Cone Calculator
Enter radius and height to calculate cone volume, slant height, and surface areas.
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Cone Formulas and Geometry
A cone is a pyramid-like 3D shape with a circular base tapering to a single point called the apex. Cone calculations appear in engineering, manufacturing, and everyday life — from ice cream cones to traffic cones to rocket nose cones.
Volume
V = (1/3)πr²h. A cone holds exactly one-third the volume of a cylinder with the same base radius and height. This relationship was proved by Archimedes. For a waffle cone with r=3 cm and h=12 cm: V ≈ 113.1 cm³.
Slant Height
The slant height l is the distance from the apex straight down to the edge of the base circle: l = √(r² + h²). This comes from the Pythagorean theorem applied to the right triangle formed by r, h, and l. For r=3 and h=8: l = √(9+64) = √73 ≈ 8.544.
Lateral Surface Area
Lateral SA = πrl. Imagine unrolling the cone's side into a flat sector of a circle — its area is πrl. This is the amount of material needed for the cone's curved surface, excluding the base.
Total Surface Area
Total SA = πr² + πrl = πr(r + l). The base circle (πr²) plus the lateral surface (πrl). For packaging or manufacturing, total SA determines the material required to create the complete shape.
Worked Example: Ice Cream Cone
An ice cream cone has a base radius of 3 cm and a height of 10 cm. Find the slant height, lateral surface area, and volume.
Slant height: l = √(r² + h²) = √(9 + 100) = √109 ≈ 10.44 cm.
Lateral SA: πrl = π × 3 × 10.44 ≈ 98.44 cm². This is the area of waffle material needed for the cone surface.
Volume: V = (1/3)πr²h = (1/3) × π × 9 × 10 = 30π ≈ 94.25 cm³. This is the amount of ice cream the cone can hold.
Frequently Asked Questions
V = (1/3)πr²h = (1/3)×π×9×8 = 24π ≈ 75.40 cubic units.
Slant height l = √(r² + h²). For r=3, h=8: l = √(9+64) = √73 ≈ 8.544. It's the distance from apex to any point on the base rim.
Lateral SA = πrl. For r=3 and slant height l=8.544: Lateral SA = π×3×8.544 ≈ 80.5 square units.
A cone's volume is exactly 1/3 of a cylinder with the same base and height. Fill a cone with water, pour it into the matching cylinder — you'd need 3 pours to fill the cylinder.
Slant height l ≈ 8.544. Total SA = πr² + πrl = π×9 + π×3×8.544 ≈ 28.27 + 80.50 ≈ 108.77 sq units.
Rearrange V = (1/3)πr²h to get h = 3V / (πr²). For example, if V = 150 and r = 5: h = 3 × 150 / (π × 25) = 450 / 78.54 ≈ 5.73 units.
A frustum is a cone with the top cut off — it has two circular bases with different radii (R and r) and height h. Volume = (π × h / 3) × (R² + Rr + r²). Frustums are used in buckets, lamp shades, and drinking cups.