A frustum is the solid formed when a cone is cut by a plane parallel to its base, removing the top. The result has a large circular base, a smaller circular top, and a slanted lateral surface connecting them. Everyday examples include plant pots, drinking cups, traffic cones, and bucket shapes. The word comes from the Latin for fragment or piece.

What is the volume formula for a frustum?

The volume of a frustum with large base radius R, small top radius r, and perpendicular height h is pi times h divided by 3 times (R squared plus Rr plus r squared). This formula is derived by subtracting the volume of the small cone removed from the full cone, and simplifies to the cone formula when r equals zero.

What is the lateral surface area of a frustum?

The lateral surface area (LSA) is the curved slanted surface of the frustum, not including the two circular faces. LSA equals pi times (R plus r) times the slant height l, where l equals the square root of (h squared plus (R minus r) squared). The slant height is the straight-line distance along the sloped surface from the edge of the base to the edge of the top.

Frustum Calculator

A frustum (sometimes called a truncated cone) is the solid you get when you slice the top off a cone with a cut parallel to the base. The resulting shape has a large circular base, a smaller circular top, and a sloped curved surface connecting them. Frustum shapes are extremely common in everyday life: a standard bucket is a frustum, a traffic cone with a flat top is a frustum, most plant pots are frustums, and many drinking cups taper from a wider rim to a narrower base in the frustum form. Engineers use frustum calculations to size hoppers and funnels, determine the amount of material in a conical stockpile with a levelled top, estimate concrete volumes for sloped walls, and design tapered pipe reducers. This calculator takes the large base radius R, the small top radius r, and the perpendicular height h and computes the volume, the lateral surface area (the sloped outer face), the total surface area (including both circular faces), the slant height (the diagonal distance along the slope), and the apex height (how tall the original full cone would have been). Set r = 0 and you get a full cone. The default uses R = 5, r = 3, h = 4, giving vol = 205.25, LSA = 112.40, TSA = 219.21, slant = 4.47.

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Large base radius (R)

units

Small top radius (r)

units

Height (h)

units

205.25

volume (cubic units)

Lateral SA (LSA)112.40

Total SA (TSA)219.21

Slant height4.47

Apex height10.00

Slant height = sqrt(h² + (R-r)²). Set r = 0 for a full cone. R must be greater than or equal to r.

How it works

For large radius R, small radius r, and height h: slant height l = sqrt(h^2 + (R - r)^2). Volume = pi * h / 3 * (R^2 + R*r + r^2). Lateral surface area = pi * (R + r) * l. Base area = pi * R^2. Top area = pi * r^2. Total surface area = LSA + base area + top area. The apex height (full cone height) = h * R / (R - r) when R is not equal to r.

Worked example

For R = 5, r = 3, h = 4: slant = sqrt(16 + 4) = sqrt(20) = 4.47. Volume = pi * 4/3 * (25 + 15 + 9) = pi * 4/3 * 49 = 205.25. LSA = pi * (5 + 3) * 4.47 = pi * 8 * 4.47 = 112.40. Base area = pi * 25 = 78.54. Top area = pi * 9 = 28.27. TSA = 112.40 + 78.54 + 28.27 = 219.21. Apex height = 4 * 5 / (5 - 3) = 20 / 2 = 10.00. These match the defaults pre-filled above.

Related calculators

Hemisphere Calculator: volume and surface area of a hemisphere.
Torus Volume Calculator: volume and surface area of a donut shape.
Annulus Calculator: area of a flat ring (the frustum base cross-section).
Pentagon Calculator: area and dimensions of a regular pentagon.

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