Cone Volume Calculator
Calculate a cone's volume, slant height, and surface area by entering radius and height. Instant geometric results. Free math calculator, no signup.
Cone Formulas: Volume, Slant Height, and Surface Area
A cone is a three-dimensional solid with a circular base and a single apex (point) directly above the center of the base for a right cone. Key measurements: radius (r) of the base, height (h) from base to apex (perpendicular), and slant height (l) from apex to any point on the base circle.
Slant height: l = √(r² + h²) by the Pythagorean theorem. The radius, height, and slant height form a right triangle with l as the hypotenuse.
Volume: V = (1/3)πr²h. Exactly one-third the volume of a cylinder with the same base and height. If you pour water from a cone into a cylinder of equal dimensions, it fills exactly one-third.
Lateral surface area: A_lateral = πrl. This is the area of the curved side surface only (not the base). Intuitively: unroll the lateral surface and you get a sector of a circle with radius l and arc length 2πr.
Total surface area: A_total = πrl + πr² = πr(l + r). The first term is lateral area; the second is base area.
The 1/3 factor in volume is not arbitrary — calculus confirms it: V = ∫₀ʰ π(rz/h)² dz = πr²/h² × h³/3 = πr²h/3. At height z, the cone's circular cross-section has radius r×z/h (linearly scaling from 0 at apex to r at base). Integrating these circular slices from 0 to h gives the exact 1/3 result.
Cone Calculation Examples and Reference Table
Common cone calculations at various dimensions. All values use π = 3.14159265.
| Radius (r) | Height (h) | Slant Height (l) | Volume | Lateral Area | Total Surface Area |
|---|---|---|---|---|---|
| 1 | 1 | 1.414 | 1.047 | 4.443 | 7.584 |
| 3 | 4 | 5.000 | 37.699 | 47.124 | 75.398 |
| 4 | 9 | 9.849 | 150.796 | 123.840 | 173.994 |
| 5 | 12 | 13.000 | 314.159 | 204.204 | 282.743 |
| 6 | 8 | 10.000 | 301.593 | 188.496 | 301.593 |
| 7 | 24 | 25.000 | 1231.504 | 549.779 | 703.717 |
| 10 | 10 | 14.142 | 1047.198 | 444.288 | 758.447 |
| 10 | 30 | 31.623 | 3141.593 | 993.459 | 1307.623 |
The 3-4-5 right triangle (r=3, h=4, l=5) is a classic example — a cone with these dimensions has exactly integer slant height. Similarly r=6, h=8, l=10 is a 3-4-5 triangle scaled by 2. When designing cone-shaped components, choosing dimensions that form Pythagorean triples simplifies calculations.
Types of Cones and Related Shapes
Understanding the different types of cones and related solids expands your ability to solve real-world geometry problems.
A right cone (the standard type) has its apex directly above the center of the base. All slant heights are equal. Our calculator assumes a right cone.
An oblique cone has a displaced apex — not directly above the center. The lateral surface is asymmetric. Volume is still (1/3)πr²h by Cavalieri's principle (where h is the perpendicular height), but the lateral surface area calculation becomes more complex.
A truncated cone (frustum) is a cone with the apex cut off by a plane parallel to the base, leaving two parallel circular faces of radii R (bottom) and r (top). Volume = (πh/3)(R² + Rr + r²). Slant height = √(h² + (R−r)²). Lateral area = π(R+r)l. Common shapes: bucket, cup, funnel, flower pot, speaker cabinet.
A double cone (bicone) is two cones joined at their bases. Volume = 2 × (1/3)πr²h = (2/3)πr²h. An hourglass shape is approximately a bicone. Spinning tops and certain aircraft nose shapes use bicone geometry.
The frustum volume formula (πh/3)(R² + Rr + r²) appears in the ancient Egyptian Moscow Papyrus (~1850 BCE) — Problem 14 calculates the volume of a frustum with specific dimensions. This is one of the most remarkable mathematical achievements of antiquity: accurate calculation of a sophisticated 3D volume 4,000 years ago.
Cones in Engineering, Design, and Nature
Cone shapes appear throughout engineering and nature for functional and mathematical reasons. Recognizing cone geometry in real objects helps you apply the formulas appropriately.
Traffic cones: A traffic cone with base radius 15 cm and height 70 cm has volume = (1/3)π(0.15)²(0.70) ≈ 0.0165 m³ = 16.5 liters. Knowing volume helps manufacturers determine material usage and weight.
Ice cream cones: A standard waffle cone is approximately a frustum (slightly tapered from bottom to top, with the bottom closed). A 5 cm base radius, 12 cm height cone holds V = (1/3)π(5)²(12) ≈ 314 cm³ = 314 mL of ice cream. Double scoop means more than double the delight.
Funnels and hoppers: Industrial hoppers for grain, sand, or powder are inverted frustums. Volume calculations determine capacity; slant angle must exceed the material's angle of repose to ensure free flow. For dry sand (repose angle ~35°), the cone half-angle must exceed 35°, meaning h/r < 1/tan(35°) ≈ 1.43.
Rocket and aircraft nosecones: The nose of a rocket or supersonic aircraft uses a conical (or ogive) shape to minimize aerodynamic drag. At supersonic speeds, a conical nose creates an oblique shock wave that stays attached to the tip, reducing drag. The optimal cone half-angle depends on Mach number — typically 7–15° for cruise missiles.
Speaker cones: Loudspeaker diaphragms are conical to improve rigidity and directional frequency response. The cone's angle and material affect how well it radiates sound at different frequencies. Larger woofer cones (25–38 cm diameter) reproduce low frequencies; small tweeter domes handle high frequencies.
Natural cones: Volcanic cinder cones form when lava fragments accumulate around a vent, creating nearly perfect right cone shapes. The angle of repose of loose volcanic material (~30–35°) determines the cone's slope. Mount Fujiyama is approximately conical with base radius ~25 km and height 3.776 km.
Cone Volume and the One-Third Rule in Geometry
The 1/3 rule applies to all pyramids and cones regardless of base shape: Volume = (1/3) × base area × height. This is one of the most elegant generalizations in elementary geometry.
Square pyramid: V = (1/3)s²h. Rectangular pyramid: V = (1/3)lwh. Triangular pyramid (tetrahedron): V = (1/3) × base triangle area × h. Regular polygonal pyramid: V = (1/3) × regular polygon area × h. Right cone: V = (1/3)πr² × h (circle is the limiting case of a regular polygon with infinite sides).
Archimedes proved that a sphere inscribed in a cylinder has volume exactly 2/3 of the cylinder, and the cone inscribed in the same cylinder has volume 1/3. So sphere = 2 × cone (for the same base circle and height equal to diameter). Archimedes was so proud of this result that he requested a sphere-in-cylinder be carved on his tomb.
Cavalieri's principle justifies the 1/3 rule: two solids have the same volume if every horizontal cross-sectional slice has the same area at the same height. For a cone of height h and base radius R: at height z, the radius is R(h−z)/h, giving cross-sectional area π R²(h−z)²/h². A pyramid with appropriate base also scales quadratically with height, giving the same volume formula.
The elegant generalization to higher dimensions: an n-dimensional simplex has volume (1/n!) × base^(n-1) × height (roughly). In 3D: 1/3! = 1/6 for a tetrahedron's specific formula, but the pyramid result 1/3 comes from a slightly different derivation. The 1/n! factor appears in the n-dimensional volume formula for hyperpyramids.
Cones as Conic Sections: The Full Picture
Cones are not just geometric solids — they're the source of the most important curves in mathematics. The four conic sections arise from intersecting a double cone with a plane at different angles:
| Conic Section | Plane Orientation | Equation Form | Applications |
|---|---|---|---|
| Circle | Perpendicular to axis | x² + y² = r² | Wheels, gears, orbits |
| Ellipse | Tilted but not touching generator | x²/a² + y²/b² = 1 | Planetary orbits, elliptical mirrors |
| Parabola | Parallel to one generator line | y = ax² | Projectile paths, satellite dishes |
| Hyperbola | Cuts both nappes (steep angle) | x²/a² − y²/b² = 1 | Cooling towers, navigational systems |
Parabolic reflectors focus parallel incoming rays to a single point (the focus) — used in satellite dishes, radio telescopes, car headlights, and solar concentrators. The parabola's equation y = x²/(4f) determines the shape for a given focal length f. A large radio telescope like Arecibo (before its collapse) used a spherical approximation with active feed corrections. Conic sections unify the geometry of cones with the physics of optics, orbital mechanics, and acoustics in a remarkably elegant way.
Planetary orbits are ellipses with the Sun at one focus (Kepler's first law, 1609). The eccentricity of an ellipse determines how elongated it is: 0 for a circle, approaching 1 for a highly elongated ellipse. Earth's orbit has eccentricity 0.017 (nearly circular); Halley's Comet has eccentricity 0.967 (very elongated).
Frequently Asked Questions
Why is a cone's volume 1/3 of a cylinder?
A cone and cylinder with the same base and height: if you fill the cone with water and pour it into the cylinder, you'll fill exactly 1/3. This can be proven with calculus (integration of circular slices) or demonstrated experimentally. Three cones fill one cylinder — a result Archimedes proved geometrically over 2,200 years ago.
What is the slant height and how do I find it?
The slant height (l) is the distance from the apex to any point on the base edge, measured along the lateral surface. By Pythagoras: l = √(r² + h²). For a cone with r = 3, h = 4: l = √(9+16) = √25 = 5. The slant height is the hypotenuse of the right triangle formed by the radius, height, and lateral edge.
What is a frustum?
A frustum is a truncated cone — the shape left when a cone is cut by a plane parallel to its base. Buckets, drinking cups, and flower pots are common frustum shapes. Volume = (πh/3)(R² + Rr + r²) where R and r are the bottom and top radii, h is height. Lateral area = π(R+r)l where l = √(h² + (R−r)²).
How do I calculate the volume of an ice cream cone?
Measure the cone's base radius r and height h. Volume = (1/3)πr²h. For a cone with 3 cm radius and 12 cm height: V = (1/3) × π × 9 × 12 ≈ 113.1 cm³ (mL). Note: the cone's rim where ice cream sits is the wider top, so the pointed end is at the bottom when eating.
What is the angle of repose and how does it relate to cones?
The angle of repose is the maximum slope at which loose material (sand, grain, snow) remains stable. Natural cone-shaped piles of material form at this angle. Sand (~35°): natural piles are steep cones. Snow (~60° when wet, ~35° dry). This principle is used in hopper design — the hopper angle must exceed the material's angle of repose for free flow.
What's the surface area formula and when do I need it?
Total surface area = πrl + πr² = πr(l + r), where l = slant height = √(r² + h²). The first term (πrl) is the lateral (curved) surface; the second (πr²) is the circular base. You need total surface area when calculating material needed to make a cone (e.g., sheet metal for a funnel, fabric for a hat, paint for a traffic cone).
Is the volume formula the same for oblique cones?
Yes — V = (1/3)πr²h where h is the perpendicular height (not the slant height). This is proven by Cavalieri's principle: for any horizontal cut at height z, the oblique cone's cross-section has the same area as the equivalent right cone's cross-section. Equal cross-sections at all heights means equal volumes.
How do I convert between cone volume in different units?
Since volume = length³, converting requires cubing the linear conversion factor. If radius and height are in cm, volume is in cm³ = mL. To convert cm³ to liters, divide by 1000. To convert to m³, divide by 1,000,000 (since 1 m = 100 cm, 1 m³ = 10⁶ cm³). To convert to cubic inches, 1 in = 2.54 cm, so 1 in³ = 16.387 cm³.
What is the relationship between a cone and a sphere?
For a cone inscribed in a sphere (base touching the equator, apex at top): height h = 2r (diameter of sphere), base radius = r (sphere radius). For a cone and cylinder with the same base and height: sphere volume = 2 × cone volume, cylinder = 3 × cone. Archimedes proved: Sphere = 2/3 × circumscribed cylinder (one of his proudest results).
How are cones used in 3D printing?
In 3D printing (FDM), conical supports are often used to support overhanging features. The slicer software calculates the volume of support material (approximately conical frustums) to estimate material usage and print time. Cone geometry also appears in drill bits, thread tapers (NPT pipe threads are conical, not cylindrical), and chamfer cuts.