Cone Volume & Surface Area Calculator Formula

Understand the math behind the cone volume & surface area calculator. Each variable explained with a worked example.

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Formulas Used

Volume

volume = (1/3) * pi * pow(r, 2) * h

Lateral Sa

lateral_sa = pi * r * slant

Total Sa

total_sa = pi * r * slant + pi * pow(r, 2)

Slant Height

slant_height = slant

Variables

VariableDescriptionDefault
rBase Radius4
hHeight9
slantDerived value= sqrt(pow(r, 2) + pow(h, 2))calculated

How It Works

Cone Formulas

Volume

V = (1/3) × pi × r² × h

Slant Height

l = sqrt(r² + h²)

Surface Area

  • Lateral area = pi × r × l
  • Total area = pi × r × l + pi × r² = pi × r × (l + r)

    • Worked Example

    Cone with radius 4 and height 9.

    r = 4h = 9
    1. 01Slant = √(16+81) = √97 ≈ 9.8489
    2. 02Volume = (1/3)×pi×16×9 = 48pi ≈ 150.7964
    3. 03Lateral SA = pi×4×9.8489 ≈ 123.7168
    4. 04Total SA = 123.7168 + pi×16 ≈ 173.9460

    Frequently Asked Questions

    How is a cone related to a cylinder?

    A cone with the same base and height as a cylinder has exactly 1/3 the volume of that cylinder.

    What is the slant height?

    The slant height is the distance from the apex (tip) to any point on the base circle, measured along the surface. It is found using the Pythagorean theorem.

    What is a truncated cone (frustum)?

    A frustum is a cone with the top cut off by a plane parallel to the base. Its volume requires both radii and the height.

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