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# i dont understand this geometry problem :(

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1. In a certain regular square pyramid, all of the edges have length \(12\). Find the volume of the pyramid.

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Thank you!

Apr 16, 2020

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Call the very top vertex of the pyramid A.

Call one of the corners of the pyramid B.

Call the midpoint of the base C..

Triangle(ABC) is a right triangle with angle(C) the right angle.

Side(AC) is the height of the right triangle.

Side(AB) is one of the edges of the pyramid and is the hypotenuse of triangle(ABC); it has length 12.

The first problem is to find the distance from C to B.

This distance is one-half of the distance from B to its opposite vertex of the base.

Since the base of the pyramid is a square, each of its sides is 12. The distance from one corner of the square to its opposite corner is 12·sqrt(2).  [You can use the Pythagorean Theorem to find this length.]

So, the distance from C to B is one-half of 12·sqrt(2), which is 6·sqrt(2).

To find the height of the pyramid, we can use the Pythagorean Theorem on triangle(ABC).

[ 6·sqrt(2) ]2 + [ CA ]2  =  [ 12 ]2

36·2 + [ CA ]2  =  144

72 + [ CA ]2  =  144

[ CA ]2  =  72

CA  =  sqrt(72)

CA  =  6·sqrt(2)     <---   height of the pyramid

The area of the base of the pyramid  =  12 · 12  =  144

Volume of the pyramid  =  (1/3) · Area of the Base · Height

=  (1/3) · 144 · 6·sqrt(2)

=  288 · 6·sqrt(2)

Apr 16, 2020