if i draw a square pyramid with all sides equal to each other, what is the lenght of the sides so that the volume and the total area of the pyramid are the same?
Assuming the edges of the triangular faces are included in " all sides " ......
Let's get the slant height and height in terms of s using the Pythagorean theorem.
slant height = √[ s2 - (s/2)2 ]
= √[ s2 - s2/4 ] Factor out an s2 .
= √[ s2(1 - 1/4) ]
= √3s / 2
height = √[ (√3s / 2)2 - (s/2)2 ]
= √[ 3s2 / 4 - s2/4 ]
= √[ s2(3/4 - 1/4) ]
volume = (1/3)(area of base)(height)
volume = (1/3)(s2)( √2s/2 )
volume = (√2/6)s3
surface area = (1/2)(perimeter of base)(slant height) + area of base
surface area = (1/2)(4s)( √3s / 2 ) + s2
surface area = √3s2 + s2
What does s equal when volume = surface area ?
What does s equal when (√2/6)s3 = √3s2 + s2 ?
(√2/6)s3 = √3s2 + s2 Divide through by s2
(√2/6)s = √3 + 1 Multiply through by 6/√2
s = 3√6 + 3√2