Frustum of a cone calculator
This frustum of a cone calculator needs the bottom radius, the top radius and the height. With r1 = 5, r2 = 3 and h = 6 the volume is 307.88, the slant height 6.32 and the surface area 265.76, with the steps.
Example: Bottom radius r1 5 · Top radius r2 3 · Height h 6 → Result: Volume V = 307.8761 · Surface area S = 265.7676. Source: Wikipedia – Kegelstumpf. Updated: .
How it is calculated
A frustum of a cone is what remains when you cut the tip off a right cone parallel to its base. It has two parallel circular faces with radii r1 (bottom) and r2 (top) at distance h. Buckets, flower pots and lampshades are typical examples.
The formulas
- Volume: V = π · h ÷ 3 · (r1² + r1 · r2 + r2²)
- Slant height: s = √((r1 − r2)² + h²)
- Lateral surface area: L = π · (r1 + r2) · s
- Surface area: S = π · r1² + π · r2² + L
- Apex height above the top face: h · r2 ÷ (r1 − r2)
How to use it
Enter both radii and the height. Example: r1 = 5, r2 = 3 and h = 6 give V = π · 2 · 49 = 307.88, s = 6.32, L = 158.95 and S = 265.76. Set r2 = 0 for a full cone; with r1 = r2 you get a cylinder. The unit is up to you; areas are in its square, the volume in its cube.
Frequently asked questions
How do you calculate the volume of a frustum of a cone?
V = π · h ÷ 3 · (r1² + r1 · r2 + r2²). For r1 = 5, r2 = 3 and h = 6 that is 307.88.
How do you calculate its lateral area?
L = π · (r1 + r2) · s with the slant height s = √((r1 − r2)² + h²). Here that gives 158.95.
How do you calculate the total surface area?
Lateral area plus bottom and top: S = L + π · r1² + π · r2². For the example values that is 265.76.
What is the difference between a cone and a frustum?
A frustum lacks the cut-off tip; instead of a point there is a second circular face on top. With r2 = 0 it becomes a cone.
How tall is the imaginary apex?
Above the top face the apex lies at h · r2 ÷ (r1 − r2). For r1 = 5, r2 = 3 and h = 6 that is 9 units.
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