 | 78.53981633974483 |
| Lateral area | pi x 5 x 13 | 204.20352248333654 |
| Total surface area | 204.20352248333654 + 78.53981633974483 | 282.7433388230814 |
Result: A cone with radius 5 and height 12 has a slant height of exactly 13, a base area of 78.53981633974483, a lateral area of 204.20352248333654, and a total surface area of 282.7433388230814.
Example 2: A cone with radius 3 and height 4
- Radius
- 3
- Height
- 4
| Step | Calculation | Result |
|---|---|---|
| Radius squared | 3 x 3 | 9 |
| Height squared | 4 x 4 | 16 |
| Slant height | sqrt(9 + 16) | 5 |
| Base area | pi x 3 x 3 | 28.274333882308138 |
| Lateral area | pi x 3 x 5 | 47.12388980384689 |
| Total surface area | 47.12388980384689 + 28.274333882308138 | 75.39822368615503 |
Result: A cone with radius 3 and height 4 has a slant height of 5, a lateral area of 47.12388980384689, and a total surface area of 75.39822368615503 including the base.
Example 3: A cone with radius 7 and height 24, side only
- Radius
- 7
- Height
- 24
| Step | Calculation | Result |
|---|---|---|
| Radius squared | 7 x 7 | 49 |
| Height squared | 24 x 24 | 576 |
| Slant height | sqrt(49 + 576) | 25 |
| Base area | pi x 7 x 7 | 153.93804002589985 |
| Lateral area | pi x 7 x 25 | 549.7787143782137 |
| Lateral area as a multiple of the base | 549.7787143782137 / 153.93804002589985 | 3.571428571428571 |
Result: A cone with radius 7 and height 24 has a slant height of 25 and a lateral area of 549.7787143782137, which is 3.571428571428571 times the base area of 153.93804002589985. The total, if the base is closed, comes to 703.7167544041135.
Calculator
Total surface area
282.7433
- Slant height
- 13
- Lateral (curved) area
- 204.2035
- Base area
- 78.5398
- Curved side as a share of the total
- 72.22%
Values update as you type. This calculator covers the single scenario its formula assumes — see Common Mistakes for what it leaves out.
Prefer a full-width tool? Open the The Surface Area Of A Cone calculator page.
Common Mistakes
Using the vertical height instead of the slant height
The lateral formula needs the distance along the side, not straight down. For a radius of 5 and height of 12, using 12 gives 188.4955592153876 instead of the correct 204.20352248333654.
Forgetting the base
Many questions ask for total surface area, which means the flat circle is included. Leaving it out drops 78.53981633974483 from the answer.
Including the base when only the lateral area is wanted
An open-topped funnel has no base, so pi times radius times slant height alone is the answer. Reading the question carefully decides which of the two you need.
Squaring the slant height with the height by mistake
The slant height is the square root of radius squared plus height squared. Adding the two squares without the square root gives 169, not 13.
Doubling the base instead of the lateral area
A cone has one curved side and one flat base. The total is lateral plus base, not lateral plus two bases.
Applying the cylinder formula to a cone
A cone tapers to a point, so its lateral area is pi times radius times slant height rather than 2 pi times radius times height. The two are not interchangeable.
FAQ
What is the formula for the surface area of a cone?
Total surface area is pi times radius times slant height plus pi times radius squared. The first term is the curved side and the second is the flat base.
What is the slant height of a cone?
It is the distance measured along the outside of the cone from the rim to the point, equal to the square root of the radius squared plus the height squared.
What is the lateral area of a cone?
Pi times the radius times the slant height, which covers only the curved side. For a radius of 5 and a slant height of 13 it is 204.20352248333654.
What is the total surface area of a cone with radius 5 and height 12?
The slant height is 13, the lateral area is 204.20352248333654, the base is 78.53981633974483, and the total is 282.7433388230814.
How is a cone's surface area different from a cylinder's?
A cone tapers to a point so it has a lateral area of pi times radius times slant height plus one base. A cylinder has a lateral area of 2 pi times radius times height plus two bases.
Does the formula work for an oblique cone?
No. These formulas assume a right cone with the point directly above the centre. An oblique cone needs integration or a correction factor for the same base and height.
References
- [1]Wikipedia, Cone — https://en.wikipedia.org/wiki/Cone
- [2]Wikipedia, Slant height — https://en.wikipedia.org/wiki/Slant_height
- [3]MathWorld, Circular cone — https://mathworld.wolfram.com/CircularCone.html