Calculate right circular cone volume, total surface area, lateral (conical) surface area, slant height, and base area with step-by-step steps.
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A right circular cone is a three-dimensional geometric shape that tapers smoothly from a flat circular base to a point called the apex or vertex, positioned directly above the center of the base.
l = \sqrt{r^2 + h^2}, \quad V = \frac{1}{3}\pi r^2 h, \quad A_{\text{lat}} = \pi r l, \quad A_{\text{total}} = \pi r(r + l)Slant height l is the hypotenuse of the right triangle formed by radius and height. Volume is exactly one-third the volume of a cylinder with identical radius and height.
Scenario: Find volume and slant height of a cone with radius r = 3 cm and height h = 4 cm
Slant height: l = √(3² + 4²) = √(9 + 16) = √25 = 5 cm
Volume: V = ⅓ × π × 3² × 4 = ⅓ × π × 9 × 4 = 12π ≈ 37.6991 cm³
Lateral area: A_lat = π × 3 × 5 = 15π ≈ 47.1239 cm²
Total Surface Area: A_total = π × 3 × (3 + 5) = 24π ≈ 75.3982 cm²
Vertical height (h) is the perpendicular distance from the center of the base to the apex. Slant height (l) is the distance from the base perimeter to the apex along the curved surface.
Calculus and Archimedes’ exhaustion method prove that any pyramid or cone occupies exactly one-third the volume of a prism or cylinder with matching base and height.
A cone frustum is the solid remaining after the top apex section is sliced off by a plane parallel to the base.
Use the Pythagorean theorem on the cross-sectional triangle: l = √(r² + h²).
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