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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units
units
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.
