Explanation:
I think the problem definition degraded a bit, when you transferred it over to here.
so, let me try to rephrase this properly, and I hope that my interpretation fits the original problem.
the base of a 3-sided prism is an isoceles triangle with a baseline length of 80 cm, and a side length of 41 cm.
calculate the surface area of the prism, if the length of the height of the prism is equal to the height of the base triangle to its baseline.
let's go through this step by step (even the ones you skipped) :
first : isoceles means that both legs are equally long. so, the prism base is a 80 - 41 - 41 triangle.
the height to its baseline is therefore splitting the baseline in the middle (into 2 equal parts) : 2×40 cm.
therefore it is also splitting the whole main triangle into 2 equal right-angled triangles (half of the baseline, a leg of the main triangle, and the height of the main triangle to the baseline).
now that we have right-angled triangles, we can use Pythagoras to calculate the height, as the leg of the main triangle is now the baseline of the "half" right-angled triangle :
41² = 40² + height²
1681 = 1600 + height²
81 = height²
height = 9 cm
so, the height to the baseline of the main triangle is 9 cm, and so is per definition the height of the prism.
that means, no, you were not correct. I honestly don't know what you did there at the beginning with the height calculation.
but let's continue.
so, the base and the top of the prism are such equal isoceles triangles.
the sides of the prism are (3) rectangles :
one is 80×9 cm²
and the other two are 41×9 cm² each
the area of a triangle is
baseline × height / 2
in our case
80 × 9 / 2 = 40 × 9 = 360 cm²
we have 2 such triangles (base and top) :
2 × 360 = 720 cm²
the areas of the rectangles are
80 × 9 = 720 cm²
41 × 9 × 2 = 738 cm²
so, the total surface area of the prism is
720 + 720 + 738 = 2178 cm²
FYI
if you were looking for the volume of the prism, that would be
base area × prism height = 360 × 9 = 3240 cm³