Cone Surface Area Calculator
Press Calculate to see the area, the working and other units.
Find the surface area of a cone from its radius and either its height or its slant height. The result splits into the sloping side and the circular base, so you can use just the part you need.
A cone's surface has two parts: a flat circle underneath and a sloping side that narrows to a point. Many real jobs want only one of them. A conical roof, a funnel and a paper party hat have no base to cover, so the lateral area is the useful number, while a solid plug or a closed container needs the total. This calculator assumes a right cone, with its tip directly above the center of the base.
Formula
A = π × r × (r + l)
- A
- the total surface area
- r
- the radius of the circular base
- l
- the slant height, from the rim up the side to the tip
The lateral area is π × r × l and the base area is π × r².
Surface area from radius and height
The height runs straight up the middle, but the formula needs the slant height along the side. The radius, the height and the slant form a right triangle, so the Pythagorean theorem gives the slant. A cone with a radius of 3 and a height of 4 has a slant of √(9 + 16) = 5. Its lateral area is π × 3 × 5 ≈ 47.12, its base is π × 9 ≈ 28.27, and the total is about 75.40.
- l = √(r² + h²)
- A = π × r × (r + l)
Surface area from radius and slant height
The slant is often the easier measurement, because you can lay a tape along the outside of the cone without reaching its center. A radius of 5 with a slant of 13 gives a lateral area of π × 5 × 13 ≈ 204.20 and a base of π × 25 ≈ 78.54, for a total of about 282.74. The height follows from the same triangle: √(169 − 25) = 12.
The slant has to be longer than the radius. In the right triangle it is the longest side, so a radius of 6 with a slant of 6 or less cannot make a cone. The calculator reports that instead of a meaningless result.
- h = √(l² − r²)
Where the lateral area formula comes from
Cut a paper cone up one side and it opens into a flat slice of a circle. The radius of that slice is the slant height, and its curved edge is the rim of the cone, 2πr long. The area of such a slice is half its curved edge times its radius, so ½ × 2πr × l = πrl.
Making a cone from a flat sheet
The same unrolling gives you a cutting pattern. Draw a circle whose radius is the slant height, then keep a slice with an angle of 360° × r ÷ l. For a party hat with a radius of 8 cm and a height of 15 cm, the slant is 17 cm and the slice angle is 360 × 8 ÷ 17 ≈ 169.4°. The paper left in the hat covers π × 8 × 17 ≈ 427.26 cm². Allow a little extra for the overlap where the seam is glued.
- Slice angle = 360° × r ÷ l
Surface area and volume
A cone's volume uses the true height, not the slant: one third of the base area times the height. The cone with a radius of 3 and a height of 4 holds π × 9 × 4 ÷ 3 ≈ 37.70 cubic units. The volume calculator handles other solids and the sphere surface area calculator covers rounded ones.
- V = π × r² × h ÷ 3
Frequently asked questions
What is the formula for the surface area of a cone?
A = π × r × (r + l), where r is the base radius and l is the slant height. If you only know the height, get the slant first from l = √(r² + h²).
What is the lateral surface area of a cone?
It is the sloping side alone, without the base: π × r × l. A cone with a radius of 3 and a slant of 5 has a lateral area of about 47.12.
How do I find the slant height of a cone?
Take the square root of the radius squared plus the height squared. A radius of 6 and a height of 8 give a slant of √(36 + 64) = 10.
What if I know the diameter instead of the radius?
Halve it first. A cone 10 ft across has a radius of 5 ft, and that is the number to enter here.
Does this work for a leaning cone?
No. These formulas assume the tip sits directly above the center of the base. In a leaning, or oblique, cone the slant changes around the rim, and the lateral area has no simple formula. The base area and the volume formula still apply.
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