Cone Calculator
Calculate the slant height, surface area, lateral surface area, and volume of a right circular cone from its radius and height.
About this Calculator
Calculate the slant height, surface area, lateral surface area, and volume of a right circular cone from its radius and height.
Formula & Calculations
Formula
Slant Height: s = √(r² + h²); Volume = ⅓πr²h; Lateral Area = πrs; Total Surface Area = πr² + πrs = πr(r+s)Where:
- r=Radius of the cone's base
- h=Height of the cone (perpendicular from base to apex)
- s=Slant height (distance from any point on the base edge to the apex)
- V=Volume of the cone
- LA=Lateral (curved) surface area
- SA=Total surface area (base + lateral)
Assumptions
- The cone is a right circular cone with the apex directly above the center of the base.
- π is approximated as 3.1415926536 (Math.PI).
- All measurements use consistent units.
Calculation Examples
Example 1
s = √(16 + 81) ≈ 9.8489. V = ⅓π × 16 × 9 ≈ 150.7964. LA = π × 4 × 9.8489 ≈ 123.7682. SA = LA + π × 16 ≈ 174.0341.
Example 2
s = √(9 + 16) = 5. This is the classic 3-4-5 right triangle, making the slant height a clean integer.
Frequently Asked Questions
What is slant height and how is it different from the regular height?
Slant height (s) is the distance from the apex to any point on the edge of the base, measured along the cone's surface. The regular height (h) is the perpendicular distance from the apex straight down to the base. They form a right triangle together with the radius: s² = r² + h².
Why is the cone volume one-third of a cylinder with the same dimensions?
A cone with the same base radius and height as a cylinder has exactly one-third the volume. This relationship was proven by Eudoxus and later by Archimedes using the method of exhaustion. The factor of ⅓ occurs because the cross-sectional area tapers linearly from the base to zero at the apex.