What is the formula for the volume of a cone?

The volume of a cone is one third of pi times radius squared times perpendicular height.

Cone Volume Calculator: Cone, Frustum, Surface Area & Radius

Cone, frustum and missing-dimension solver

Calculate Cone or Truncated Cone Volume and Surface Area

Calculate a full cone or frustum from radius or diameter, solve for a missing radius or height, estimate slant height, lateral and total surface area, fill volume, mass and material cost.

Geometry mode

Full cone Truncated cone / frustum Solve a missing dimension

Measurement unit Base input Base radius / diameter Vertical height Fill percentage (%) Material density (kg/m³, optional) Material price per m³ ($)

Cone calculation result

Cone and frustum volume formulas

A full cone has volume πr²h/3. A frustum has volume πh(R² + Rr + r²)/3, where R is the base radius and r is the top radius. When the top radius is zero, the frustum formula becomes the cone formula.

Surface area and slant height

For a cone, slant height is the square root of r² + h². For a frustum, use the radius difference: √((R−r)²+h²). Lateral surface area equals π(R+r)s for a frustum.

Solving a missing dimension

Height can be isolated directly from the volume formula. Cone radius can be solved from volume and height. Frustum reverse solving is not offered unless enough dimensions are known.

Can I enter diameter instead of radius?

Yes. The calculator divides diameter by two before applying the formulas.

How do I calculate liters or gallons?

The selected length unit is converted to cubic meters internally, then to liters and US gallons.

Does fill percentage change surface area?

No. It changes the filled volume, mass and material cost, not the complete cone’s geometric surface area.

Cone Volume Calculator: formula-based step-by-step solution

This Cone Volume Calculator worked derivation uses the published default state. Its live formula panel replaces these values whenever an input, unit, mode, date, material, or scenario changes.

Governing formula or algorithm — exact production relationships:

R = radius(Base Value,Base Input)
R = Mode = 'frustum'?radius(Top Value,Base Input):0
H = Height
Volume = π × h × (R × R+R × r+r × r) ÷ 3
Slant = hypot(R-r,h)
Lateral = π × (R+r) × slant
Base Area = π × R × R
Top Area = π × r × r
Total Area = lateral+baseArea+(r>0?topArea:0)
Fill = Fill Percent ÷ 100
Filled Volume = volume × fill

1. Governing formulas and numeric substitution

Formula 1: R = radius(Base Value,Base Input)
Default-value substitution: R = radius((5),Base Input)
Formula 2: R = Mode = 'frustum'?radius(Top Value,Base Input):0
Default-value substitution: R = Mode = 'frustum'?radius((2),Base Input):0
Formula 3: H = Height
Default-value substitution: H = (12)
Intermediate evaluation: 12
Formula 4: Volume = π × h × (R × R+R × r+r × r) ÷ 3
Default-value substitution: Volume = π × (12) × (R × R+R × r+r × r) ÷ 3
Formula 5: Slant = hypot(R-r,h)
Default-value substitution: Slant = hypot(R-r,(12))
Formula 6: Lateral = π × (R+r) × slant
Default-value substitution: Lateral = π × (R+r) × slant | current data: Measurement unit=cm; Base input=radius; Base radius / diameter=5; Top radius / diameter=2; Vertical height=12; Known volume=314.159; Known volume unit=unit3; Unknown=height
Formula 7: Base Area = π × R × R
Default-value substitution: Base Area = π × R × R | current data: Measurement unit=cm; Base input=radius; Base radius / diameter=5; Top radius / diameter=2; Vertical height=12; Known volume=314.159; Known volume unit=unit3; Unknown=height
Formula 8: Top Area = π × r × r
Default-value substitution: Top Area = π × r × r | current data: Measurement unit=cm; Base input=radius; Base radius / diameter=5; Top radius / diameter=2; Vertical height=12; Known volume=314.159; Known volume unit=unit3; Unknown=height
Formula 9: Total Area = lateral+baseArea+(r>0?topArea:0)
Default-value substitution: Total Area = lateral+baseArea+(r>0?topArea:0) | current data: Measurement unit=cm; Base input=radius; Base radius / diameter=5; Top radius / diameter=2; Vertical height=12; Known volume=314.159; Known volume unit=unit3; Unknown=height
Formula 10: Fill = Fill Percent ÷ 100
Default-value substitution: Fill = (100) ÷ 100
Intermediate evaluation: 1
Formula 11: Filled Volume = volume × fill
Default-value substitution: Filled Volume = (314.159) × (1)
Intermediate evaluation: 314.159

2. Variables and measurement units used

Measurement unit = cm
Base input = radius
Base radius / diameter = 5
Top radius / diameter = 2
Vertical height = 12
Known volume = 314.159
Known volume unit = unit3
Unknown = height
Fill percentage (%) = 100 %
Material density (kg/m³, optional) = 0 kg/m³, optional
Material price per m³ ($) = 0 $

3. Cone Volume Calculator final evaluation: this calculator’s production engine evaluates the substituted relationship with unrounded internal precision; its subject-specific result cards, table, and visual report the rounded final answer and verification quantities.

Cone Volume Calculator: formula, steps, units and verification

cone volume calculator is designed for a transparent calculation rather than a black-box answer. Cone Volume Calculator: calculate cone or frustum volume, slant height, surface area, liters, gallons, mass and cost, or solve a missing cone radius.

How to use the cone volume calculator in 5 steps

Choose the required mode. Select the calculation path that matches the quantity you know and the result you need.

Enter source values. Use measured, documented or assignment values rather than rounded estimates whenever possible.

Confirm every unit. The cone volume calculator converts supported units before formula substitution, so each selector must describe the entered number.

Run the calculation. Review the displayed formula, normalized values and numeric substitution before accepting the final result.

Verify the answer. Reproduce the substitution manually and check whether the output is reasonable for the stated assumptions.

Cone Volume Calculator formula and unit checks

The cone volume calculator keeps source inputs, canonical calculation units and displayed output units separate. This prevents a correct formula from producing a wrong answer because feet were treated as meters, percentages as decimals, or time values as the wrong interval.

For an independent check, copy the formula shown by the calculator, substitute the unrounded canonical values, preserve full precision through intermediate operations and round only the final result. The verified answer should match both the displayed number and its measurement unit.

How to interpret the cone volume calculator result

A result is useful only when its assumptions match the real problem. Compare the answer with expected ranges, inspect any warning or boundary message, and test a nearby input to confirm that the output changes in the direction predicted by the governing relationship.

The cone volume calculator provides an educational and planning result. For regulated, medical, structural, financial, laboratory or safety-critical decisions, verify the inputs and method against the applicable professional requirements before acting.

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Cone Volume Calculator quick verification checklist

Before saving or reporting a result from the cone volume calculator, confirm the input source, unit selections, formula mode, intermediate substitution, final unit and rounding rule. These checks make the calculation reproducible and easier to audit.

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