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Cone Calculator

Volume equals one-third pi r squared h

Solution

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Calculate Cone Volume from Radius and Height

Use this form when the base radius and height of a right circular cone are known. The cone holds exactly one-third the volume of a cylinder with the same base and height.

V = (1/3) π r² h

Calculate Total Surface Area of a Cone

Use this form when you need the entire exterior surface — circular base plus the curved lateral side. Common for material-takeoff on conical roofs, funnels, and party hats.

S = π r (r + √(r²+h²))

Calculate Lateral Surface Area of a Cone

Use this form when only the slanted side matters — paint coverage on a tipi or silo cone, wraps and labels, or the unrolled sheet metal needed to form a cone.

L = π r √(r² + h²)

Calculate Cone Radius from Volume and Height

Use this rearrangement when a target volume and fixed height are known, and you need the base radius. Common when sizing a conical hopper to fit a vertical drop.

r = √(3V / (π h))

Calculate Cone Height from Volume and Radius

Use this rearrangement when the base radius is fixed by available stock and you need the height that gives a target volume.

h = 3V / (π r²)

How It Works

This cone calculator uses V = (1/3) π r² h plus the surface-area pair S = π r (r + s) and L = π r s, where s = √(r² + h²) is the slant height. Pick the unknown with the solve-for toggle, enter the remaining two values in any supported length or volume unit, and the calculator converts to SI internally before computing all related quantities — including diameter d = 2r, base circumference C = 2π r, and the slant height s — so a single page covers funnels, hoppers, party hats, traffic cones, and conical roofs.

Example Problem

A traffic-cone-shaped hopper has a base radius of 3 m and a height of 4 m. What volume does it hold, and what is its lateral surface area?

  1. Identify the dimensions: r = 3 m and h = 4 m.
  2. Compute the slant height s = √(r² + h²) = √(9 + 16) = √25 = 5 m.
  3. Volume: V = (1/3) π r² h = (1/3) · π · 9 · 4 = 12π ≈ 37.699 m³.
  4. Lateral surface area: L = π r s = π · 3 · 5 = 15π ≈ 47.124 m².
  5. Total surface area (with the base): S = π r (r + s) = π · 3 · 8 = 24π ≈ 75.398 m².
  6. Round trip: from V ≈ 37.699 and h = 4, the inverse r = √(3V/(πh)) recovers r = 3 m, confirming the volume.

The 3-4-5 right triangle makes this example arithmetic clean. In real applications, slant height s is usually computed from r and h rather than measured directly.

When to Use Each Variable

  • Solve for Volume — when r and h are known and you need the enclosed volume of a hopper, funnel, or conical container.
  • Solve for Total Surface Area — when you need the full exterior including the base — e.g., sheet-metal estimates for a closed cone.
  • Solve for Lateral Surface Area — when only the slanted side matters — paint, label wraps, or the unrolled flat pattern needed to roll a cone.
  • Solve for Radius — when a target volume and a fixed height are known and you need the base radius.
  • Solve for Height — when the base radius is fixed and you need the height for a target volume.

Key Concepts

A right circular cone is the solid formed by rotating a right triangle around one of its legs — equivalently, a circular base joined to a single apex point. The geometry is fully determined by two numbers: the base radius r and the height h. From those, the slant height s = √(r² + h²) is the hypotenuse of the cross-section triangle and is what scales the lateral surface area. A cone holds exactly one-third the volume of a cylinder with the same base and height, a result that goes back to Eudoxus and Archimedes.

Applications

  • Hoppers and funnels: size conical chutes for grain, granular material, or fluid drains
  • Coatings and wraps: estimate paint or vinyl needed to cover a cone-shaped column, roof, or product packaging
  • Manufacturing: compute the flat sheet-metal pattern needed to roll a conical part
  • Geometry instruction: classic textbook shape for volume and surface-area problems

Common Mistakes

  • Forgetting the 1/3 factor in V = (1/3) π r² h — the cone is one-third of the equivalent cylinder, not the full cylinder
  • Using the cone height h in place of the slant height s in the lateral-area formula — L = π r s uses the slant, not the perpendicular height
  • Mixing radius and diameter — real-world specs often quote diameter; divide by 2 before plugging in
  • Confusing total surface area (includes the base) with lateral area (curved side only)

Frequently Asked Questions

How do you calculate the volume of a cone?

Use V = (1/3) π r² h. Square the base radius, multiply by the height, multiply by π, and divide by 3. For example a cone with r = 3 m and h = 4 m has V = (1/3) · π · 9 · 4 = 12π ≈ 37.7 m³.

What is the formula for the surface area of a cone?

Total surface area S = π r (r + s), where s = √(r² + h²) is the slant height. Lateral surface area (curved side only) is L = π r s.

What is the slant height of a cone?

The slant height s is the distance from the apex to any point on the base edge, measured along the curved surface. By the Pythagorean theorem s = √(r² + h²), so for a cone with r = 3 m and h = 4 m, s = √(9+16) = 5 m.

How do you find the radius of a cone given its volume?

Rearrange V = (1/3) π r² h to r = √(3V / (π h)). For a cone with V = 37.7 m³ and h = 4 m, r = √(3 · 37.7 / (4π)) = √9 = 3 m.

Why is a cone's volume one-third of a cylinder's?

The cone of radius r and height h fits inside a cylinder of the same base and height, and three identical cones can be assembled to fill the cylinder exactly. This result is one of the foundational theorems of solid geometry, proven by Eudoxus and made rigorous by Archimedes.

What is the difference between total surface area and lateral surface area for a cone?

Lateral surface area is the curved side only: L = π r s. Total surface area adds the circular base: S = L + π r² = π r (r + s). For an open-top cone like a traffic cone or party hat, use only the lateral area.

How is a cone unrolled into a flat sheet?

The lateral surface unrolls into a circular sector with radius equal to the slant height s and arc length equal to the base circumference 2π r. The sector angle is θ = 2π r / s radians, or 360° · r / s degrees. This is the pattern a sheet-metal shop cuts to roll a cone.

Does the cone formula work for oblique cones?

Volume V = (1/3) π r² h still holds for oblique cones (where the apex is not directly above the base center) — only the perpendicular height matters. Surface-area formulas, however, assume the apex is directly above the center; oblique cones need a different lateral-surface calculation.

Reference: Weisstein, Eric W. "Cone." MathWorld — A Wolfram Web Resource. https://mathworld.wolfram.com/Cone.html

Worked Examples

Hopper Sizing

How do you calculate the volume of a conical grain hopper?

A grain silo terminates in a conical hopper with base radius 3 m and height 4 m. Use V = (1/3)πr²h to find its capacity.

  • Knowns: r = 3 m, h = 4 m
  • Formula: V = (1/3) π r² h
  • V = (1/3) · π · 9 · 4 = 12π ≈ 37.699 m³
  • Convert to liters: 37.699 m³ × 1000 ≈ 37,699 L

Volume ≈ 37.7 m³ ≈ 37,699 liters

Real hoppers have a discharge opening at the apex — for working volume subtract a small cone or use an effective height to the bottom of the cylindrical section above.

Sheet Metal

How much sheet metal is needed to roll a 3 m × 4 m cone?

A fabricator needs to roll a 3-m-radius, 4-m-tall cone from sheet metal. The amount of material is the lateral surface area L = π r s.

  • Knowns: r = 3 m, h = 4 m
  • Slant height: s = √(r² + h²) = √25 = 5 m
  • Formula: L = π r s
  • L = π · 3 · 5 = 15π ≈ 47.124 m²

Lateral surface area ≈ 47.1 m² (~507 ft²)

When rolled flat, the cone unfolds into a circular sector of radius s = 5 m and arc length 2π r = 6π ≈ 18.85 m — that's the pattern the shop cuts.

Inverse Solve

What base radius does a 10-liter cone with fixed 0.5 m height need?

A funnel must hold 10 L (0.01 m³) of fluid with a fixed slope height of 0.5 m. Solve r = √(3V / (π h)) for the required base radius.

  • Knowns: V = 0.01 m³, h = 0.5 m
  • Formula: r = √(3V / (π h))
  • r = √(3 · 0.01 / (π · 0.5)) = √(0.06 / π)
  • r ≈ √0.01910 ≈ 0.1382 m (≈ 13.8 cm)

Radius ≈ 0.138 m → diameter ≈ 0.276 m

For pouring or draining funnels, the practical base radius is set by the discharge spout, so this calculator is most useful for sizing the upper rim or transition.

Cone Formulas

All cone properties follow from two dimensions: the base radius r and the height h. From those, slant height, volume, both surface areas, diameter, and base circumference all derive:

Volume equals one-third pi r squared hLateral surface area equals pi r times the square root of r squared plus h squaredTotal surface area equals pi r times the quantity r plus square root of r squared plus h squaredRadius equals the square root of three V divided by pi hHeight equals three V divided by pi r squared
Right circular cone with labeled radius r and height hrh

Where:

  • V — enclosed volume (m³, L, gal, ft³)
  • S — total surface area: base + lateral (m², ft², in²)
  • L — lateral surface area: curved side only, unrolls into a circular sector
  • r — base radius (m, cm, in, ft, yd)
  • h — perpendicular height from apex to base center
  • s — slant height: s = √(r² + h²), the apex-to-rim distance along the cone's surface
  • d — base diameter (d = 2 r); common spec dimension
  • C — base circumference (C = 2π r), equal to the arc length when the cone is unrolled flat

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