What is the formula for the volume of a conical frustum?

The volume of a conical frustum is: V = (πh/3)(R² + Rr + r²), where R is the larger base radius, r is the smaller top radius, and h is the perpendicular height between the two parallel faces. This formula arises from subtracting the volume of the smaller cone (which was cut off) from the volume of the full cone: V = V_full − V_cut = (π/3)(R²H) − (π/3)(r²(H−h)), where H is the height of the complete original cone. Using similar triangles to express H in terms of R, r, and h, and simplifying using the factorisation R³ − r³ = (R − r)(R² + Rr + r²), the result is the compact frustum formula.

What is the slant height of a frustum?

The slant height ℓ of a conical frustum is the straight-line distance along the curved lateral surface from any point on the rim of the smaller circle to the corresponding point on the rim of the larger circle. It is calculated using the Pythagorean theorem applied to a right triangle formed by the height h (vertical leg), the difference in radii (R − r) (horizontal leg), and the slant height (hypotenuse): ℓ = √(h² + (R − r)²). The slant height is needed for computing the lateral and total surface areas but not for the volume. It is always larger than the height h unless R = r (a cylinder), in which case ℓ = h.

What is the lateral surface area of a conical frustum?

The lateral (curved) surface area of a conical frustum — the area of the outer sloping surface, excluding the two circular bases — is: A_L = π(R + r)ℓ, where ℓ = √(h² + (R−r)²) is the slant height. This formula is analogous to the cone's lateral area (A = πRℓ) but uses the sum of the two radii. The total surface area including both circular bases is: A_T = π(R² + r² + (R + r)ℓ) = A_L + πR² + πr². These formulas are essential for calculating the amount of material needed to construct conical frustum-shaped objects like funnels, lampshades, or sheet-metal hoppers.

What is the volume of a frustum when r = 0?

When the smaller radius r = 0, the frustum becomes a complete cone (the 'cut-off' piece has zero size). Substituting r = 0 into the frustum formula: V = (πh/3)(R² + R×0 + 0²) = (πh/3)(R²) = (1/3)πR²h. This is exactly the well-known formula for the volume of a right circular cone. Similarly, the slant height formula gives ℓ = √(h² + R²), the standard cone slant height, and the lateral surface area becomes A_L = πRℓ — the standard cone lateral area. So the conical frustum formula is the most general case that reduces to a cone as a special case.

What happens when R = r in the frustum formula?

When R = r, both radii are equal and the frustum becomes a right circular cylinder. Substituting R = r into the frustum formula: V = (πh/3)(R² + R·R + R²) = (πh/3)(3R²) = πR²h. This is exactly the volume of a cylinder. The slant height becomes ℓ = √(h² + 0²) = h (straight vertical walls, no slope), and the lateral surface area becomes A_L = π(R + R)h = 2πRh — the standard cylinder lateral surface area formula. This is a reassuring check that the frustum formula is correct.

What are real-world examples of conical frustums?

Conical frustums appear everywhere in engineering and daily life: (1) Buckets and pails — almost all plastic or metal buckets taper from a wider base to a narrower top for stackability; (2) Funnels — the wide opening collects fluid and the narrow end channels it; (3) Traffic cones and bollards — frustum shape provides stability and reduces wind resistance; (4) Flower planters and terracotta pots; (5) Lampshades; (6) Grain hoppers and industrial silos; (7) Disposable paper/plastic cups — many taper from top to bottom; (8) Cooling towers (approximately); (9) Engine intake cones in jet aircraft; (10) 3D-printed objects — slicing software calculates frustum fill volumes layer by layer.

How is the frustum volume formula derived?

The derivation uses similar triangles and the difference-of-cubes factorisation. (1) A frustum with base radius R, top radius r, and height h was originally a complete cone of height H and base radius R, with a smaller cone of height (H−h) and radius r cut from the top. (2) By similar triangles: r/R = (H−h)/H, giving H = Rh/(R−r). (3) Volume of full cone: V_full = πR²H/3. Volume of cut cone: V_cut = πr²(H−h)/3. (4) After substitution and simplification: V = V_full − V_cut = πh(R³−r³)/[3(R−r)]. (5) Factorising R³−r³ = (R−r)(R²+Rr+r²), the (R−r) factors cancel, giving V = (πh/3)(R²+Rr+r²).

Volume Formula Surface Areas Slant Height Step-by-Step

Conical Frustum Volume Calculator

Q: What is a conical frustum?

A conical frustum (also called a truncated cone or frustum of a cone) is the solid geometric shape obtained by cutting a right circular cone with a plane parallel to its base. The result is a shape with two parallel circular faces (a larger base of radius R and a smaller top of radius r) connected by a curved lateral surface. The word 'frustum' comes from the Latin for 'morsel' or 'piece'. Common everyday examples include: standard buckets and pails, drinking cups (tapered), funnels, traffic cones, lamp shades, and flower planters. It is formally classified as a solid of revolution — generated by rotating a trapezoid around one of its parallel sides.

Instantly calculate the volume, slant height, lateral surface area, and total surface area of any conical frustum (truncated cone). Enter the two radii and height — our calculator handles all unit conversions and shows a complete step-by-step solution.

Includes full formula derivation, 4 real-world presets, 3 worked examples, and a comprehensive guide to frustum geometry used in engineering, architecture, and everyday design.

Open the calculator See the derivation

The Core Formulas

Volume of a Conical Frustum $$V = \frac{\pi h}{3}\left(R^2 + Rr + r^2\right)$$

Slant Height & Surface Areas $$\ell = \sqrt{h^2 + (R-r)^2}$$ $$A_L = \pi(R+r)\ell \qquad A_T = \pi\!\left(R^2+r^2+(R+r)\ell\right)$$

where $R$ = larger base radius, $r$ = smaller top radius, $h$ = height, $\ell$ = slant height.

Conical Frustum Calculator

Quick Presets

Dimensions

Larger Radius (R)

Radius of the larger (base) circle — must be ≥ r

Smaller Radius (r)

Radius of the smaller (top) circle — set r = 0 for a cone

Height (h)

Perpendicular height between the two circular faces

Output Unit

Volume

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Step-by-Step Solution

What Is a Conical Frustum?

A conical frustum — often called a truncated cone or simply a frustum of a cone — is the three-dimensional solid formed when a plane parallel to the base of a cone slices through the cone, removing the top portion. The result is a shape with two parallel, circular faces of different sizes (radii R and r, with R ≥ r) connected by a smooth curved lateral surface.

The name "frustum" comes from the Latin frustum, meaning "a morsel" or "a cut portion." In formal geometry, it belongs to a broader family called frustums of pyramids, where the base is a polygon rather than a circle. The conical frustum is by far the most common in engineering and everyday life because circles are the natural cross-section of many manufacturing processes (turning, casting, rolling).

What makes the frustum particularly interesting mathematically is that it generalises both a cone and a cylinder as limiting cases — when the smaller radius r approaches zero you get a cone, and when r = R you get a cylinder. Its volume formula is a beautiful algebraic expression that unifies all three shapes.

Everyday Examples of Conical Frustums

Household Objects

Standard plastic or metal buckets and pails — taper so they stack efficiently
Flower planters and terracotta pots
Funnels — wide mouth captures liquid, narrow spout directs it
Many drinking cups and paper cones
Lampshades

Engineering & Industry

Grain hoppers and industrial silos with tapered bottoms
Traffic cones and road bollards
Sheet-metal ductwork reducers
Engine intake cones in jet aircraft
3D-printed parts — slicing software tiles frustums layer by layer

Deriving the Frustum Volume Formula

The frustum volume formula is derived by treating the frustum as the difference between two complete cones — the original large cone and the smaller cone that was removed from the top.

Step 1 — Set up similar triangles. Let the original cone have base radius $R$ and total height $H$. The small cone removed from the top has base radius $r$ and height $H - h$, where $h$ is the frustum's height. Since both cones share the same apex, their dimensions are proportional by similar triangles:

$$\frac{r}{R} = \frac{H - h}{H} \quad \Rightarrow \quad H = \frac{Rh}{R - r} \quad (R \neq r)$$

Step 2 — Compute both cone volumes.

$$V_{\text{full}} = \frac{\pi R^2 H}{3} = \frac{\pi R^3 h}{3(R-r)} \qquad V_{\text{cut}} = \frac{\pi r^2 (H-h)}{3} = \frac{\pi r^3 h}{3(R-r)}$$

Step 3 — Subtract and factorise. The frustum volume is:

$$V = V_{\text{full}} - V_{\text{cut}} = \frac{\pi h}{3(R-r)}\!\left(R^3 - r^3\right)$$

Using the difference-of-cubes factorisation $R^3 - r^3 = (R-r)(R^2 + Rr + r^2)$, the factor $(R - r)$ cancels:

Conical Frustum Volume Formula — Derived $$\boxed{V = \frac{\pi h}{3}\!\left(R^2 + Rr + r^2\right)}$$

This elegant factorisation is why the middle term $Rr$ appears — it is the cross-term from $(R-r)(R^2+Rr+r^2)$. The formula works even when R = r (cylinder case) because the derivation through similar triangles breaks down, but the formula itself remains valid as a continuous limit.

Complete Formula Reference

1 — Volume $$V = \frac{\pi h}{3}\left(R^2 + Rr + r^2\right)$$

Alternatively, using the base areas $A_1 = \pi R^2$ and $A_2 = \pi r^2$ — the Prismatoid formula:

$$V = \frac{h}{3}\!\left(A_1 + A_2 + \sqrt{A_1 A_2}\right) = \frac{h}{3}\!\left(\pi R^2 + \pi r^2 + \pi Rr\right)$$

2 — Slant Height $$\ell = \sqrt{h^2 + (R - r)^2}$$

The slant height $\ell$ is the length along the sloping lateral face from the rim of the small circle to the rim of the large circle. It is derived from the Pythagorean theorem: the right triangle formed has legs $h$ (vertical height) and $(R - r)$ (horizontal distance between the two circular edges).

3 — Lateral Surface Area (curved side only) $$A_L = \pi(R + r)\ell$$

4 — Total Surface Area (including both circular bases) $$A_T = \pi R^2 + \pi r^2 + \pi(R+r)\ell = \pi\!\left(R^2 + r^2 + (R+r)\ell\right)$$

3 Fully Worked Examples

Example 1 — Classic Textbook Problem

Given: $R = 5\ \text{m},\quad r = 3\ \text{m},\quad h = 4\ \text{m}$

Slant height:

$$\ell = \sqrt{4^2 + (5-3)^2} = \sqrt{16 + 4} = \sqrt{20} = 2\sqrt{5} \approx 4.472\ \text{m}$$

Volume:

$$V = \frac{\pi \times 4}{3}(5^2 + 5 \times 3 + 3^2) = \frac{4\pi}{3}(25 + 15 + 9) = \frac{4\pi}{3} \times 49 = \frac{196\pi}{3} \approx \boxed{205.25\ \text{m}^3}$$

Lateral Surface Area: $A_L = \pi(5+3)(2\sqrt{5}) = 16\pi\sqrt{5} \approx 112.37\ \text{m}^2$

Total Surface Area: $A_T = \pi(25 + 9 + (8)(2\sqrt{5})) = \pi(34 + 16\sqrt{5}) \approx \pi \times 69.78 \approx 219.21\ \text{m}^2$

Example 2 — Engineering: Grain Hopper

Given: A grain hopper with larger opening $R = 3\ \text{m}$, smaller outlet $r = 1\ \text{m}$, depth $h = 5\ \text{m}$. Find its capacity in litres.

Slant height:

$$\ell = \sqrt{5^2 + (3-1)^2} = \sqrt{25 + 4} = \sqrt{29} \approx 5.385\ \text{m}$$

Volume:

$$V = \frac{\pi \times 5}{3}(9 + 3 + 1) = \frac{5\pi}{3} \times 13 = \frac{65\pi}{3} \approx 68.07\ \text{m}^3 = \boxed{68{,}070\ \text{litres}}$$

Lateral SA: $A_L = \pi(3+1)(5.385) = 4\pi \times 5.385 \approx 67.66\ \text{m}^2$

Interpretation: The hopper holds 68,070 litres of grain — equivalent to about 18 standard tanker truck loads (each ~3,800 L). This calculation is used by agricultural engineers to specify hopper capacity and structural loading requirements.

Example 3 — Household: 5-Gallon Bucket

Given: A standard 5-gallon bucket with top (larger) inner radius $R = 14.8\ \text{cm}$, bottom (smaller) inner radius $r = 12.5\ \text{cm}$, height $h = 35.5\ \text{cm}$.

Volume (in cm³ then litres):

$$V = \frac{\pi \times 35.5}{3}(14.8^2 + 14.8 \times 12.5 + 12.5^2) = \frac{35.5\pi}{3}(219.04 + 185 + 156.25)$$ $$= \frac{35.5\pi}{3} \times 560.29 = \frac{19{,}890.3\pi}{3} \approx 20{,}825\ \text{cm}^3 = \boxed{20.83\ \text{litres} \approx 5.5\ \text{US gal}}$$

Insight: The nominal "5-gallon" bucket is actually slightly larger than 5 US gallons (18.93 L) at ~20.8 L inner volume. The extra volume is intentional headroom. This type of calculation is used in packaging engineering to verify fill capacity from dimensional drawings.

Special Cases: When a Frustum Becomes a Cone or Cylinder

The conical frustum formula is the most general form — the cone and cylinder formulas are both special cases of it.

When r = 0 → Cone

$$V = \frac{\pi h}{3}(R^2 + 0 + 0) = \frac{1}{3}\pi R^2 h$$

The slant height becomes $\ell = \sqrt{h^2 + R^2}$, the standard cone slant height. The lateral surface area reduces to $A_L = \pi R \ell$.

When R = r → Cylinder

$$V = \frac{\pi h}{3}(R^2 + R^2 + R^2) = \pi R^2 h$$

The slant height becomes $\ell = h$ (vertical walls). The lateral surface area gives $A_L = 2\pi R h$ — exactly the cylinder formula.

The Geometric Mean Link

$$V = \frac{h}{3}\!\left(A_1 + \sqrt{A_1 A_2} + A_2\right)$$

The cross-term $Rr = \sqrt{R^2 r^2} = \sqrt{A_1 A_2}/\pi$ is the geometric mean of the two base radii — the frustum's volume is the arithmetic mean of three values: top area, base area, and their geometric mean.

Frustum vs Cone vs Cylinder — Formula Comparison

Property	Cone (r=0)	Cylinder (R=r)	Frustum (R > r > 0)
Volume	$\frac{1}{3}\pi R^2 h$	$\pi R^2 h$	$\frac{\pi h}{3}(R^2+Rr+r^2)$
Slant height	$\sqrt{h^2+R^2}$	$h$ (vertical)	$\sqrt{h^2+(R-r)^2}$
Lateral SA	$\pi R \ell$	$2\pi R h$	$\pi(R+r)\ell$
Total SA	$\pi R(\ell+R)$	$2\pi R(h+R)$	$\pi(R^2+r^2+(R+r)\ell)$
% of cylinder volume	33.3%	100%	33.3% – 100%
Special condition	r = 0 (apex)	R = r (equal radii)	R > r > 0

Quick volume check: The volume of a frustum with R = 5, r = 4, h = 6 can be verified mentally. A cylinder with r = 5, h = 6 has volume 150π. The frustum with R = 5, r = 4 has volume (6π/3)(25 + 20 + 16) = 2π × 61 = 122π. That is 122/150 = 81.3% of the circumscribed cylinder — a useful sanity check since a non-trivial frustum should be 33–100% of the cylinder.

Real-World Applications of Conical Frustum Geometry

The conical frustum is one of the most practically useful solid shapes in engineering and design. Its combination of a wide base (stability) and narrow top (material efficiency / directional output) makes it ideal across many fields.

Packaging and container design — Buckets, pails, and storage containers are universally tapered frustums. The taper allows thousands of empty containers to stack (nest) into the space of just one, reducing shipping costs by 80–90% compared to cylindrical containers of the same volume.
Civil engineering — hoppers and silos — Agricultural and industrial hoppers use frustum geometry to ensure gravity-driven flow. Engineers calculate frustum volume to determine the mass of stored material and the resulting structural loads on the hopper walls and support structure.
Traffic safety equipment — Standard road traffic cones are frustums: the wide base provides stability against wind and traffic turbulence, while the narrow top reduces weight and material use. ASTM F2956 specifies minimum base dimensions for highway-speed zones.
Fluid dynamics and funnels — Funnel geometry determines flow rate characteristics via the frustum's internal volume and the narrowing cross-sectional area. Flow rate through the exit orifice is governed by Torricelli's theorem and is affected by the frustum's capacity above the exit.
Architecture and aesthetics — Lamp shades, decorative planters, and architectural column capitals (echinus) are frustums. Architects compute frustum volumes and surface areas to estimate material quantities (fabric, ceramic, stone) and costs.
3D printing and CNC machining — Slicing software for FDM printers calculates the volume of each layer (a thin frustum when support structures or tapered features are present). Material consumption per layer is the tiny frustum volume multiplied by density.
Geology and mining — Open-pit mines and quarries are often approximated as inverted frustums. Mining engineers calculate frustum volumes to estimate ore reserves and plan extraction sequences.
Astronomy — The Apollo Command Module was a conical frustum shape. Mission engineers calculated frustum volumes for fuel and oxygen tank sizing and mass budgets critical to spacecraft trajectory planning.

Frequently Asked Questions

What is a conical frustum?

A conical frustum (truncated cone) is the solid formed by cutting a right circular cone with a plane parallel to its base. It has two parallel circular faces of radii R (larger) and r (smaller), and a curved lateral surface connecting them. The word "frustum" comes from Latin for "a cut piece." When r → 0 the shape becomes a complete cone; when R = r it becomes a cylinder. It appears in everyday objects — buckets, funnels, flower pots, traffic cones — and in engineering: grain hoppers, industrial reducers, and aircraft intake cones.

How do I find the volume of a frustum of a cone?

Use the formula $V = \frac{\pi h}{3}(R^2 + Rr + r^2)$, where R is the larger base radius, r is the smaller top radius, and h is the perpendicular height between the two circular faces. All three must be in the same unit before calculating. The result is in cubic units. Example: R = 6 cm, r = 4 cm, h = 8 cm → V = (π × 8 / 3)(36 + 24 + 16) = (8π/3)(76) = 608π/3 ≈ 636.9 cm³.

What units does the calculator use?

Each dimension (R, r, h) can be entered in any supported unit independently: metres, centimetres, millimetres, inches, feet, or kilometres. The calculator converts all values to metres before computing, then converts the result to your selected output unit. Volume output options include: m³, cm³, mm³, in³, ft³, Litres, and US Gallons. Surface areas are always reported in square metres (m²) in the step-by-step solution.

What is the slant height of a conical frustum?

The slant height $\ell$ is the distance along the sloping lateral face — from the edge of the smaller circle to the edge of the larger circle. Calculated as $\ell = \sqrt{h^2 + (R-r)^2}$. It is the hypotenuse of a right triangle with vertical leg h and horizontal leg (R − r). The slant height is needed to compute lateral and total surface areas but not the volume. For a cylinder (R = r), the slant height equals the perpendicular height since the walls are vertical.

What is the difference between the lateral and total surface area?

Lateral surface area (A_L = π(R+r)ℓ) is the area of just the curved outer surface — what you would paint if you painted only the sides of a bucket, not the top or bottom. Total surface area (A_T = π(R² + r² + (R+r)ℓ)) adds both circular faces (the base area πR² and the top area πr²). Use lateral SA when calculating material for the sides only (sheet metal, fabric). Use total SA when calculating full enclosure material (painting, coating, or moulding the entire surface).

Can the smaller radius be zero?

Yes — when r = 0, the frustum becomes a complete cone. The formula V = (πh/3)(R² + 0 + 0) = (πR²h)/3 reduces exactly to the standard cone volume formula. The slant height becomes ℓ = √(h² + R²), and the lateral surface area becomes πRℓ. Our calculator accepts r = 0 as a valid input representing a cone (not a frustum of a cone). The two faces become the base circle and a point (the apex).

Why is the formula V = (πh/3)(R² + Rr + r²) and not just the average of the two base areas times height?

The "average" approach V = h × π × ((R² + r²)/2) would be incorrect (it averages linearly but ignores the curved-taper geometry). The correct formula uses the arithmetic mean of three quantities: R², Rr, and r². The middle term Rr is the geometric mean of R² and r² — it accounts for how the radius changes along the height (linearly, not as a step function). If you average only the top and bottom areas linearly, you over-estimate when R >> r. The correct formula, derived via difference of cubes, gives the average of the base area, top area, AND the geometric mean area — a three-term average that exactly captures the linear taper.

Is a frustum the same as a trapezoid rotated around an axis?

Yes — a right circular frustum is the solid of revolution generated by rotating a right trapezoid (a trapezoid with one vertical leg) around its vertical leg. The two parallel sides of the trapezoid become the radii R and r of the circular faces, the vertical leg becomes the height h, and the slanted side becomes the slant height ℓ. This perspective using Pappus's centroid theorem gives an alternative surface area derivation: A_L = 2π × (centroid distance) × (slant height) = 2π × ((R+r)/2) × ℓ = π(R+r)ℓ.

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