Cone Calculator
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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What Is a Cone Calculator?
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.
How to Use This Calculator
- Enter base radius (r) and vertical height (h).
- View the calculated Volume, Slant Height (l), Total Surface Area, and Lateral Area.
- Review the step-by-step derivation using Pythagorean theorem and circular sectors.
Cone Volume, Slant Height & Surface Area Formulas
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.
Worked Example
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
Tips & Key Notes
- A cone has exactly one-third the volume of a cylinder with the same radius and height.
- Slant height (l) is always strictly greater than vertical height (h).
- When the lateral surface of a cone is flattened out, it forms a sector of a larger circle of radius l.
Frequently Asked Questions
What is the difference between height and slant height?
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.
Why is the volume of a cone 1/3 of a cylinder?
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.
What is a truncated cone (frustum)?
A cone frustum is the solid remaining after the top apex section is sliced off by a plane parallel to the base.
How do you find the slant height of a cone?
Use the Pythagorean theorem on the cross-sectional triangle: l = √(r² + h²).
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