Cone
Definition, properties, and formulas for cone calculations
Definition and Properties
A cone is a three-dimensional body with a circular base and a single apex point. The side surface connects all points of the base circle with the apex.
- r = base radius, d = 2r = base diameter
- h = vertical height, L = slant height
- If apex is above center: right cone; otherwise: oblique cone
For a right cone: \(L^2 = h^2 + r^2\). This relation is used in many cone formulas.
Formulas
Radius
\(\displaystyle r=\sqrt{\frac{A}{\pi}}\)
Diameter
\(\displaystyle d=\sqrt{\frac{4A}{\pi}}=2r\)
Height
\(\displaystyle h=\sqrt{L^2-r^2}=\frac{3V}{A}\)
Slant height
\(\displaystyle L=\sqrt{h^2+r^2}\)
Lateral surface
\(\displaystyle S_l=\frac{d\pi L}{2}=\frac{PL}{2}\)
Total surface
\(\displaystyle S=S_l+A\)
Volume
\(\displaystyle V=\frac{Ah}{3}=\frac{\pi r^2h}{3}\)
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