3D Mensuration • Topic 4 of 4

Frustum

A frustum is what remains when a cone is sliced by a plane parallel to its base and the small top cone is removed — think of a bucket, a lampshade or a drinking glass. With top radius r, bottom radius R and vertical height h, its volume is (1/3)πh(R² + Rr + r²). The slant height runs along the sloping side: l = √(h² + (R − r)²), using the difference of radii, not their sum. The curved (lateral) surface area is πl(R + r), and the total surface area adds the two circular ends: πl(R + r) + πR² + πr². For an open bucket you drop the top face πr². The reliable CAT method is to treat the frustum as a big cone minus a small cone: because the two cones are similar, r/R = h_small/(h_small + h), which lets you recover any missing dimension. Always pair the slant difference (R − r) with the slant formula and the radius sum (R + r) with the area formula — mixing them up is the standard trap.

✅ Solved examples

1. A frustum has radii 8 cm and 4 cm and height 6 cm. Find its volume (π).
V = (1/3)π×6×(8² + 8×4 + 4²) = 2π(64 + 32 + 16) = 2π×112 = 224π cm³.
2. A frustum has R = 10 cm, r = 4 cm and height 8 cm. Find its slant height.
l = √(h² + (R−r)²) = √(8² + 6²) = √(64+36) = √100 = 10 cm.
3. For the frustum above (R=10, r=4, l=10), find the curved surface area (π).
CSA = πl(R + r) = π×10×(10 + 4) = 140π cm².
4. A bucket (open top) has radii 14 cm and 7 cm and slant height 25 cm. Find its total external surface area (π = 22/7).
Open top ⇒ CSA + bottom face. CSA = πl(R+r) = (22/7)×25×21 = 1650 cm². Bottom = πr² = (22/7)×49 = 154 cm². Total = 1650 + 154 = 1804 cm².

✏️ Practice — try these, take hints as needed

1. A frustum has radii 6 cm and 3 cm and height 4 cm. Find its volume (π).
V = (1/3)πh(R²+Rr+r²).
(1/3)π×4×(36+18+9).
(4/3)π×63.
84π cm³
2. A frustum has R = 12 cm, r = 5 cm, height 24 cm. Find its slant height.
l = √(h²+(R−r)²).
√(24²+7²).
√(576+49)=√625.
25 cm
3. A frustum has R = 9 cm, r = 6 cm and slant height 5 cm. Find its curved surface area (π).
CSA = πl(R+r).
π×5×15.
75π.
75π cm²
4. A frustum-shaped glass has radii 4 cm (top) and 2 cm (bottom) and height 9 cm. Find its capacity (π = 22/7).
V = (1/3)πh(R²+Rr+r²).
(1/3)(22/7)(9)(16+8+4).
(1/3)(22/7)(9)(28).
264 cm³
5. The radii of a frustum are 7 cm and 3 cm with slant height 6 cm. Find its total surface area including both ends (π = 22/7).
TSA = πl(R+r) + πR² + πr².
πl(R+r) = (22/7)(6)(10).
Add (22/7)(49) and (22/7)(9).
370.86 cm² (≈ 2596/7 cm²)

📝 Topic test — 8 questions

Auto-graded with full solutions; saved to your dashboard. Use the calculator and formula sheet (top-right) any time.

Loading questions…
← Sphere & Hemisphere Take the chapter test →