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Cone Surface Area Calculator

Total and lateral surface area of a cone from radius and height

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How to use

  1. 1.Enter the base radius (r) of the cone in the first field.
  2. 2.Enter the vertical height (h) in the second field — the calculator derives the slant height automatically.
  3. 3.Read the total surface area instantly, along with the slant height, lateral area (πrl), and base area (πr²).

About Cone Surface Area Calculator

The total surface area of a right circular cone is A = πr² + πrl, where r is the base radius and l is the slant height. This calculator returns that answer the moment you type a radius and height, and it also breaks the result into its two parts so you can see exactly where the number comes from. Enter r and h and you instantly get the slant height, the base area, the lateral area, and the total surface area — no formula rearranging on your part.

A cone has two surfaces: the flat circular base and the curved side that wraps up to the apex. The base is just a circle, so its area is πr². The curved side is called the lateral surface, and its area is πrl. Add them together and you get the total surface area, which factors neatly to πr(r + l). If you only need the wrapper or the paper cone — an open cone with no bottom — use the lateral area πrl on its own and ignore the base.

The slant height l is the key value most people are missing. It is the straight-line distance from the edge of the base up to the apex, measured along the sloped side, and it is longer than the vertical height h. Because the height, the base radius, and the slant height form a right triangle, the Pythagorean theorem gives l = √(r² + h²). This calculator computes l for you, so you can start from the height (which is usually what you measure) rather than having to know the slant length up front. If your problem already gives you l directly, note that the lateral area is simply πrl without the square root step.

These formulas describe a right circular cone — one whose apex sits directly above the center of the base. Surface area comes out in square units of whatever unit you use for the inputs: measure r and h in centimetres and the areas are in cm²; use inches and you get in². Keep both inputs in the same unit.

Common uses include figuring out how much sheet metal, paper, or fabric is needed to build a cone, calculating paint or coating for a conical roof or hopper, sizing an ice-cream cone or funnel, and checking geometry homework where the total-versus-lateral distinction matters. All calculations run entirely in your browser, so nothing you type is uploaded.

Frequently asked questions

What is the surface area formula for a cone?
The total surface area of a right circular cone is πr² + πrl, which factors to πr(r + l). Here r is the base radius and l is the slant height, found from l = √(r² + h²). The πr² term is the flat base and the πrl term is the curved side.
How do you find the slant height of a cone?
The slant height is l = √(r² + h²), where r is the base radius and h is the vertical height. This comes from the Pythagorean theorem, because the radius, the height, and the slant height form a right triangle. The slant height is always longer than the vertical height.
What is the difference between lateral and total surface area?
The lateral surface area (πrl) covers only the curved side of the cone. The total surface area (πr² + πrl) adds the flat circular base. Use lateral area for an open cone like a paper cone or party hat, and total area when the cone is closed at the bottom.
Why is the lateral surface area πrl and not πr²?
If you unroll the curved side of a cone, it flattens into a circular sector with radius l (the slant height) and arc length equal to the base circumference, 2πr. The area of that sector works out to πrl. The πr² formula is only for the flat circular base, not the slanted side.
What units does the cone surface area come in?
Surface area is in square units of whatever unit you enter for the radius and height. If r and h are in centimetres, the areas are in cm²; if they are in inches, the areas are in in². Always keep the radius and height in the same unit before calculating.

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