SOLUTION: The base of a right pyramid is a square of side 20 cm. The lateral faces of the pyramid are all equilateral triangles. Find the surface area and volume of this pyramid.

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Question 1159770: The base of a right pyramid is a square of side 20 cm. The lateral faces of the pyramid are all equilateral triangles. Find the surface area and volume of this pyramid.

Answer by greenestamps(13200) (Show Source):
You can put this solution on YOUR website!

The area of the square base is the square of the length of the side. That one is easy.

For each face that is an equilateral triangle, the altitude of the triangular face (from the peak of the pyramid to the center of the base) divides the equilateral triangle into two 30-60-90 right triangles. Use your knowledge of 30-60-90 right triangles to find the length of that altitude; then use area equals one-half base times height to find the area of each triangular face.

Or there is a concise formula you can use for the area of an equilateral triangle in terms of the length of a side: . I chose not to use that formula, because we are going to need the length of the altitude of each triangular face to determine the height of the pyramid, which I need for calculating the volume of the pyramid.

That takes care of how to find the total surface area.

The volume is one-third the area of the base, multiplied by the height. So we need to determine the height.

Consider the right triangle formed by the peak of the pyramid, the center of the square base, and the center of the base of one of the triangular faces. We know the lengths of one leg and the hypotenuse; the other leg is the height of the pyramid.

So we know how to find the height of the pyramid; that means we can find its volume.

I leave the relatively simple calculations to you....