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?
+9466 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) ]
= √2s/2
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
