Question

area? (2.5 points)
here is a right square pyramid. round answers in this problem to the neares
if necessary.
what is the height of the pyramid? (1 point)
what is the volume of the pyramid? (1 point)

Explanation:

Height of the Pyramid

Step1: Identify the right triangle

The base of the square pyramid is a square with side length 6, so the distance from the center of the base to the midpoint of a base edge is half of 6, which is 3. The slant edge (from the apex to the midpoint of a base edge) is 5. We can use the Pythagorean theorem \( a^2 + b^2 = c^2 \) where \( c = 5 \) (the slant edge), \( b = 3 \) (half of the base side), and \( a \) is the height \( h \) of the pyramid.

Step2: Apply the Pythagorean theorem

We have \( h^2 + 3^2 = 5^2 \). So \( h^2 = 5^2 - 3^2 = 25 - 9 = 16 \). Then \( h=\sqrt{16} = 4 \).

Step1: Recall the volume formula for a pyramid

The volume \( V \) of a pyramid is given by \( V=\frac{1}{3}Bh \), where \( B \) is the area of the base and \( h \) is the height.

Step2: Calculate the base area

The base is a square with side length 6, so the area of the base \( B = 6\times6=36 \).

Step3: Substitute values into the volume formula

We know the height \( h = 4 \) from the previous calculation. So \( V=\frac{1}{3}\times36\times4 \).

Step4: Compute the volume

\( \frac{1}{3}\times36 = 12 \), and \( 12\times4 = 48 \).

Answer:

The height of the pyramid is 4.

Volume of the Pyramid