Frustum Calculator
Find the volume and surface of a right circular frustum (a cone with the tip sliced off) from the two radii and the height, or from the slant, or of a square frustum from the two sides. The circular formulas are V = (1/3)πh(R² + Rr + r²), slant l = √(h² + (R − r)²), and lateral π(R + r)l.
Lateral 141.371669412 · total S = 282.743338823
- Slant l = √(h² + (R − r)²) = √(16 + 9) = 5.
- Volume V = (1/3)πh(R² + Rr + r²) = (1/3) × 3.14159265359 × 4 × 63 = 263.893782902.
- Lateral = π(R + r)l = 141.371669412. Total including both bases = 282.743338823.
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How to Use the Frustum Calculator
From the two radii and the height (or the slant) grows a right circular frustum. From two square sides and the height grows a square frustum.
- Choose From R, r, and h; From R, r, and l; or From square A, a, and h.
- Enter non-negative lengths. The slant must be at least |R − r|.
- Read the volume, the slant, the lateral area, and the total including both bases.
A full party hat (top radius 0) also belongs on the Cone Calculator (V = (1/3)πR²h). Equal radii also belong on the Cylinder Calculator (V = πR²h). A full square tent belongs on the Pyramid Calculator (V = (1/3)A²h). Equal square sides belong on the Rectangular Prism Calculator (V = A²h). Converting cubic units belongs on the Volume Converter.
Volume and Surface
A frustum is what remains when you slice a cone or pyramid with a plane parallel to the base and throw away the tip. For a right circular cone the two bases are circles of radii R and r. Subtracting the small cone from the large one and simplifying gives V = (1/3)πh(R² + Rr + r²). The formula is symmetric in R and r, so swapping top and bottom does not change the volume.
A vertical cut is a trapezoid. The non-parallel side is the slant l = √(h² + (R − r)²). Unrolling the side gives a sector-ring whose area is π(R + r)l. Add both lids πR² and πr² for the total surface.
A square frustum uses the same one-third pattern with areas instead of circles: V = (1/3)h(A² + Aa + a²). The face slant (apothem of each trapezoid) is m = √(h² + ((A − a)/2)²), and four trapezoids give lateral 2(A + a)m. From the slant, h = √(l² − (R − r)²) provided l ≥ |R − r|. Height is perpendicular. An oblique frustum is out of scope.
Frustum Calculator Examples
Radii 6 and 3, height 4
l = 5. V = 84π ≈ 263.893782902. Lateral = 45π ≈ 141.371669412. Total = 90π ≈ 282.743338823.
Top radius 0 — a cone
R = 6, r = 0, h = 4. V = 48π ≈ 150.796447372, matching the Cone Calculator for radius 6 and height 4.
Equal radii — a cylinder
R = r = 3, h = 5. V = 45π ≈ 141.371669412, matching the Cylinder Calculator default can. The slant collapses to the height.
Radii 6 and 3, slant 5
h = √(25 − 9) = 4, the first example recovered from the wrapper.
Square sides 6 and 4, height 3
V = (1/3) × 3 × (36 + 24 + 16) = 76. Face slant m = √(9 + 1) = √10 ≈ 3.16228. Lateral ≈ 63.246. Total ≈ 115.246.
A slant that cannot reach
l = 2 cannot span |6 − 3| = 3. The calculator rejects it instead of inventing a height.
Cone and Cylinder Limits
Set r = 0. Then V = (1/3)πh R² and l = √(h² + R²), the cone formulas. Set R = r. Then V = πR²h and l = h, the cylinder formulas. A frustum interpolates between a party hat and a can. Do not use (1/3)πR²h on a bucket that still has a top lid — that ignores the top radius.
The same story holds for squares: a = 0 is a pyramid of side A; A = a is a square prism. This page reports both the circular and the square cases. A rectangular frustum (four different pairwise sides) is out of scope.
Common Frustum Mistakes
Using (1/3)πR²h and ignoring r
That is a cone, on the Cone Calculator. A frustum keeps both radii: (1/3)πh(R² + Rr + r²).
Using πR²h with only the bottom radius
That is a cylinder of radius R. Equal radii belong on the Cylinder Calculator. Unequal radii need the extra Rr + r² terms.
Averaging the radii and then using a cone or a can
V is not (1/3)πh((R+r)/2)², and it is not πh((R+r)/2)². Expand R² + Rr + r² instead.
Using a face edge as the height
Volume wants the perpendicular height. The slant l is for the wrapper.
Treating an oblique bucket as a right frustum
A leaning frustum still has volume with perpendicular height, but its lateral area is not π(R + r)l. That case is out of scope.
Converting liters on this page
V is already in the cube of your input unit. Use the Volume Converter for liters and gallons.
Frustum Calculator FAQs
What is the volume of a conical frustum?
V = (1/3)πh(R² + Rr + r²). For R = 6, r = 3, and h = 4, V = 84π ≈ 263.893782902 cubic units.
What is the surface area of a conical frustum?
Lateral area is π(R + r)l with l = √(h² + (R − r)²). Total surface adds both lids. For the 6-3-4 frustum those are 45π and 90π.
When is a frustum a cone or a cylinder?
Top radius 0 is a cone. Equal radii are a cylinder. This page recovers both limits.
What is the volume of a square frustum?
V = (1/3)h(A² + Aa + a²). For sides 6 and 4 and height 3, V = 76.
Does this treat an oblique frustum or convert liters?
No. This page is a right frustum. Convert liters on the Volume Converter.
Are my numbers sent to a server?
No. The calculation runs locally in your browser.
Related Calculators
Explore More Geometry Tools
Continue with cones, cylinders, and solid-geometry tools in the SolverCalc collection.
Reviewed by Prof. Dr. Khalil Mudassar (PhD) · Last updated: [mc_modified_date]
Sources
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Disclaimer
This calculator is intended for general educational and practical use. Review important results before relying on them in professional work.