Hollow Shell • V_mat = (πh/3)(R² - r²)

Hollow Cone Volume Calculator: Material & Void Solver

Calculate the solid material volume and internal void capacity of a hollow conical shell. Ideal for estimating material weight, wall thickness, metal fabrication requirements, and 3D printing filament consumption.

Input Dimensions

Adjust values below to compute instant results in real time.
R
r
h
r R t = R - r Material shell: $V_{ ext{mat}} = rac{pi h}{3}(R^2 - r^2)$
Calculated Result
0.0000 cm³
Secondary Metric 1
—
Secondary Metric 2
—
Equivalent Unit Conversions
Litres (L) —
Millilitres (mL) —
Cubic Metres (m³) —
US Gallons (gal) —
UK Imp Gallons —
Cubic Feet (ft³) —
Step-by-Step Proof

Mathematical Calculation Steps

How the Formula Works & Geometric Derivation

In Euclidean solid geometry, a cone is formed by connecting a planar circular base to an apex vertex point via a continuous curved lateral surface. The governing equation for this calculator is:

Governing Equation
$$V_{\text{material}} = \frac{\pi h}{3} (R^2 - r^2)$$

Material volume equals the full outer cone volume minus the hollow inner void volume.

This mathematical relationship originates from integral calculus. By slicing the conical volume into infinitesimal horizontal circular disks of thickness $dz$ from the apex ($z = 0$) down to the base altitude ($z = h$), each disk at level $z$ possesses a radius $r(z) = rac{r}{h}z$. The volume integral is evaluated as:

$$V = \int_0^h \pi [r(z)]^2 dz = \pi \frac{r^2}{h^2} \int_0^h z^2 dz = \pi \frac{r^2}{h^2} \left[ \frac{z^3}{3} \right]_0^h = \frac{1}{3} \pi r^2 h$$

Variables and Parameters Reference

Below is a breakdown of every mathematical symbol and physical dimension used across this calculator:

Symbol Parameter Name Definition & Geometric Meaning
V_mat Material Volume Volume of the solid shell material (metal, plastic, concrete)
V_out Outer Volume Full volume enclosed by outer dimensions ((1/3)πR²h)
V_in Inner Void Volume Hollow interior void capacity ((1/3)πr²h)
R Outer Base Radius Radius to the outer perimeter
r Inner Void Radius Radius to the inner hollow wall (must be < R)
t Wall Thickness Radial shell thickness: t = R - r

Worked Example: Steel Funnel Material Volume

Problem Statement: A steel funnel cone has outer radius R = 10 cm, inner radius r = 8 cm, and height h = 15 cm. Find its steel material volume.

  1. Identify dimensions: R = 10 cm, r = 8 cm, h = 15 cm.
  2. Compute outer volume: V_out = (1/3) × π × (10)² × 15 = 500π ≈ 1570.80 cm³.
  3. Compute inner void volume: V_in = (1/3) × π × (8)² × 15 = 320π ≈ 1005.31 cm³.
  4. Subtract inner from outer: V_mat = 1570.80 - 1005.31 = 565.49 cm³.
  5. Verify with formula: V_mat = (π × 15 / 3) × (10² - 8²) = 5π × 36 = 180π ≈ 565.49 cm³.
  6. Weight estimate (steel density ~7.85 g/cm³): 565.49 × 7.85 ≈ 4.44 kg.

Practical Applications & Real-World Use Cases

Conical mathematics is widely applied across aerospace engineering, industrial manufacturing, food science, civil earthworks, and architectural design:

  • Metal spinning and stamping: calculating blank sheet mass and spun cone material volume.
  • 3D printing & additive manufacturing: determining resin or filament requirements for hollow conical models.
  • Precast concrete: sizing hollow conical piling tips and storm drain catch basins.
  • Plastics injection molding: estimating shot volume and wall thickness uniformity.
Contextual Solvers

Related Cone Calculators

Explore sister tools designed to compute surface area, tank hoppers, slant angles, and conversions.

cone volume calculator

Cone Volume Calculator

This precision cone volume calculator computes the exact volume of any right circular cone. Input your base ra...

Open Calculator → cone tank volume calculator

Cone Tank Capacity Calculator

Calculate the total volumetric capacity and dip level volumes of industrial cone-bottom storage tanks, hoppers...

Open Calculator → frustum cone volume formula calculator

Frustum of a Cone Volume Calculator (Truncated Cone)

A conical frustum (truncated cone) is formed by slicing off the top apex of a cone with a plane parallel to it...

Open Calculator → partial cone volume calculator

Partial Cone & Tank Fill Calculator (Liquid Depth Solver)

Calculate liquid volume, percentage full, and remaining ullage capacity in a conical tank or hopper oriented w...

Open Calculator →
FAQ Reference

Frequently Asked Questions

Authoritative answers on cone formulas, common pitfalls, and calculation methods.

For a hollow conical shell with matching inner and outer heights, subtract the inner cone's volume from the outer cone's volume. The material volume is V = πh(R² − r²) / 3, where R is the outer radius and r is the inner radius.

Internal volume is the empty space inside the cone, which can represent its capacity. Material volume is the solid material used to construct its walls. The two values are different and should be calculated separately.

Use the internal radius and internal perpendicular height in V = ⅓πr²h. If the internal dimensions are measured in centimetres, divide the calculated cubic centimetres by 1,000 to convert the capacity to litres.

Yes. Divide each diameter by two to obtain the corresponding radius. Calculate the outer and inner cone volumes and subtract the inner volume from the outer volume, provided the dimensions describe compatible nested cones.

For uniform radial wall thickness, the inner radius is smaller than the outer radius. The difference between the outer and inner cone volumes gives the material volume when their heights and geometry are compatible. Real shells with varying thickness or different inner and outer apex positions may require a more specific model.

Yes, if the material volume and material density are known. Multiply the material volume by density to calculate mass. Make sure the volume and density units are compatible, and remember that coatings, openings, and structural components may affect the actual weight.

Not necessarily. A hollow cone describes a shell with an inner void, while a frustum describes a truncated solid with two parallel circular ends. A hollow frustum needs its own inner and outer geometry to calculate material volume accurately.

For a simple nested cone model, you need the inner and outer radii and their compatible perpendicular heights. For a shell with different inner and outer heights or an open truncated end, additional dimensions may be required.

Dedicated Geometry Solvers

Explore Other Cone Calculators

Need a specialized solver for slant heights, tank hoppers, liquid dip fill levels, or non-standard frustums? Choose from our 15 other dedicated, step-by-step cone calculators below.

Looking for the Full Geometry Suite Hub?

Visit our central Cone Calculators Suite Hub featuring formula breakdowns, geometry cheat sheets, and technical FAQs across all 16 cone solvers.

Explore All 16 Calculators Hub →