Right cone: definitions and formulas

A right cone is a solid of revolution from rotating a right triangle around one leg. Characterized by base radius r, height h, slant height l. Apex perpendicular to base center.

• Volume: V = (1/3) pi r^2 h.
• Slant height: l = sqrt(r^2 + h^2).
• Lateral surface: S_lat = pi r l.
• Base surface: S_base = pi r^2.
• Total surface: S_tot = pi r (l + r).

Numerical example

r = 3, h = 4: l = 5, V = 12 pi ~ 37.70, S_lat = 15 pi, S_tot = 24 pi ~ 75.40. Apex angle 2 atan(0.75) = 73.74 deg.

Volume derivation

Cross-sections at height y from apex have radius r(y) = (r/h) y. V = integral pi (r y/h)^2 dy from 0 to h = (1/3) pi r^2 h. Cavalieri principle: cone has 1/3 of circumscribed cylinder volume. Eudoxus ~370 BC, Archimedes ~250 BC.

Cone development (unrolling)

Cut along slant: circular sector of radius l, arc length 2 pi r. Sector angle theta = 360 r/l deg. Plus base circle of radius r.

Frustum (truncated cone)

Truncated by parallel plane: V = (pi h/3)(r^2 + R^2 + rR). l_f = sqrt((r-R)^2 + h^2). S_lat = pi (r+R) l_f. Special cases: R = 0 cone, R = r cylinder.

Right vs oblique cone

Right: axis perpendicular to base. Oblique: not. SAME volume (Cavalieri), DIFFERENT lateral surface (no closed form). Calculator assumes right cone.

Apex angle and base angle

• Apex angle: 2 atan(r/h).
• Base angle: atan(h/r) (between base and slant).
• Narrow cone (h >> r): small apex.
• Wide cone (r >> h): large apex.

Inverse formulas

• From V and r: h = 3V/(pi r^2).
• From V and h: r = sqrt(3V/(pi h)).
• From S_lat and r: l = S_lat/(pi r), then h = sqrt(l^2 - r^2).
• Mass: m = density * V.
• Centroid: h/4 from base.
• Moment of inertia (axis): I = (3/10) m r^2.

Generalizations

• Elliptical cone: V = (pi a b h)/3.
• Right pyramid: V = (1/3) base * height.
• n-dimensional cone: V_n = h * V_{n-1}/n.
• Double cone: 2 cones at base.
• Conic sections: circle, ellipse, parabola, hyperbola from plane cuts.

Applications

• Civil engineering: conical roofs, chimneys.
• Architecture: domes, spires.
• Hydraulics: funnels, nozzles.
• Astronomy: solid angle 2 pi (1-cos alpha).
• Aerospace: ogives, reentry capsules.
• Geology: volcanoes, dunes.
• Industrial: hoppers.
• Food: ice cream cones, funnels.

Conic sections

Plane cuts give: circle (perpendicular to axis), ellipse (oblique), parabola (parallel to slant), hyperbola (parallel to axis through both nappes). Apollonius of Perga ~262-190 BC studied 8 books on conics. Kepler applied to planets, Newton to gravity.

History

Eudoxus ~370 BC: first proof V_cone = V_cylinder/3 by exhaustion. Archimedes ~250 BC: formalized cones and frustums. Apollonius: conic sections treatise. Cavalieri 1635: indivisibles modern proof. Calculus (Newton, Leibniz 17th c.): rigorous integral derivation. Pappus theorems ~300 AD: rotation surfaces.

Calculator modes

Five modes: all (V + surfaces + slant + apex angle); volume; total/lateral surface; slant l; frustum with two radii. Inputs: r, h, R. Output: all notable cone quantities.

Why the cone is exactly one third of the cylinder

The most memorable fact about a cone is that its volume is precisely a third of the cylinder that encloses it, sharing the same base radius and height. This is not a coincidence but a deep geometric truth, and it belongs to a family: a pyramid is likewise a third of the prism around it. The Greeks proved it long before calculus. Eudoxus, around 370 BC, used the method of exhaustion, slicing the solid into ever thinner layers and bounding the total between inscribed and circumscribed stacks until the gap vanished, and Archimedes refined the argument a century later.

The modern derivation makes the factor of a third transparent. Imagine the cone standing point-up and cut it into thin horizontal disks; at height y from the apex the radius grows in proportion to y, so the disk area grows as y squared. Summing these areas, which calculus does by integrating y squared from zero to the height, produces exactly one third of the height cubed times the constant, and hence one third of the full cylinder. Cavalieri reached the same result in 1635 with his principle of indivisibles, showing that two solids with matching cross-sectional areas at every level must have equal volume, a beautiful bridge between the ancient and the calculus proofs.

The unrolled cone and its applications

Cut a cone along a straight line from apex to base and flatten it, and the curved lateral surface opens into a flat circular sector whose radius is the slant height. The arc of that sector equals the circumference of the base, so the sector spans an angle of two pi times the ratio of radius to slant height. This unrolling is the practical heart of sheet-metal and paper work: to build a real cone you cut exactly this sector from a flat sheet and join its straight edges, and the same construction produces funnels, party hats, filter papers and the conical roofs of towers.

Cones and their truncated cousins, the frustums, are everywhere in engineering because they channel and concentrate. A loudspeaker cone couples a small driver to a large volume of air, a rocket's nose cone parts the airflow at high speed, and machine tool tapers and pencil points exploit the way a cone comes smoothly to a single point. The frustum, a cone with its tip sliced off, models buckets, lampshades, traffic cones and cooling towers, and the calculator's frustum mode returns its volume and surface directly from the two radii and the height, saving the work of subtracting one cone from another.

Updated July 2026