Cone Surface Area Calculator
Calculate the total surface area of a right circular cone, including the base. Uses A = pi x r x (r + l) where l is the slant height. Enter radius and height.
How to use this tool
- Enter base radius and height in the fields above.
- Results update instantly as you type — or click Calculate.
- Read your total surface area and the full breakdown beneath it.
Formula
A = pi x r x (r + l), where l = sqrt(r^2 + h^2)
How it works
Find slant height l = √(r² + h²). Lateral area = πrl. Total area = πr(r + l).
Worked example
Cone r=3, h=4 (3-4-5 triangle)
- Slant height l = sqrt(9+16) = sqrt(25) = 5
- Lateral = pi x 3 x 5 = 15 pi = 47.1238...
- Base = pi x 9 = 28.2743...
- Total = 47.1238 + 28.2743 = 75.3981...
Total Surface Area = 75.3982 square units
Common mistakes to avoid
- Using the vertical height h instead of the slant height l directly in the formula — you must compute l = sqrt(r^2 + h^2) first before substituting.
- Forgetting to include the circular base area (pi * r^2) and computing only the lateral surface pi * r * l, giving a result that is too small.
- Confusing the radius with the diameter — entering the full width of the cone base instead of half of it doubles the computed area.
Key terms
- What is slant height?
- The straight-line distance from the apex of the cone to any point on the base circle edge.
- What is lateral area?
- The area of the cone's curved surface, not including the circular base.
Frequently asked questions
- What is the slant height and how is it different from the cone's vertical height?
- The vertical height h is the perpendicular distance from the apex to the centre of the base. The slant height l is the distance along the side of the cone from the apex to the base edge, calculated as l = sqrt(r^2 + h^2). The surface area formula requires l, not h.
- Does the formula include the base of the cone?
- Yes. The total surface area A = pi * r * (r + l) expands to pi*r^2 (base) + pi*r*l (lateral face). If you only need the lateral surface area, use pi * r * l alone.
- What units does the result use?
- The result is in square units matching your input. If radius and height are entered in centimetres, the surface area is in square centimetres (cm^2).