Cone — Volume + Surface Area
Right circular cone: V = ⅓πr²h, slant ℓ = √(r² + h²), lateral SA = πrℓ, total SA = πr(r + ℓ).
Result
Volume
37.6991
⅓πr²h = 37.6991. Slant ℓ = 5.0000.
- Radius3
- Height4
- Slant height (ℓ)5.000000
- Volume (⅓πr²h)37.699112
- Lateral SA (πrℓ)47.123890
- Base area (πr²)28.274334
- Total SA (πr(r+ℓ))75.398224
Step-by-step
- Slant height ℓ = √(r² + h²) = √(9 + 16) = 5.0000.
- V = ⅓ × π × r² × h = ⅓ × π × 9 × 4 = 37.6991.
- Lateral SA = πrℓ = π × 3 × 5.0000 = 47.1239.
- Total SA = πr² + πrℓ = 28.2743 + 47.1239 = 75.3982.
How to use this calculator
- Enter base radius and perpendicular height.
- Read slant height, volume, and surface areas.
About this calculator
A right circular cone has one circular base and tapers to a single apex perpendicular to the base. Volume V = ⅓πr²h — exactly ⅓ the volume of a cylinder with the same base and height (provable by Cavalieri / integration). Slant height ℓ = √(r² + h²) — the Pythagorean distance from base edge to apex. Lateral surface (the curved side, unrolled = a sector of a circle) = πrℓ. Total SA = πr² + πrℓ = πr(r + ℓ). Source: Stewart, "Calculus" §6.2; Larson §11.
Frequently asked
Calculus integration of a linearly shrinking circle from 0 (apex) to r (base) over height h yields ⅓πr²h. Three identical cones fit exactly inside a cylinder of the same base + height.
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V = (1/3)πr²h. One-third of the cylinder of same base and height.
Cone Surface Area Calculator
SA = πr² + πrl, where l = √(r² + h²) is the slant height.
Cylinder — Volume + Surface Area
Right circular cylinder: V = πr²h, lateral SA = 2πrh, total SA = 2πr(r + h).
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Square pyramid: V = ⅓a²h, slant ℓ = √((a/2)² + h²), lateral SA = 2aℓ, total SA = a² + 2aℓ.
Sphere — Volume + Surface Area
Sphere: V = (4/3)πr³, SA = 4πr². The most-symmetric solid; same SA = SA of cylinder of equal r and h = 2r.
Triangular Pyramid (Regular Tetrahedron) — Volume + Surface Area
Regular tetrahedron: V = a³/(6√2), SA = √3·a², height = a√(2/3). All four faces are equilateral triangles.