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Conical Frustum — Definition, Formula & Examples

A conical frustum is the solid shape that remains when you slice the top off a cone with a cut parallel to its base. It looks like a truncated cone — a bucket or lampshade shape — with two circular faces of different sizes connected by a slanted surface.

A conical frustum (or truncated cone) is the portion of a right circular cone contained between two parallel planes that intersect the cone. It is bounded by two circular bases of radii RR (larger) and rr (smaller) and a lateral surface. Its height hh is the perpendicular distance between the two bases, and its slant height ll is the distance along the lateral surface from one base circle to the other.

Key Formula

V = \frac{\pi h}{3}\left(R^2 + Rr + r^2\right)$$ $$A_{\text{lateral}} = \pi(R + r)\,l$$ $$A_{\text{total}} = \pi(R + r)\,l + \pi R^2 + \pi r^2$$ $$l = \sqrt{h^2 + (R - r)^2}
Where:
  • RR = Radius of the larger (bottom) base
  • rr = Radius of the smaller (top) base
  • hh = Perpendicular height between the two bases
  • ll = Slant height along the lateral surface
  • VV = Volume of the frustum
  • AlateralA_{\text{lateral}} = Lateral (side) surface area
  • AtotalA_{\text{total}} = Total surface area including both bases

How It Works

To find the volume of a conical frustum, you use a formula that averages the areas of both circular bases along with their geometric mean. The surface area calculation has two parts: the areas of the two circular ends and the lateral (side) surface area, which depends on the slant height. You can find the slant height from the height and the difference in radii using the Pythagorean theorem: l=h2+(R−r)2l = \sqrt{h^2 + (R - r)^2}. These formulas come from subtracting a smaller cone from the original larger cone, but they save you from having to reconstruct the missing tip.

Worked Example

Problem: A conical frustum has a bottom radius of 6 cm, a top radius of 3 cm, and a height of 8 cm. Find its volume and total surface area.
Step 1: Find the slant height: Use the Pythagorean relationship with the height and the difference in radii.
l=82+(6−3)2=64+9=73≈8.544 cml = \sqrt{8^2 + (6 - 3)^2} = \sqrt{64 + 9} = \sqrt{73} \approx 8.544 \text{ cm}
Step 2: Calculate the volume: Plug the radii and height into the volume formula.
V=π(8)3(62+6⋅3+32)=8π3(36+18+9)=8π3(63)=168π≈527.8 cm3V = \frac{\pi(8)}{3}(6^2 + 6 \cdot 3 + 3^2) = \frac{8\pi}{3}(36 + 18 + 9) = \frac{8\pi}{3}(63) = 168\pi \approx 527.8 \text{ cm}^3
Step 3: Calculate the lateral surface area: Use the sum of the radii and the slant height.
Alateral=π(6+3)73=9π73≈241.5 cm2A_{\text{lateral}} = \pi(6 + 3)\sqrt{73} = 9\pi\sqrt{73} \approx 241.5 \text{ cm}^2
Step 4: Calculate the total surface area: Add the areas of both circular bases to the lateral area.
Atotal=9π73+π(6)2+π(3)2=9π73+36π+9π≈241.5+113.1+28.3≈382.9 cm2A_{\text{total}} = 9\pi\sqrt{73} + \pi(6)^2 + \pi(3)^2 = 9\pi\sqrt{73} + 36\pi + 9\pi \approx 241.5 + 113.1 + 28.3 \approx 382.9 \text{ cm}^2
Answer: The volume is 168π≈527.8168\pi \approx 527.8 cm³ and the total surface area is approximately 382.9 cm².

Another Example

Problem: A frustum-shaped bucket has a top diameter of 40 cm, a bottom diameter of 30 cm, and a depth of 24 cm. How many liters of water can it hold?
Step 1: Identify the radii: Convert diameters to radii: R = 20 cm (top), r = 15 cm (bottom), h = 24 cm.
Step 2: Apply the volume formula: Substitute into the frustum volume formula.
V=π(24)3(202+20⋅15+152)=8π(400+300+225)=8π(925)=7400π≈23,248 cm3V = \frac{\pi(24)}{3}(20^2 + 20 \cdot 15 + 15^2) = 8\pi(400 + 300 + 225) = 8\pi(925) = 7400\pi \approx 23{,}248 \text{ cm}^3
Step 3: Convert to liters: Since 1 liter = 1,000 cm³, divide by 1,000.
23,2481,000≈23.25 liters\frac{23{,}248}{1{,}000} \approx 23.25 \text{ liters}
Answer: The bucket holds approximately 23.25 liters of water.

Why It Matters

Conical frustums appear constantly in engineering and everyday life — buckets, drinking cups, lampshades, and dam structures all have this shape. In high school geometry and AP Calculus, you need the frustum formula to solve solid-of-revolution and composite-volume problems. Civil engineers use frustum calculations when estimating earthwork volumes for road and dam construction.

Common Mistakes

Mistake: Using the perpendicular height instead of the slant height in the lateral surface area formula.
Correction: The lateral area formula requires the slant height ll, not the vertical height hh. Calculate l=h2+(R−r)2l = \sqrt{h^2 + (R - r)^2} first, then use it in π(R+r)l\pi(R + r)l.
Mistake: Averaging the two base areas and multiplying by height for volume, as if it were a prism.
Correction: The correct formula is V=πh3(R2+Rr+r2)V = \frac{\pi h}{3}(R^2 + Rr + r^2), which includes the cross-term RrRr. Simply averaging πR2\pi R^2 and πr2\pi r^2 then multiplying by hh overestimates the volume.

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