<PRE><B>Question 1133461</B>
&lt;br&gt;
The response from the other tutor uses 10 as the SLANT height of the pyramid; I suspect the 10 is supposed to be the full height of the pyramid.&lt;br&gt;
That response also uses an approximation to find the area of the base; I suspect an exact answer is required.&lt;br&gt;
The base is an equilateral triangle with side length 30.  The area of an equilateral triangle with side length s is&lt;br&gt;
{{{s^2*sqrt(3)/4}}}&lt;br&gt;
So the area of the base of this pyramid is&lt;br&gt;
{{{30^2(sqrt(3))/4 = 225*sqrt(3)}}}&lt;br&gt;
To find the area of the triangular faces, we need to find the slant height, using the height of the pyramid and the length of each side of the base.&lt;br&gt;
An altitude of the equilateral triangle base divides the triangle into two 30-60-90 right triangles, each with hypotenuse 30 and one leg 15; the length of the other leg (the altitude of the triangular base) is 15*sqrt(3).&lt;br&gt;
The length of the apothem of the triangle (from the center of the base perpendicular to each edge) is one-third the length of the altitude, which is 5*sqrt(3).&lt;br&gt;
Then the altitude of the pyramid, the apothem of the base, and the slant height of a triangular face form a right triangle.  With the height of the pyramid 10 and the length of the apothem 5*sqrt(3), the slant height of each triangular face is 5 (it&#39;s another 30-60-90 right triangle).&lt;br&gt;
So the area of each triangular face is one-half base times height:&lt;br&gt;
{{{(1/2)(30)(5) = 75}}}&lt;br&gt;
And so finally the total surface area of the pyramid is the area of the base plus the area of the three triangular faces:&lt;br&gt;
{{{225*sqrt(3)+225}}} or {{{225(1+sqrt(3))}}}</PRE>