the base of the pyramid is a square and its faces one equilateral triangles if 'a' is the side of the base ,show that its volume is ✓2/6a*3
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Answer:
If the base of a pyramid is a square, then the lateral faces will have the same area, I assume that the top of pyramid is right on the mass center of square
Draw a pyramid with base of square with length of s and height of t, then find the height of lateral faces, we find it as sqrt(t^2 -1/4 s^2) which is the same for all other lateral faces, so, the lateral faces have the same area
For equilateral triangle, draw the pyramid with base of equilateral triangle with length of s and height of t, I assume that the top of pyramid is on the mass center of triangle, which is (s/2, s sqrt(3)/6), from this, we find that the lateral side is equal for all lateral sides and since the base side is also equal for all base side, then the angle between lateral sides and base sides must be equal too, so, the lateral area must be equal for all lateral faces
Remember, these are if the top of a pyramid is right on the mass center
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