mathz question!

A tetrahedron has 4 planes on its surface in the shape of 4 equivalent congruent triangles. Each triangle on each plane has all sides of 3 cm length.

Please draw a 3-dimension diagram (diagram 1) to show such tetrahedron for easy understanding of the following illustration. Then, draw a line (line 1) showing the height (h) of the tetrahedron. Afterwards, draw a line (line 2) to form a triangle with a side of the tetrahedron and the line 1 as its 2 sides. Finally, we draw another line (line 3) to form a triangle with another side of the tetrahedron and the line 1 as its 2 sides.

Let the length of line 2 and line 3 be 'x'.

We then draw another diagram to show only the base of the tetrahedron (i.e. only a congruent triangle on a plane - diagram 2).

We find that all the angles inside the base triangle equal 60 degree.

In the triangle forming by line 2, line 3 and a side of the tetrahedron, their inside angles are 120, 30 and 30 degree.

Since length of line 2 and 3 equal x, we have:

(sin 120) / 3 = (sin 30) / x
x (sin 60) = 3 (1/2)
hence, x = (root 3)

We then look at the triangle formed by line 1, line 2 and a side of the tetrahedron.
Since line 1 shows the height of the tetrahedron, the angle between line 1 and line 2 and 90 degree.

Hence, by 畢氏定理,
square of h + (square of x) = square of 3
square of h + 3 = 9
square of h = 6
h = (root 6) cm

Hence, the height is (root 6) cm.

Hope that the above can help you. If required and available, I can email a Word file to show the aforesaid diagrams and lines for your easy reference.

你有無教過tetrahedron的公式？
height= (side/3) x 開方6
你的tetrahedron的side剛好是3，故
height= (3/3) x 開方6
= 1x 開方6
= 開方6 (root 6)

條公式是通過筆氏定理及三角形度數(即cos, sin那些呢)計出。因為我不知道你form幾，未知要解釋多少... 希望你明啦！

其實你知道regular tetrahedron是怎樣的嗎？它是由四個相同的三角形組成的錐體，我以前唸書時好像叫三角錐體，你講的正四面體，起初我還以為是正方形的...