1. In a certain regular square pyramid, all of the edges have length \(12\). Find the volume of the pyramid.
(There was no image provided for this question.)
Thank you!
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)