YLL討論網

正4面體
邊長1
求體積

Let O be the Centroid of base triangle ABC, the apex is D:

We can draw AO to cut BC at E
By congruent triangle, we get that ∠AEB=∠ AEC=90°
By Pythagoras, we get AE=√3/2

Since O is the centroid, so AO:OE=2:1
So, OA=√3/3

We get that base area = (1/2)(AE)(BC) = √3/4

By Pythagoras Theorem:
DO²+OA²=AD²
DO²+1/3=1
DO= √(2/3)

So, the volume = (1/3)(Base area)(height)=√2/12