Example 1
\(r = 3, h = 4 \rightarrow l = \sqrt{3^2 + 4^2} = 5, V = \frac{1}{3}\pi(3^2)(4) \approx 37.70\) units³, \(SA = \pi(3^2) + \pi(3)(5) \approx 75.40\) units²
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Theory & Formula
A cone is a three-dimensional shape with a circular base that tapers smoothly to a point called the apex or vertex. The volume of a cone is exactly one-third the volume of a cylinder with the same base and height.
The surface area consists of the base area (πr²) and the lateral surface area (πrl), where l is the slant height - the distance from the apex to any point on the circle's edge. The slant height can be found using the Pythagorean theorem: l = √(r² + h²).
Properties and Applications
- Volume of cone = 1/3 × Volume of cylinder with same base and height
- The slant height is always greater than the vertical height
- When unrolled, the lateral surface forms a sector of a circle
- Used to calculate volumes of ice cream cones, traffic cones, funnels, and conical structures
- A cone with apex directly above the center of the base is a right circular cone
\(V = \frac{1}{3}\pi r^2 h, SA = \pi r^2 + \pi rl, l = \sqrt{r^2 + h^2}\)
Worked Examples
Example 2
\(r = 5, h = 12 \rightarrow l = \sqrt{5^2 + 12^2} = 13, V = \frac{1}{3}\pi(5^2)(12) \approx 314.16\) units³, \(SA = \pi(5^2) + \pi(5)(13) \approx 282.74\) units²