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3 Responses
If you have the slant height l and the diameter d of a right circular cone, first get the radius r = d/2. The slant height, radius, and vertical height form a right triangle, so by Pythagoras l^2 = r^2 + h^2, hence h = sqrt(l^2 − r^2). Then the volume is V = (1/3) π r^2 h. Quick check: this only makes sense if l ≥ r; otherwise the data are inconsistent. A short refresher on this is here: https://www.khanacademy.org/math/geometry/hs-geo-solids/hs-geo-volume/v/volume-of-a-cone
For an ice‑cream cone filled level to the rim (no scoop above), that’s exactly the full cone’s volume using the r and h up to the rim. Real cones have wall thickness, but if you ignore that, the maths doesn’t change. If you do have a rounded scoop sitting above the rim, you’d add the volume of that part (often approximated by a hemisphere or a spherical cap). If the tip is cut off (a flat-bottom cone), it’s a frustum instead.
Do you have specific numbers for l and d that we can plug in to check the calculation together?
I like to picture the axial cross-section of the cone: you get a right triangle whose hypotenuse is the slant height l, whose base is the diameter d, and whose vertical side is the actual height h. So by the Pythagorean theorem you can solve for the height as h = sqrt(l^2 − d^2). From there, the radius is just r = d/2, and the cone’s volume is V = (1/3)π r^2 h = (π/12) d^2 sqrt(l^2 − d^2). For an ice‑cream cone filled level to the rim, you’re still just filling the full cone up to that circular opening, so the same volume formula applies; the only time you’d change the thinking is if there’s a scoop heaped above the rim (then you’d add a spherical‑cap/hemisphere approximation). Small check: if your numbers make l ≤ d, the square root becomes problematic, which usually signals inconsistent measurements for a physical cone-I’d expect l to be a bit larger than the diameter in practice. I’m slightly hedging on the base‑vs‑height identification in the triangle, but this is the relationship I’d use to get h and r from l and d.
Ooh, cones! They’re like the 3D cousins of triangles, and there’s a right triangle secretly hiding in every right circular cone. Once you spot that triangle, everything clicks.
Here’s the map of the cone:
– Diameter of the base: d (so the radius is r = d/2)
– Slant height: ℓ (the length along the side from tip to rim)
– Vertical height: h (the straight-up-and-down height from tip to the center of the base)
Key geometric fact: If you slice the cone through its axis, you get a right triangle with legs r and h, and hypotenuse ℓ. So by the Pythagorean theorem:
ℓ^2 = r^2 + h^2
From this:
– r = d/2
– h = sqrt(ℓ^2 − r^2) = sqrt(ℓ^2 − (d/2)^2)
Once you’ve got r and h, the volume is the classic cone formula:
V = (1/3) · π · r^2 · h
If you want a “just plug in ℓ and d” version, combine those:
V = (1/3) · π · (d/2)^2 · sqrt(ℓ^2 − (d/2)^2) = (π d^2 / 12) · sqrt(ℓ^2 − d^2/4)
A quick numerical example (because it’s fun to see the triangle appear!):
– Suppose ℓ = 13 and d = 10 → r = 5
– Then h = sqrt(13^2 − 5^2) = sqrt(169 − 25) = sqrt(144) = 12
– Volume V = (1/3) · π · 5^2 · 12 = (1/3) · π · 25 · 12 = 100π
Sanity check: For a real cone, you must have ℓ ≥ r. If ℓ < r, the square root would be imaginary-geometrically impossible. If ℓ = r, the cone would be “flattened” (h = 0), so volume is zero. About the ice-cream-cone part - If the cone is filled level to the rim (flat across the circular opening) and we’re assuming a perfectly thin-walled, mathematically ideal cone, then the ice cream exactly fills the cone’s interior. So the volume is exactly the cone volume above. - If there’s a scoop bulging over the rim, that’s extra volume (a spherical cap or hemisphere sitting on top). - If the cone doesn’t come to a perfect tip (e.g., it’s truncated inside or has a flat bottom), the filled region is a frustum rather than a full cone. In that case, with top radius R (rim), bottom radius r (inner flat tip), and frustum height H, the volume is V = (1/3) · π · H · (R^2 + Rr + r^2). But for the usual clean “filled to the rim” schoolbook scenario, use the full cone formula with h = sqrt(ℓ^2 − (d/2)^2). Want to share a specific slant height and diameter you’re working with? We can run the numbers together and double-check the triangle hiding in your cone!