Slant height

Geometry Level 3

The volume of a regular pyramid with square base is 512 3 \frac{512}{3} . If the height is equal to length of the base edge, find the length of the slant height. If your answer can be expressed as a \sqrt{a} , give your answer as a a .


The answer is 80.

A Former Brilliant Member by A Former Brilliant Member

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1 solution

Let the height=base edge = x. Then

V = 1 3 A b h V=\dfrac{1}{3}A_bh

512 3 = 1 3 ( x 2 ) ( x ) \dfrac{512}{3}=\dfrac{1}{3}(x^2)(x)

512 = x 3 512=x^3

x = 8 x=8

Note that the slant height is the altitude of one face. So,

L = 8 2 + 4 2 = L=\sqrt{8^2+4^2}= 80 \boxed{\sqrt{80}}

Finally. a = 80 \color{#D61F06}\boxed{\large a=80}

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