Hong Life CalculatorsUnderstanding the cone and frustum calculator — concept and principle
A cone has a circular base that tapers to a single point; a frustum is a cone with its tip cut off by a plane parallel to the base, like a bucket, a lampshade or a flowerpot. Enter the bottom radius R, the top radius r (0 for a full cone) and the perpendicular height h, and the calculator returns the volume, the slant height, the lateral (sloping side) area and the total surface area. It is used for hoppers, funnels, sand piles and sheet-metal patterns.
How it is calculated
- Volume: V = πh(R² + Rr + r²) ÷ 3; with r = 0 this is πR²h ÷ 3
- Slant height: l = √(h² + (R − r)²)
- Lateral area: π(R + r)l; with r = 0 this is πRl
- Total surface: π(R + r)l + π(R² + r²)
All lengths share one unit; areas are in that unit squared and volume in that unit cubed.
The principle
A cone holds exactly one third of the cylinder with the same base and height. Euclid proved this by exhaustion, and in calculus it follows because each slice’s area grows with the square of the distance from the tip, and the integral of x² is x³/3. A frustum is the large cone minus the small cone removed from the top; working out that difference with similar triangles gives h(R² + Rr + r²)/3 times π. The slant height is the hypotenuse of a right triangle with legs h and R − r. Unrolled, a cone’s side is a sector of radius l and arc 2πR, with area ½ × l × 2πR = πRl.
Worked example
- Inputs: R = 4, r = 0 (full cone), h = 9.
- Volume: π × 9 × (16 + 0 + 0) ÷ 3 = 48π ≈ 150.7964474.
- Slant height: √(81 + 16) = √97 ≈ 9.848857802.
- Lateral area: π × 4 × 9.848858 ≈ 123.7643973.
- Total surface: 123.7644 + π × 16 = 123.7644 + 50.2655 ≈ 174.0298797.
| Bottom radius R | Volume |
|---|---|
| 1 | 9.424777961 |
| 2 | 37.69911184 |
| 3 | 84.82300165 |
| 4 | 150.7964474 |
| 5 | 235.619449 |
| 6 | 339.2920066 |
| 8 | 603.1857895 |
How to read the result
The cylinder with the same base and height would hold 144π, three times the cone’s 48π, which is a quick check. The lateral area is always larger than the base area, because the slant height is longer than the radius; for a frustum, the total surface counts both the top and bottom circles, so remove one if the container is open.
Common mistakes
- Using the slant height instead of the perpendicular height in the volume.
- Averaging the two radii for a frustum, π((R + r)/2)²h, which underestimates the volume.
- Including a top circle for an open bucket or funnel.
Frequently asked questions
How do I find the volume of a pile of sand?
Treat it as a cone: measure the base circumference C and height h, then use R = C ÷ 2π and V = πR²h ÷ 3.
How do I cut a paper cone?
Cut a sector of radius l with central angle 360° × R ÷ l. For R = 4 and l ≈ 9.85, that is about 146°.
Does the frustum formula work when r is larger than R?
Yes. The formula is symmetric in R and r, so an upside-down frustum has the same volume and surface.
Sources: Euclid, Elements, Book XII, Proposition 10 (a cone is one third of the cylinder with the same base and height) · Moscow Mathematical Papyrus, Problem 14 (volume of a frustum of a square pyramid)