Equilateral cone

Text

An equilateral cone (whose axial cross-section is an equilateral triangle) has base radius $6$ cm. Find the height, the slant height, the total surface and the volume.

Solution

In an equilateral cone the slant height is twice the radius: $a=2r=12$ cm. The height is $h=\sqrt{a^2-r^2}=\sqrt{144-36}=\sqrt{108}=6\sqrt3\approx10.39$ cm. Total surface: $S_t=\pi r(r+a)=\pi\cdot6\cdot18=108\pi\approx339.29$ cm$^2$. Volume: $V=\dfrac13\pi r^2 h=\dfrac13\pi\cdot36\cdot6\sqrt3=72\sqrt3\,\pi\approx391.78$ cm$^3$. $\boxed{h=6\sqrt3\approx10.39;\ a=12\ \text{cm};\ S_t=108\pi;\ V=72\sqrt3\,\pi\approx391.78\ \text{cm}^3}$