SOLUTION: Hi! My question is: I need to find the total surface area of a regular triangular pyramid has base edges of length 30 and height 10. I already tried 1558.845...m^2 but it is

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Question 1133461: Hi!
My question is:
I need to find the total surface area of a regular triangular pyramid has base edges of length 30 and height 10.
I already tried 1558.845...m^2 but it is wrong.
Thanks for helping!
Found 2 solutions by addingup, greenestamps:
Answer by addingup(3677) (Show Source):
You can put this solution on YOUR website!
(30*10)/2 = 150 each of 3 sides has an area of 150.
Thus:
150*3 = 450 this is the surface area of the top 3 triangles.
Now we need the area of the triangular base, which is an equilateral triangle with sides of length 30. Draw a line perpendicular from one of the vertex to the middle
of the side. Now we have two right triangles with one side 15 and the hypotenuse 30. Pythagoras says:
sqrt(30^2 - 15^2) = 26 this is the height of the triangle.
Area:
bh/2 = 30*26/2 = 390
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Total area:
450 + 390 = 840

Answer by greenestamps(13198) (Show Source):
You can put this solution on YOUR website!

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.

That response also uses an approximation to find the area of the base; I suspect an exact answer is required.

The base is an equilateral triangle with side length 30. The area of an equilateral triangle with side length s is

So the area of the base of this pyramid is

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.

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).

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).

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's another 30-60-90 right triangle).

So the area of each triangular face is one-half base times height:

And so finally the total surface area of the pyramid is the area of the base plus the area of the three triangular faces:

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