+0

# Prymid

0
218
2
+157

A regular tetrahedron is a triangular pyramid in which each face is an equilateral triangle. If the height of a regular tetrahedron is 20 inches then what is the length of each edge of the tetrahedron?

Sort:

#1
+18956
0

A regular tetrahedron is a triangular pyramid in which each face is an equilateral triangle.

If the height of a regular tetrahedron is 20 inches then what is the length of each edge of the tetrahedron?

For a regular tetrahedron of edge length a:
Height of pyramid  $${\displaystyle h={\frac {\sqrt {6}}{3}}a={\sqrt {\frac {2}{3}}}\,a\,}$$

$$\begin{array}{|rcll|} \hline h &=& \sqrt {\frac {2}{3}}\,a \\ a &=& \frac{h}{\sqrt {\frac {2}{3}}} \\ a &=& \sqrt {\frac {3}{2}}\,h \\ \mathbf {a} &\mathbf {=}& \mathbf {\sqrt {1.5}\,h } \\ \hline \end{array}$$

$$\begin{array}{rcll} a &=& \sqrt{1.5}\cdot 20 \\ a &=& 24.4948974278\, \text{inch} \\ \end{array}$$

The length of each edge of the tetrahedron is 24.5 inch

heureka  Sep 19, 2017
#2
+82897
+1

Let us find the edge length

Draw a segment from the apex of the tetrahedron perpendicular to the base.....this is the height....call the point where this segment intersects the base, A

Draw  a segment  from one of the bottom vertexes (call this point  B)  to  A

Find the midpoint  of one of the sides connecting to B....label this point  C

And BC  = (1/2) the side length  = (1/2)s

So.....we have the right triangle ABC lying on the base

And we have this relationship

cos 30°  =  ( BC) / (BA)

cos (30°) = (1/2)s / (BA)

BA  =  (1/2)s / [√ 3 / 2]  =  s / √ 3

So......we have another right triangle...one leg is the height, one leg is BA and the hypotenuse is the side length......and we have......

√ [ 20^2 + BA^2]  = s

√ [ 20^2  + (s/√3)^2  ]  = s

√ [(1200 + s^2) / 3 ]  = s

[1200 + s^2] / 3  = s^2

1200 + s^2  = 3s^2

1200  = 2s^2

600  = s^2

√ 600  = s

10√ 6   = s  ≈  24.49  inches

CPhill  Sep 19, 2017
edited by CPhill  Sep 19, 2017
edited by CPhill  Sep 19, 2017

### 1 Online Users

We use cookies to personalise content and ads, to provide social media features and to analyse our traffic. We also share information about your use of our site with our social media, advertising and analytics partners.  See details