Truncated Cone Calculator (Frustum)
Calculate the volume, surface area, lateral area, and slant height of any truncated cone (frustum) — the solid formed by cutting a cone parallel to its base. Complete step-by-step working with cone-subtraction verification for every calculation.
Enter R, r, h — plus optionally l or V for verification/alternative input:
Slide to see how the truncated cone transitions between a full cone (r=0) and a cylinder (r=R). Base: R=6, h=8 cm.
A truncated cone (frustum) = large full cone minus small full cone. Enter R, r, h to see the subtraction method.
What Is a Truncated Cone (Frustum)?
A truncated cone — also called a frustum — is the solid formed when a cone is cut by a plane parallel to its base, removing the top portion. The result is a three-dimensional shape with two circular bases of different sizes: a large base of radius R at the bottom and a small base of radius r at the top, connected by a slanted lateral surface.
The word frustum comes from Latin meaning "morsel" or "piece cut off." Truncated cones appear everywhere in everyday life: buckets, drinking cups, traffic cones, paper cups, wine glasses, lampshades, cooling towers, and even the frustum shape of many mountains. This truncated cone calculator handles all frustum calculations including volume of a truncated cone, surface area of a truncated cone, and slant height.
Key distinction: When r = 0, the truncated cone becomes a full cone (volume = (πh/3)R²). When r = R, it becomes a cylinder (volume = πR²h). Any value 0 < r < R gives a true frustum — a truncated cone.
Truncated Cone Formula — Volume, Surface Area & Slant Height
All truncated cone formulas depend on three variables: R (large base radius), r (small base radius), and h (perpendicular height). The slant height l is derived from these using the Pythagorean theorem.
| Property | Symbol | Formula | Notes |
|---|---|---|---|
| Volume | V | (πh/3)(R²+Rr+r²) | Three-term expression — never forget Rr |
| Slant height | l | √(h²+(R−r)²) | (R−r) is horizontal reach |
| Lateral area | LA | π(R+r)l | Curved surface only |
| Total surface area | SA | π(R²+r²+(R+r)l) | Lateral + both circles |
| Bottom area | A_bot | πR² | Large base circle |
| Top area | A_top | πr² | Small base circle |
Why the truncated cone volume formula has three terms
The expression (R² + Rr + r²) in the volume formula V = (πh/3)(R² + Rr + r²) comes from the algebraic expansion of V_large_cone − V_small_cone. By similar triangles, the full large cone has height H = hR/(R−r) and the removed small cone has height (H−h) = hr/(R−r). Subtracting gives exactly (πh/3)(R² + Rr + r²). The middle term Rr is a cross-term from the expansion — the most commonly forgotten term in the truncated cone formula.
How to Calculate Truncated Cone Volume — Step-by-Step
Follow four steps to find the volume of a truncated cone reliably:
- Identify R (large base radius), r (small base radius), and h (perpendicular height)
- Compute the three-term expression: R² + Rr + r²
- Multiply by πh/3
- Verify: if r=0, should give (πh/3)R²; if r=R, should give πR²h
Example 1 — R=6, r=3, h=4 cm (truncated cone)
- l = √(h²+(R−r)²) = √(16+9) = √25 = 5 cm (3-4-5 triple!)
- V = (π×4/3)(6²+6×3+3²) = (4π/3)(36+18+9) = (4π/3)(63) = 84π ≈ 263.894 cm³
- LA = π(6+3)×5 = 45π ≈ 141.372 cm²
- SA = π(36+9+45) = 90π ≈ 282.743 cm²
Example 2 — Bucket: R=15, r=10, h=20 cm
- l = √(400+25) = √425 ≈ 20.616 cm
- V = (π×20/3)(225+150+100) = (20π/3)(475) = 9500π/3 ≈ 9948.4 cm³ ≈ 9.95 L
- LA = π(15+10)×20.616 = 515.4π ≈ 1619.0 cm²
Example 3 — Very shallow frustum: R=10, r=9, h=2 cm
- l = √(4+1) = √5 ≈ 2.236 cm
- V = (π×2/3)(100+90+81) = (2π/3)(271) = 542π/3 ≈ 567.6 cm³
Surface Area of a Truncated Cone
The surface area of a truncated cone (total SA) consists of three parts: the large base circle (πR²), the small base circle (πr²), and the lateral curved surface (π(R+r)l). Combined: SA = π(R² + r² + (R+r)l).
The lateral area formula LA = π(R+r)l can be understood geometrically: when the lateral surface of a truncated cone is "unrolled," it forms a portion of an annulus (ring-shaped sector). The factor π(R+r) is related to the average circumference of the two bases — it represents the arithmetic mean of the two base perimeters multiplied by the slant height l.
Surface area breakdown: R=6, r=3, l=5 cm
- Bottom circle: πR² = 36π ≈ 113.10 cm²
- Top circle: πr² = 9π ≈ 28.27 cm²
- Lateral surface: π(R+r)l = π×9×5 = 45π ≈ 141.37 cm²
- Total SA = (36+9+45)π = 90π ≈ 282.74 cm²
Truncated Cone vs Full Cone — Special Cases
The truncated cone formula is a generalization that reduces to the full cone and cylinder formulas at the extremes:
| Case | Condition | Volume formula reduces to | Verification |
|---|---|---|---|
| Full cone | r = 0 | (πh/3)(R²+0+0) = (1/3)πR²h | ✓ Standard cone formula |
| Cylinder | r = R | (πh/3)(R²+R²+R²) = (πh/3)(3R²) = πR²h | ✓ Standard cylinder formula |
| Frustum | 0 < r < R | (πh/3)(R²+Rr+r²) | Interpolates between cone and cylinder |
Quick verification trick: Always check your truncated cone answer by testing the special cases. Set r=0 in your formula — does it give the cone formula? Set r=R — does it give the cylinder formula? If both checks pass, the formula is correct.
Real-World Examples of Truncated Cones
Plastic Bucket: R=15cm, r=12cm, h=25cm
- V = (π×25/3)(225+180+144) = (25π/3)(549) = 4575π ≈ 14,373 cm³ = 14.37 L
- LA = π(15+12)×√(625+9) = 27π×√634 ≈ 2136 cm²
Paper Drinking Cup: R=4cm, r=3cm, h=9cm
- l = √(81+1) = √82 ≈ 9.055 cm
- V = (π×9/3)(16+12+9) = 3π×37 = 111π ≈ 348.7 cm³ ≈ 349 mL
Traffic Cone: R=15cm, r=3cm, h=45cm
- l = √(2025+144) = √2169 ≈ 46.57 cm
- V = (π×45/3)(225+45+9) = 15π×279 = 4185π ≈ 13,148 cm³
Common Mistakes With Truncated Cone Calculations
Mistake 1 — Forgetting the Rr middle term
- ❌ Wrong: V = (πh/3)(R²+r²) — missing the Rr term
- ✅ Correct: V = (πh/3)(R²+Rr+r²) — all three terms are required
- For R=6,r=3,h=4: Wrong gives (4π/3)(45)=60π; Correct gives (4π/3)(63)=84π — 40% error!
Mistake 2 — Wrong slant height: using h instead of (R−r)
- ❌ Wrong: l = √(h²+R²) — treats R as the horizontal reach
- ✅ Correct: l = √(h²+(R−r)²) — (R−r) is the horizontal difference between radii
Mistake 3 — Swapping R and r (forgetting R ≥ r)
- Convention: R is always the LARGER base radius (bottom). r is the smaller (top). The formula is symmetric in some ways but keep R≥r for correct slant height: (R−r) must be ≥ 0.
Mistake 4 — Forgetting both base circles in total SA
- ❌ Wrong (lateral only): LA = π(R+r)l — gives curved surface only
- ✅ Correct (total): SA = πR² + πr² + π(R+r)l — must add both circle bases
Mistake 5 — Using cone formula V=(1/3)πR²h for a frustum
- The cone formula only applies when r=0. For any truncated cone with r>0, you must use V=(πh/3)(R²+Rr+r²). Using the cone formula underestimates the true frustum volume.