About the Frustum Calculator
A frustum is what is left when the top of a cone or pyramid is sliced off parallel to its base — the shape of a bucket, lampshade, flower pot, drinking cup or tapered column. This frustum calculator handles both a truncated circular cone (enter the bottom and top radii) and a truncated square pyramid (enter the bottom and top side lengths), plus the vertical height.
It returns the volume, the slant height, the lateral (side) surface area and the total surface area including both ends. Pick a length unit and the volume is also converted to litres and US gallons, which is handy for estimating how much a tapered bucket, planter or hopper holds.
The height must be the perpendicular height between the two parallel faces, not the length measured along the sloping side. If you only know the slant height, use the Pythagorean theorem to convert it first.
With the default inputs, the volume is 527.7876. Change any value above to recalculate instantly.
How to use the frustum calculator
- 1Choose a truncated cone or a truncated square pyramid.
- 2Enter the bottom and top radii (or side lengths).
- 3Enter the perpendicular height between the two faces.
- 4Pick your length unit to see the capacity in litres and gallons.
- 5Read the volume, slant height and surface areas.
Formula and method
The volume of any frustum is h/3 × (A₁ + √(A₁A₂) + A₂), where A₁ and A₂ are the areas of the two parallel faces. For a circular frustum that simplifies to πh(R² + Rr + r²)/3, and for a square frustum to h(a² + ab + b²)/3. It equals the big cone or pyramid minus the small one cut from the top.
The slant height is the hypotenuse of a right triangle whose legs are the height and the horizontal step between the edges (R − r, or (a − b)/2 for the square face). The lateral area of a cone frustum is π(R + r)s; a square frustum has four trapezoid faces of area (a + b)s/2 each. Total surface area adds both end faces.
- R, r
- Bottom and top radii (cone frustum)
- a, b
- Bottom and top side lengths (square frustum)
- h
- Perpendicular height
- s
- Slant height
- L
- Lateral surface area
Worked examples
Cone frustum R = 6, r = 3, h = 8
V = π × 8 × (36 + 18 + 9) / 3 = 168π ≈ 527.79 cubic units. The slant height is √(3² + 8²) ≈ 8.544, so the side area is π × 9 × 8.544 ≈ 241.58, and adding the two circles (36π + 9π) gives about 382.95.
Bucket 30 cm across the base, 24 cm at the top, 30 cm tall
With R = 15 cm, r = 12 cm and h = 30 cm the volume is π × 30 × (225 + 180 + 144)/3 ≈ 17,247 cm³. Since 1,000 cm³ is a litre, the bucket holds about 17.25 L, or roughly 4.56 US gallons.
Square frustum: base 10, top 4, height 6
V = 6 × (100 + 40 + 16)/3 = 312 cubic units. Each trapezoid face slopes over a horizontal step of 3, so the slant height is √(9 + 36) ≈ 6.708, the four sides total 2 × 14 × 6.708 ≈ 187.83, and adding 100 + 16 for the ends gives about 303.83.
Frequently asked questions
What is a frustum?+
A frustum is the part of a cone or pyramid that remains after cutting off the top with a plane parallel to the base. Buckets, lampshades, plant pots and many cups are frustum-shaped.
How do you find the volume of a truncated cone?+
Use V = πh(R² + Rr + r²)/3, where R and r are the bottom and top radii and h is the vertical height. For R = 6, r = 3 and h = 8 the volume is 168π ≈ 527.8 cubic units.
How do I get the height if I only know the slant height?+
The height, the slant height and the radius difference form a right triangle, so h = √(s² − (R − r)²). For a square frustum replace R − r with half the difference of the side lengths.
How many litres does my bucket hold?+
Measure the inside radii and inside height in centimetres, choose centimetres as the unit and read the litres output — 1,000 cubic centimetres equal one litre. Real buckets are rarely filled to the brim, so allow some headroom.
What happens if the top radius is zero?+
With r = 0 the frustum becomes a complete cone and the formula reduces to V = πR²h/3. Likewise a top side of 0 gives a full square pyramid with V = a²h/3.