Question

A prism with an altitude of 15 cm has an octagonal base inscribed in a square with side 10 cm. Find the volume of the prism.

Answer 1

Regular ooctagon with 15 cm altitude is inscribed in a 10X10 cm. square.

To find it's side length, square has to be cut, such that all sides of octagon are maximum and equal.

inscribed in square ABCD.

Let$BF = BG = CH = CI = X, so GH = 10 - 2X.$

By pythagoras theorem,

$FG = √2X and GH = FG =10 - 2X,$ therefore,

$√2X = 10 - 2X => X(2+√2) = 10 => X = 10/(2+√2)$

side of the octagon

$= 10 - 20/(2+√2) (20 + 10√2 - 20)/(2+√2) = 10√2/(2+√2$ )

Area of regular octagon = 2(1+√2)S^2

Volume of prism =>

$V = 15×2(1+√2)[(10√2/(2+√2)]^2$ $V = (30+30√2)[200/(6+4√2)]$

$V = 6000(1+√2)/(6+4√2) = 1242.64 cm^3$