Question

2. SABC is a regular triangular pyramid with all sides equal to 2 cm. Calculate the pyramid: 1) the length of the apothem; 2) the length of the apex.

Answer 1

Given:

SABC is a regular triangular pyramid with all sides equal to 2 cm.

That is,

$s=2cm$

To find:

1) The length of the apothem.

2) The length of the apex.

Explanation:

1)

Since it is a regular triangular pyramid.

So, it has 3 equilateral triangular faces.

The apothem "a" formula is given by,

$a=\frac{s}{2\sqrt{3}}$

Substituting the side length, we get

$\begin{gathered} OM=\frac{2}{2\sqrt{3}} \\ OM=\frac{1}{\sqrt{3}}cm \end{gathered}$

2)

Here, the slant height is,

$l=2cm$

The length of the apothem is,

$a=OM=\frac{1}{\sqrt{3}}cm$

By Pythagoras theorem,

$\begin{gathered} SC^2=MC^2+SM^2 \\ 2^2=(\frac{2}{2})^2+SM^2 \\ 4=1+SM^2 \\ SM^2=3 \\ SM=\sqrt{3}cm \end{gathered}$

Next, we find the length of the apex from O.

That is the length of SO.

Using the Pythagoras theorem,

$\begin{gathered} SM^2=OM^2+SO^2 \\ (\sqrt{3})^2=(\frac{1}{\sqrt{3}})^2+SO^2 \\ 3=\frac{1}{3}+SO^2 \\ SO^2=3-\frac{1}{3} \\ SO^2=\frac{8}{3} \\ SO=\sqrt{\frac{8}{3}} \\ =\frac{2\sqrt{2}}{\sqrt{3}} \\ =\frac{2\sqrt{2}}{\sqrt{3}}\times\frac{\sqrt{3}}{\sqrt{3}} \\ SO=\frac{2\sqrt{6}}{3}cm \end{gathered}$

Therefore, the length of the apex is,

$\frac{2\sqrt{6}}{3}cm$

Final answer:

The length of the apothem is,

$a=OM=\frac{1}{\sqrt{3}}cm$

The length of the apex is,

$\frac{2\sqrt{6}}{3}cm$