Frustum Calculator | Volume, Area & Slant Height
Calculate volume, slant height, lateral area, and total surface area of a frustum (truncated cone). Supports 5 units, 6 presets, SVG diagram, and step-by-step working.
Set R₂ = 0 for a complete cone, or R₁ = R₂ for a cylinder.
Press Enter to calculate · Esc to reset
What Is the Frustum Calculator | Volume, Area & Slant Height?
A frustum of a cone is the solid formed when a smaller cone is cut from the apex of a larger cone by a plane parallel to the base. The word comes from the Latin frustum, meaning “morsel” or “piece cut off.” It is one of the most practically useful geometric solids, it describes everything from a drinking cup to a volcanic cone to a concrete footing.
Understanding the Volume Formula
The volume formula V = (πh/3)(R₁² + R₁R₂ + R₂²) is a special case of the prismatoid volume formula. It can be derived by subtracting the volume of the removed small cone from the original large cone:
- ›Large cone volume: V_big = π·R₁²·H / 3 (H = full cone height)
- ›Small cone volume: V_small = π·R₂²·(H−h) / 3
- ›By similar triangles: H = R₁·h/(R₁−R₂) when R₁ ≠ R₂
- ›Substituting and simplifying yields the prismatoid form with the cross-term R₁R₂
The middle term R₁R₂ is the geometric mean of the two base areas, it accounts for the smoothly tapering cross-sections between the two bases. Omitting this term is a common mistake that underestimates volume, especially when R₁ and R₂ are close.
Slant Height vs. Perpendicular Height
The slant height l is the distance measured along the sloped surface, while h is the true vertical height. They are related by a simple right triangle: the horizontal leg has length |R₁ − R₂| and the vertical leg is h, giving l = √(h² + (R₁−R₂)²). The slant height is what you would measure with a tape along the outside of a traffic cone or bucket.
Formula
| Symbol | Name | Description |
|---|---|---|
| R₁ | Bottom radius | Radius of the larger circular base (must be ≥ R₂ by convention, though formula works either way) |
| R₂ | Top radius | Radius of the smaller circular base; set to 0 for a complete cone, equal to R₁ for a cylinder |
| h | Perpendicular height | Vertical distance between the two circular bases, measured perpendicular to both |
| l | Slant height | Length of the shortest path along the lateral surface from bottom edge to top edge |
| V | Volume | Space enclosed inside the frustum, computed using the prismatoid (Heronian mean) formula |
| L | Lateral surface area | Area of the curved side surface only, excluding the two circular bases |
| S | Total surface area | Sum of lateral area and both circular base areas |
| H | Original cone height | Height of the complete cone from which the frustum was cut; only defined when R₁ ≠ R₂ |
How to Use
- 1Try a preset: Click a real-world example (coffee cup, bucket, traffic cone, lampshade, grain hopper, funnel) to load typical dimensions immediately.
- 2Select unit: Choose mm, cm, m, in, or ft from the unit dropdown. All inputs and outputs use the same unit.
- 3Enter R₁: Type the bottom (larger) radius. For a funnel or bucket, this is the widest opening.
- 4Enter R₂: Type the top (smaller) radius. Set to 0 for a complete cone, or equal to R₁ for a cylinder.
- 5Enter h: Type the perpendicular height, the vertical distance between the two circular faces.
- 6Calculate: Press Calculate or hit Enter. Volume, all surface areas, slant height, and the original cone height appear instantly.
- 7Read the diagram: A labeled schematic shows R₁, R₂, h, and l to confirm your inputs match the shape you have in mind.
- 8Copy results: Click "Copy all results" to get every measurement as formatted text for pasting into reports, spreadsheets, or notes.
Example Calculation
Example 1, Bucket: R₁=18 cm, R₂=13 cm, h=30 cm
Example 2, Traffic Cone: R₁=15 cm, R₂=3 cm, h=60 cm
Example 3, Degenerate case: R₂=0 (complete cone) and R₁=R₂ (cylinder)
Understanding Frustum | Volume, Area & Slant Height
Frustum vs. Cone vs. Cylinder, Comparison
| Shape | Volume | Lateral Area | Condition |
|---|---|---|---|
| Cone | πR²h/3 | πRl | R₂ = 0 |
| Frustum | πh/3 × (R₁²+R₁R₂+R₂²) | π(R₁+R₂)×l | 0 < R₂ < R₁ |
| Cylinder | πR²h | 2πRh | R₁ = R₂ = R |
Pyramidal Frustums
The same concept applies to pyramids. A frustum of a square pyramid (a truncated square pyramid) has two square bases rather than two circular bases. The volume formula is analogous: V = (h/3)(A₁ + √(A₁·A₂) + A₂), where A₁ and A₂ are the base areas. The geometric mean term √(A₁·A₂) plays the same role as R₁R₂ in the conical case, it accounts for intermediate cross-sections. Ancient Egyptians used this formula to calculate the volume of pyramid frustums in construction; it appears in the Moscow Mathematical Papyrus (c. 1850 BCE).
Real-World Applications
- ›Civil engineering, embankment and excavation volume estimates: earthworks are modelled as frustums when the cross-section changes gradually between two levels
- ›Manufacturing, tapered containers (cups, buckets, funnels, hoppers) need exact volume for capacity labelling and material cost estimation
- ›Architecture, frustum shapes appear in column entasis, cooling tower bases, and decorative capitals; surface area determines cladding quantity
- ›Agriculture, conical grain hoppers in silos are frustums; volume calculation determines storage capacity per bin
- ›3D graphics, the camera view frustum in 3D rendering is a four-sided pyramidal frustum defining the visible region in a scene
- ›Road safety, traffic cone material requirements (for the rubber body) are a frustum lateral surface area calculation
Why the Volume Formula Has a Cross-Term
Integrating the cross-sectional area A(y) = π·r(y)² from 0 to h, where r(y) varies linearly from R₁ to R₂, gives a quadratic integral. When expanded, this quadratic produces three terms: π∫₀ʰ [R₁ + (R₂−R₁)y/h]² dy = πh/3 × (R₁² + R₁R₂ + R₂²). The cross-term R₁R₂ disappears only in the degenerate cases R₂=0 (cone) or R₁=R₂ (cylinder), which is why naive averaging of the two base areas underestimates the volume of a frustum.
All calculations run entirely in your browser using IEEE 754 double-precision floating-point arithmetic. No data is uploaded to any server. Results are accurate to at least 10 significant figures for inputs within the normal engineering range. π is taken as Math.PI = 3.141592653589793.
Frequently Asked Questions
What is a frustum and how does it differ from a cone?
A frustum is the portion of a cone that remains after cutting off the apex with a plane parallel to the base. The result has two circular faces of different sizes connected by a curved lateral surface.
- • A cone has one circular base and tapers to a single point (apex)
- • A frustum has two circular bases (radii R₁ and R₂) and a flat top, no apex
- • Setting R₂ = 0 in the frustum formulas recovers exactly the cone formulas
The terms “frustum” and “truncated cone” are interchangeable. “Frustum” is the formal mathematical term; “truncated cone” is more descriptive.
Why does the volume formula include the R₁×R₂ cross-term?
The cross-term arises from integrating the quadratic cross-sectional area along the height. The radius varies linearly from R₁ to R₂, so the area varies quadratically, not linearly.
- • Simple average of the two base areas: A_avg = π(R₁²+R₂²)/2, this overestimates volume because it ignores the concave taper
- • Correct formula uses the Heronian mean: π(R₁²+R₁R₂+R₂²)/3, this accounts for intermediate cross-sections
- • Example: R₁=6, R₂=3 → naive method gives 47π per unit height, correct formula gives 21π per unit height (off by 12%)
This cross-term is why it's important to use the proper frustum formula rather than averaging the two base areas.
What is the slant height and how is it different from the height?
The height (h) is the perpendicular distance between the two bases, measured straight up and down. The slant height (l) is the distance along the sloped surface from the edge of one base to the edge of the other.
- • They form a right triangle with horizontal leg |R₁ − R₂|
- • l = √(h² + (R₁−R₂)²)
- • l = h only when R₁ = R₂ (cylinder, vertical sides)
- • The slant height is what you would measure with a tape along the outside of a bucket or cone
The lateral surface area formula uses l (not h) because the curved surface unrolls into a trapezoid whose slanted sides have length l.
What is the "original cone height" shown in the results?
The original cone height H is the height of the complete, uncut cone from which the frustum was obtained. It tells you how tall the cone would be if the top cap were replaced.
- • H = R₁ × h / (R₁ − R₂) (by similar triangles)
- • The small removed cone has height H − h and base radius R₂
- • For a traffic cone with R₁=15, R₂=3, h=60: H = 15×60/12 = 75 cm, the cone tip is 15 cm above the top cap
This value is undefined (not shown) when R₁ = R₂, because parallel bases mean the original shape was already a cylinder.
How do I calculate the volume of material in a frustum-shaped container?
If the container is a frustum (e.g., a bucket or hopper), the total volume capacity equals the frustum volume computed by this calculator.
- • For a partially-filled container, you need to know the fill height h_fill and the radius at that height r_fill = R₁ + (R₂−R₁)×h_fill/h
- • Then compute the volume of the smaller frustum (R₁ to r_fill, height h_fill) using this calculator
- • 1 litre = 1,000 cm³; 1 US gallon ≈ 3,785 cm³
For example, a bucket with R₁=18 cm, R₂=13 cm, h=30 cm has a volume of about 22.84 litres, a standard 10 L bucket with these dimensions filled to 13 cm depth holds roughly 7.5 litres.
What happens when R₁ equals R₂ in this calculator?
When R₁ = R₂, the frustum becomes a cylinder. The formulas reduce exactly:
- • Volume: V = πR²h (standard cylinder formula)
- • Slant height: l = h (sides are vertical, so slant = height)
- • Lateral area: L = 2πRh (cylinder lateral surface)
- • The “original cone height” is not shown because a cylinder has no apex
The calculator handles this case cleanly, try entering equal radii to verify the cylinder formulas.
How is a frustum used in 3D computer graphics?
In 3D graphics and game engines, the “view frustum” defines what the camera can see. It is a four-sided pyramidal frustum (not circular) with:
- • Near clipping plane: small rectangle close to the camera
- • Far clipping plane: large rectangle at the maximum render distance
- • Four slanted sides connecting them
Frustum culling is the optimisation where objects outside the view frustum are not rendered, dramatically improving performance. The geometry is the same prismatoid concept as the conical frustum, just with rectangular rather than circular cross-sections.
How accurate are the calculations for very large or very small values?
The calculator uses JavaScript's IEEE 754 double-precision floating-point arithmetic (64-bit), which provides about 15–16 significant decimal digits of precision.
- • For typical engineering values (radii in mm to m, heights in mm to m) the results are accurate to at least 10 significant figures
- • Very large values (>10⁹ in any unit) may lose precision due to floating-point limits, switch to a larger unit (e.g., m instead of mm) for big structures
- • The slant height formula √(h²+(R₁−R₂)²) can lose precision when R₁ ≈ R₂ and h is much larger, the relative error remains small, but if extremely high precision is needed, use a numerically stable Pythagorean algorithm