What is the slant height x of this square pyramid? Express your answer in radical form.

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Answer 1 · 2017-05-24T04:45:23

Slant height is $3.464$ $3.464$ meters.

Consider the triangular face, which appears as follows:
enter image source here

It is apparent that while side $A B = 4$ $AB=4$ meters, $∠ B = ∠ C = 60^{\circ}$ $/_B=/_C=60^@$

Hence $∠ a = 180^{\circ} - 60^{\circ} - 60^{\circ} = 60^{\circ}$ $/_a=180^@-60^@-60^@=60^@$ and $DeltaABC$ is an equilateral triangle.

and slant height $AD=x$ , where $AD_|_BC$

Nowin $DeltasACD$ and $ABD$ , we have $AB=AC$ , $/_B=/_C$ and $AD=AD$ and from $SSS$ postulate we have $DeltaACD-=DeltaABD$

and hence $CD=BD$ and as $BC=4$ meters and hence $CD=BD=2$ meters.

Now, we have from Pythagoras theorem

$AB^2=AD^2+BD^2$

i.e. $4^2=AD^2+2^2$

or $16=AD^2+4$

i.e. $AD=12$

and $AD=sqrt12=3.464$ meters

Hence, slant height is $3.464$ meters.