Last updated: 2026-05-18

Definition

A right circular cone is a solid with a circular base and a single apex that sits directly above the center of that base. Its three defining measurements are the base radius, the perpendicular height from base to apex, and the slant height along the lateral surface. From the radius and height alone, four properties follow: the volume, the slant height, the lateral (curved) surface area, and the total surface area.

The formulas

For a cone with base radius r and perpendicular height h:

The slant height

The slant height l is the straight-line distance from the apex to any point on the rim of the base, measured along the surface. It connects the apex and the base edge in a right triangle alongside r and h — hence $l = \sqrt{r^2 + h^2}$ by the Pythagorean Theorem Calculator.

The slant height is the dimension that governs surface area: the lateral surface unrolls into a flat sector with radius l and arc length 2πr, which is why it appears in both the lateral and total area formulas.

The one-third factor

The factor 1/3 in V = (1/3)πr²h is one of the oldest results in solid geometry. Euclid proved it in the Elements (Book XII, Proposition 10): a cylinder with the same base and height can be exactly filled by three identical cones.

The relationship is easy to demonstrate physically — a conical container of water poured into a matching cylinder fills it after exactly three pours. The underlying reason is the way the cross-sectional area shrinks toward the apex: slicing the cone into thin circular disks stacked from base to apex and summing their volumes yields the same one-third ratio.

Worked example

A party hat is roughly cone-shaped with a base radius of 8 cm and a height of 24 cm.

Slant height: $l = \sqrt{8^2 + 24^2} = \sqrt{64 + 576} = \sqrt{640} \approx 25.3$ cm

Volume: $V = \tfrac{1}{3} \times \pi \times 8^2 \times 24 = \tfrac{1}{3} \times \pi \times 1536 \approx 1608.5$ cm³ (≈ 1.6 liters)

Lateral surface area (the printed part of the hat): A_lat = π × 8 × 25.3 ≈ 635.9 cm² (about the size of an A4 sheet of paper)

Total surface area (including the base): A_tot = π × 8 × (8 + 25.3) ≈ 836.9 cm²

A typical ice cream cone has r ≈ 2.5 cm and h ≈ 12 cm, giving a volume of about 78.5 cm³ — roughly 80 mL, about the same as a small liqueur glass.

When only right circular cones apply

These formulas are for right circular cones — the axis is perpendicular to the base. Oblique cones (leaning to one side) have the same volume formula (1/3)πr²h when h is the true perpendicular height, but their lateral area formula differs and is harder to compute. In practice, the vast majority of real-world cones (ice cream cones, funnels, party hats, traffic cones) are right circular, so this calculator covers almost every everyday case.