The surface area of the pyramid

Question

What is the surface area of ​​the regular triangular pyramid if the base edge is $a = 12$ cm and height $H = 8$ cm?

Answer 1

You'll want to check my arithmetic, but working in $\operatorname{cm}^2$:

The base is an equilateral triangle of side $a$ and area $a^2\sqrt{3}/4=36\sqrt{3}$. Each of the other three faces is an isosceles triangle of base $a$ and legs $\sqrt{H^2+a^2/3}=4\sqrt{7}$, since the equilateral face's centroid is a distance $a/\sqrt{3}$ from each of its vertices. Each such triangle has base $12$ and height $\sqrt{H^2+a^2/3-a^2/4}=\sqrt{H^2+a^2/12}=2\sqrt{19}$. The area of one such triangle can also be computed by Heron's formula, as$$\sqrt{(4\sqrt{7}+6)(4\sqrt{7}-6)6^2}=12\sqrt{19}.$$So the surface area is $36(\sqrt{3}+\sqrt{19})$.

Answer 2

As I understood from the information given the pyramid is formed by four triangles

So the formula of the surface area would be when we add the area of those triangles. To find the area of the triangles you should the base of the triangles and the height. So the height of the pyramid is not the same as the height of the triangles that form the sides of the pyramid. The picture below will help you understand it better.

The Surface are would be when we add the are of the base triangle, and the area of the triangles on the side. As I understood the base triangle is a equilateral triangle. The formula is: $$ A=\frac{\sqrt3}{2}*a^2$$, in your case $a=12$ The formula for the side of one triangle would be: $$A1= \frac{1}{2}*a*h$$ $h$ is the slant height. which from the picture we will find it using the Pythagorean Theorem. $$h=\sqrt{\frac{a^2}{3}+H^2}$$ here $H$ is the height of the pyramid which in your case is $8cm$.

So, in conclusion the total surface area would be $$ S=A+3*A1$$ I didn't substitute the value to give you all the solution. But, you can update with the results if you are not sure.

You are new here, but it would be good next time to provide with more information and also include what you have worked so far to solve this problem so that we would know where to help you.