Math

Cone Volume Calculator — V = (1/3)πr²h

Q: What is the formula for the volume of a cone?

The volume of a right circular cone is **V = (1/3)πr²h**, where r is the base radius and h is the perpendicular height. The 1/3 factor comes from the fact that a cone occupies exactly one-third the volume of a cylinder with the same base and height — a result proven rigorously in Euclidean geometry. For r = 5 cm and h = 12 cm: V = (1/3) × π × 25 × 12 ≈ 314.16 cm³.

Q: What is slant height and how is it different from cone height?

The **perpendicular height (h)** is the straight vertical distance from the base center to the apex. The **slant height (l)** is the distance along the outer surface from the base edge to the apex: l = √(r² + h²) by the Pythagorean theorem. For a cone with r = 3, h = 4, l = 5 exactly (a 3-4-5 right triangle). You need l to calculate lateral surface area, but only h for volume.

Q: How do I calculate the total surface area of a cone?

Total surface area = lateral area + base area: **A_total = πrl + πr²**, where l = √(r² + h²). For a cone with r = 5 cm and h = 12 cm, l = 13 cm, so A_total = π×5×13 + π×25 = 65π + 25π = 90π ≈ 282.74 cm². The lateral area alone (πrl) represents the unrolled curved surface — like the flat sector of a circle with radius l and arc length 2πr.

Q: What units does the calculator use, and can I mix centimeters and inches?

The calculator is **unit-agnostic**: if you enter r and h in centimeters, volume is in cm³ and areas in cm². If you use inches, outputs are in³ and in². Never mix units — entering r in inches and h in feet requires first converting h to inches (multiply by 12). For reference: 1 cm³ = 0.061 in³; 1 in³ = 16.387 cm³.

Q: What is the volume of a standard ice cream cone?

A typical waffle cone has a base opening radius of about **3 cm** and an interior depth of **12 cm**, giving V = (1/3)×π×9×12 ≈ **113 cm³ (≈ 113 mL)**. Sugar cones are shallower (r ≈ 2.5 cm, h ≈ 9 cm) with V ≈ **59 cm³**. These figures help confectioners calibrate single-scoop sizes and match nutritional labeling requirements.

Q: Does the formula work for oblique cones (apex not centered above the base)?

Yes — for **volume only**. Cavalieri's Principle guarantees that any oblique cone with base radius r and vertical height h has the same volume as a right cone with the same r and h: V = (1/3)πr²h. However, the lateral surface area formula **A = πrl is NOT valid** for oblique cones; computing surface area of an oblique cone requires integrating along each slant element, which varies around the circumference.

Q: What is the relationship between a cone and a cylinder with the same base and height?

A cone always has exactly **one-third** the volume of a cylinder sharing the same base radius and height. So if a cylinder holds 300 cm³, a cone of equal base and height holds exactly 100 cm³. This ratio is constant regardless of the actual dimensions — it is a fundamental result of solid geometry used in engineering sizing, container design, and standardized packaging.

Calculate cone volume, slant height, and surface area instantly. Enter radius and height — get V = (1/3)πr²h, slant height l = √(r²+h²), and total surface area in one click. Free, no sign-up.

Data verified · June 2026
Edited by Martín Rodríguez
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A cone is a three-dimensional geometric solid with a circular base of radius r and an apex directly above (or below) the center at height h. This calculator computes four key measurements: Volume (V), Slant Height (l), Lateral Surface Area, and Total Surface Area. The core formula is V = (1/3)πr²h — exactly one-third of the volume of a cylinder with the same base and height. Use this tool whenever you need precise cone measurements for engineering designs, packaging, food-science portions, architecture, or academic geometry problems. Inputs are the base radius r and the perpendicular height h; all other outputs are derived automatically.

When to use this calculator

Calculating the volume of an ice cream cone (typical waffle cone: r ≈ 3 cm, h ≈ 12 cm → V ≈ 113.1 cm³) to match filling standards or nutritional labeling.
Determining the cubic yards of concrete needed to fill traffic cone-shaped bollard molds in road construction projects.
Computing the lateral surface area of a conical roof or spire to estimate the square footage of material (sheet metal, shingles) required.
Finding the capacity of a conical hopper or funnel used in industrial grain storage or pharmaceutical powder dispensing.
Verifying geometry homework or exam answers for cone volume and surface area problems with step-by-step formula breakdowns.
Designing conical paper cups (Dixie-style) — matching a target volume (e.g., 180 mL) to find the required radius and height ratio.

Volume of Everyday Conical Objects — V = (1/3)πr²h

Typical (nominal) dimensions of common cone-shaped items and their computed volume. Real products vary by brand; use the calculator above with your own measurements.

Object	Radius r (cm)	Height h (cm)	Volume (cm³)	Volume (mL / L)
Sugar cone (ice cream)	2.5	9	59 cm³	≈ 59 mL
Waffle cone (ice cream)	3	12	113 cm³	≈ 113 mL
Conical paper water cup	3	8	75 cm³	≈ 75 mL (2.5 oz)
Snow-cone paper cup	3.5	11	141 cm³	≈ 141 mL
Birthday party hat	9	30	2,545 cm³	≈ 2.5 L
Tabletop Christmas tree	30	90	84,823 cm³	≈ 84.8 L

Computed with V = (1/3)πr²h using full-precision π. 1 cm³ = 1 mL; 1,000 cm³ = 1 L. Ice-cream-cone reference dimensions per confectionery industry standards (waffle cone ≈ 113 mL single-scoop capacity). These items approximate right circular cones; vessels with curved walls (e.g., martini glasses) deviate from the formula.

How it works

How It's Calculated

All four outputs derive from just two inputs — base radius r and perpendicular height h.

# Slant Height (l)
l = √(r² + h²)

# Volume (V)
V = (1/3) × π × r² × h

# Lateral (Side) Surface Area
A_lateral = π × r × l
           = π × r × √(r² + h²)

# Base Area
A_base = π × r²

# Total Surface Area
A_total = A_lateral + A_base
        = π × r × l + π × r²
        = π × r × (l + r)

π ≈ 3.14159265358979 (the calculator uses full double-precision π).

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Quick-Reference Table

Common cone dimensions and their computed outputs (π rounded to 5 decimal places for display):

r (cm)	h (cm)	l (cm)	V (cm³)	A_lateral (cm²)	A_total (cm²)
1	3	3.162	3.14	9.93	13.07
3	4	5.000	37.70	47.12	75.40
5	12	13.000	314.16	204.20	282.74
7	24	25.000	1,231.5	549.78	703.72
10	10	14.142	1,047.2	444.29	758.58
15	20	25.000	4,712.4	1,178.10	2,884.07
30	40	50.000	37,699	4,712.4	11,541

> Tip: When r : h = 3 : 4, the slant height l forms a perfect 3-4-5 Pythagorean triple (scaled). This is why l = 5 when r = 3, h = 4 — and l = 25 when r = 15, h = 20.

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Typical Cases

Case 1 — Waffle Ice Cream Cone

A standard waffle cone has a base opening radius of r = 3 cm and a depth of h = 12 cm.

l = √(3² + 12²) = √(9 + 144) = √153 ≈ 12.37 cm
V = (1/3) × π × 9 × 12 ≈ 113.10 cm³  (≈ 113 mL)
A_lateral = π × 3 × 12.37 ≈ 116.65 cm²
A_total   = 116.65 + 28.27 ≈ 144.92 cm²

Knowing the volume (~113 mL) helps manufacturers calibrate the scoop size to avoid overflow.

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Case 2 — Conical Roof Spire

An architect designs a conical church spire with a base radius of r = 5 m and a height of h = 12 m.

l = √(25 + 144) = √169 = 13 m   (exact Pythagorean triple: 5-12-13)
A_lateral = π × 5 × 13 ≈ 204.20 m²

The roofer needs approximately 204.2 m² of sheet copper — a calculation impossible to do accurately without the slant height.

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Case 3 — Industrial Grain Hopper

A conical hopper at the base of a grain silo has r = 2 m and h = 3 m.

V = (1/3) × π × 4 × 3 ≈ 12.57 m³

Since wheat has a bulk density of ~770 kg/m³ (USDA), the hopper holds roughly 9,679 kg (≈ 9.7 metric tons) of wheat — critical for load-bearing calculations.

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Common Errors

1. Using diameter instead of radius. The formula requires the base radius (r = d/2). Plugging in diameter doubles r, which quadruples r² and quadruples the volume — a 4× overestimate.

2. Confusing slant height with perpendicular height. The formula V = (1/3)πr²h uses the vertical (perpendicular) height h, not the slant height l. Using l instead of h inflates the volume whenever the cone is not degenerate (l > h always, except when r = 0).

3. Forgetting the 1/3 factor. Students often compute πr²h (the cylinder volume) instead of (1/3)πr²h, yielding a result exactly 3× too large.

4. Mixing units. Entering r in inches and h in centimeters without converting produces a physically meaningless result. Always ensure both inputs share the same unit; the output unit is that unit cubed (for volume) or squared (for area).

5. Applying cone formulas to oblique cones without adjustment. This calculator assumes a right circular cone (apex directly above the center). Cavalieri's Principle confirms the volume formula still holds for oblique cones, but the slant height and lateral surface area formulas do not — a common exam trap.

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Related Calculators

Cylinder Volume Calculator

Sphere Volume Calculator

Circle Area Calculator

Pythagorean Theorem Calculator

Surface Area Calculator

Worked Example: r = 3 cm, h = 4 cm

Slant height: l = √(3² + 4²) = √25 = 5 cm

Volume: V = (1/3) × π × 9 × 4 = 12π ≈ 37.70 cm³

Lateral area: A_lat = π × 3 × 5 = 15π ≈ 47.12 cm²

Total surface area: A_total = π × 3 × (3 + 5) = 24π ≈ 75.40 cm²

V = 37.70 cm³ | l = 5.00 cm | A_total = 75.40 cm²

Frequently asked questions

What is the formula for the volume of a cone?

The volume of a right circular cone is V = (1/3)πr²h, where r is the base radius and h is the perpendicular height. The 1/3 factor comes from the fact that a cone occupies exactly one-third the volume of a cylinder with the same base and height — a result proven rigorously in Euclidean geometry. For r = 5 cm and h = 12 cm: V = (1/3) × π × 25 × 12 ≈ 314.16 cm³.

What is slant height and how is it different from cone height?

The perpendicular height (h) is the straight vertical distance from the base center to the apex. The slant height (l) is the distance along the outer surface from the base edge to the apex: l = √(r² + h²) by the Pythagorean theorem. For a cone with r = 3, h = 4, l = 5 exactly (a 3-4-5 right triangle). You need l to calculate lateral surface area, but only h for volume.

How do I calculate the total surface area of a cone?

Total surface area = lateral area + base area: A_total = πrl + πr², where l = √(r² + h²). For a cone with r = 5 cm and h = 12 cm, l = 13 cm, so A_total = π×5×13 + π×25 = 65π + 25π = 90π ≈ 282.74 cm². The lateral area alone (πrl) represents the unrolled curved surface — like the flat sector of a circle with radius l and arc length 2πr.

Why is cone volume exactly 1/3 of cylinder volume?

This relationship — proven by Archimedes and later formalized via Cavalieri's Principle and calculus — holds for any cone and cylinder sharing the same base area and height. Integrating the stacked circular cross-sections from base to apex gives ∫₀ʰ π(r·x/h)² dx = πr²h/3. Experimentally, filling a cone-shaped vessel three times fills the matching cylinder exactly, which is how the 1/3 rule is often demonstrated in classrooms.

What units does the calculator use, and can I mix centimeters and inches?

The calculator is unit-agnostic: if you enter r and h in centimeters, volume is in cm³ and areas in cm². If you use inches, outputs are in³ and in². Never mix units — entering r in inches and h in feet requires first converting h to inches (multiply by 12). For reference: 1 cm³ = 0.061 in³; 1 in³ = 16.387 cm³.

What is the volume of a standard ice cream cone?

A typical waffle cone has a base opening radius of about 3 cm and an interior depth of 12 cm, giving V = (1/3)×π×9×12 ≈ 113 cm³ (≈ 113 mL). Sugar cones are shallower (r ≈ 2.5 cm, h ≈ 9 cm) with V ≈ 59 cm³. These figures help confectioners calibrate single-scoop sizes and match nutritional labeling requirements.

Does the formula work for oblique cones (apex not centered above the base)?

Yes — for volume only. Cavalieri's Principle guarantees that any oblique cone with base radius r and vertical height h has the same volume as a right cone with the same r and h: V = (1/3)πr²h. However, the lateral surface area formula A = πrl is NOT valid for oblique cones; computing surface area of an oblique cone requires integrating along each slant element, which varies around the circumference.

How much material is needed to make a conical paper cup with a volume of 180 mL?

180 mL = 180 cm³. Set V = (1/3)πr²h = 180, so r²h = 540/π ≈ 171.9. A common ratio for paper cones is h ≈ 3r: substituting gives r²(3r) = 3r³ = 171.9 → r³ ≈ 57.3 → r ≈ 3.86 cm, h ≈ 11.57 cm, l ≈ 12.19 cm. The lateral surface area (material needed) = π×3.86×12.19 ≈ 147.8 cm².

What is the relationship between a cone and a cylinder with the same base and height?

A cone always has exactly one-third the volume of a cylinder sharing the same base radius and height. So if a cylinder holds 300 cm³, a cone of equal base and height holds exactly 100 cm³. This ratio is constant regardless of the actual dimensions — it is a fundamental result of solid geometry used in engineering sizing, container design, and standardized packaging.

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Math calculator with its formula verified automatically against NIST Digital Library of Mathematical Functions – Geometric Formulas, per our editorial policy and methodology.

Updates

Updated: June 2026. Parameters are verified periodically against the cited sources.

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Indicative results. For critical decisions, consult a professional.

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APA format

Rodríguez, M. (2026). Cone Volume Calculator — V = (1/3)πr²h. Hacé Cuentas. https://hacecuentas.com/en/cone-volume-radius-height

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@misc{hacecuentas_cone_volume_radius_height_2026,
  author       = {Rodríguez, Martín},
  title        = {{Cone Volume Calculator — V = (1/3)πr²h}},
  year         = {2026},
  howpublished = {\url{https://hacecuentas.com/en/cone-volume-radius-height}},
  note         = {Hacé Cuentas}
}

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