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Conical Frustum

Calculate volume, lateral surface area, and total surface area of a conical frustum (truncated cone). Step-by-step solutions with visual diagrams.

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Why: Understanding conical frustum helps you make better, data-driven decisions.

How: Enter Large Radius (R), Small Radius (r) to calculate results.

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TRUNCATED CONE

Conical Frustum Calculator — Volume, LSA & TSA

Enter large radius (R), small radius (r), and height or slant height. Buckets, lamp shades, and cups are all frustums.

Enter Dimensions

frustum_calc.out
CALCULATED
$ calc_frustum --R=5 --r=3 --h=4
Volume
205.2507
cubic units
LSA
112.397
sq units
TSA
219.2112
sq units
Slant Height
4.4721
units
Top Area
28.2743
πr²
Bottom Area
78.5398
πR²
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Conical Frustum Calculation
R = 5, r = 3, h = 4
205.2507cubic units
LSA = 112.397TSA = 219.2112s = 4.4721
numbervibe.com/calculators/mathematics/geometry/conical-frustum-calculator

Area Breakdown (Bar)

Area Proportions (Doughnut)

Get AI-Powered Explanation

Understand the formulas, step-by-step derivation, and real-world applications of your frustum calculation.

Step-by-Step Breakdown

Given large radius (R):

55

Given small radius (r):

33

Given height (h):

44

1. Slant height:

s=h2+(R−r)2=42+(5−3)2≈4.4721s = \sqrt{h^2 + (R-r)^2} = \sqrt{4^2 + (5-3)^2} \approx 4.4721

2. Volume:

V=πh3(R2+Rr+r2)=π×43(52+5×3+32)≈205.2507V = \frac{\pi h}{3}(R^2 + Rr + r^2) = \frac{\pi \times 4}{3}(5^2 + 5\times3 + 3^2) \approx 205.2507

3. Lateral surface area:

LSA=π(R+r)s=π×(5+3)×4.4721≈112.3970LSA = \pi(R+r)s = \pi \times (5+3) \times 4.4721 \approx 112.3970

4. Total surface area:

TSA=π(R2+r2+(R+r)s)≈219.2112TSA = \pi(R^2 + r^2 + (R+r)s) \approx 219.2112

For educational and informational purposes only. Verify with a qualified professional.

📋 Key Takeaways

  • • A frustum is a cone with its top cut off by a plane parallel to the base
  • • Volume: V=fracpih3(R2+Rr+r2)V = \\frac{\\pi h}{3}(R^2 + Rr + r^2)
  • • Slant height: s=sqrth2+(R−r)2s = \\sqrt{h^2 + (R-r)^2}
  • • Lateral area: LSA=pi(R+r)sLSA = \\pi(R+r)s; Total area includes top + bottom + lateral

💡 Did You Know?

🏛️The Egyptian pyramids are truncated pyramids (frustums of pyramids). The Great Pyramid's top was originally capped, making it a frustum when viewed as a truncated solid.
🏭Cooling towers at power plants are hyperboloids, but many industrial hoppers and silos use conical frustum shapes for efficient material flow.
🌋Volcano calderas often form frustum-like depressions when the magma chamber collapses—the cross-section resembles a truncated cone.
📐"Frustum" comes from Latin frustum meaning "piece" or "morsel"—literally a piece cut off from a cone.
🪣Buckets, lampshades, and drinking cups are all frustums. The taper makes them easier to stack and pour.
🔬In 3D printing and CAD, frustum shapes are common for supports, funnels, and tapered connectors.

📖 How It Works

A conical frustum is the portion of a cone between two parallel planes. It has two circular bases: large radius R and small radius r.

From Dimensions

Enter large radius R, small radius r, and height h (or slant height s). If using slant height, derive h from h=sqrts2−(R−r)2h = \\sqrt{s^2 - (R-r)^2}.

Volume Formula

Volume comes from the difference of two cone volumes: V=fracpih3(R2+Rr+r2)V = \\frac{\\pi h}{3}(R^2 + Rr + r^2) — derived by subtracting the removed top cone from the full cone.

Surface Area

Lateral area: LSA=pi(R+r)sLSA = \\pi(R+r)s. Total area adds top (πr²) and bottom (πR²): TSA=pi(R2+r2+(R+r)s)TSA = \\pi(R^2 + r^2 + (R+r)s).

🎯 Expert Tips

Height vs Slant Height

Height (h) is perpendicular to the bases. Slant height (s) runs along the lateral surface. Always use h in the volume formula; use s for lateral area.

When r → 0

If the small radius approaches zero, the frustum becomes a full cone. Volume reduces to (1/3)πR²h.

Material Estimation

For manufacturing (e.g., sheet metal for a bucket), use TSA for total material. Add a small allowance for seams and overlap.

Unit Consistency

Keep R, r, and h in the same units. Volume will be in cubic units; area in square units. Convert before calculating if needed.

📊 Frustum vs Cone vs Cylinder vs Pyramid

ShapeVolume FormulaNotes
Frustum(πh/3)(R²+Rr+r²)Two parallel circular bases
Cone(1/3)πR²hFrustum with r=0
CylinderπR²hFrustum with R=r
Pyramid Frustum(h/3)(B₁+√(B₁B₂)+B₂)Square/rectangular bases

❓ Frequently Asked Questions

What if r = 0?

Then the frustum becomes a full cone. Volume = (1/3)πR²h, and the lateral area formula still applies with r=0.

What if R = r?

When both radii are equal, you have a cylinder. Volume = πR²h, and the lateral area is 2πRh (the curved part only).

Can I use slant height instead of height?

Yes. Select "Slant Height" mode and enter s. The calculator derives h from h = √(s² - (R-r)²).

Why must R > r?

By convention, R is the large (bottom) radius and r is the small (top) radius. If they were equal, it would be a cylinder.

How do I find the volume of a bucket?

Measure the top and bottom diameters, divide by 2 for radii, measure the height, and use this calculator.

What units should I use?

Use any consistent units (cm, m, inches). Volume will be in cubic units; area in square units.

Is a frustum the same as a truncated cone?

Yes. "Conical frustum" and "truncated cone" are synonymous—both mean a cone with its top cut off by a plane parallel to the base.

📈 Frustum by the Numbers

3
Key Dimensions (R, r, h)
π
In Every Formula
4
Main Outputs (V, LSA, TSA, s)
∞
Real-World Uses

Disclaimer: This calculator provides mathematically precise results based on standard geometric formulas. Results are limited by floating-point precision (~15 significant digits). For critical engineering or manufacturing, verify with domain-specific tools. Not a substitute for professional analysis.

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