A solid right pyramid has a square base. The base edge length is 2 cm longer than the height of the pyramid.
If the height is 6 cm, what is the volume of the pyramid?

A. \( 24 \, \text{cm}^3 \)
B. \( 48 \, \text{cm}^3 \)
C. \( 128 \, \text{cm}^3 \)
D. [tex]\( 144 \, \text{cm}^3 \)[/tex]



Answer :

To find the volume of a right pyramid with a square base, we use the formula for the volume of a pyramid:

[tex]\[ V = \frac{1}{3} \times \text{Base Area} \times \text{Height} \][/tex]

Given that the height of the pyramid is \( 6 \, \text{cm} \) and the base edge length is \( 2 \, \text{cm} \) longer than the height, we can determine the base edge length as follows:

[tex]\[ \text{Base Edge Length} = 6 \, \text{cm} + 2 \, \text{cm} = 8 \, \text{cm} \][/tex]

Next, we calculate the area of the square base. The area \( A \) of a square with side length \( s \) is given by:

[tex]\[ A = s^2 \][/tex]

So, for the base edge length of \( 8 \, \text{cm} \):

[tex]\[ \text{Base Area} = 8 \, \text{cm} \times 8 \, \text{cm} = 64 \, \text{cm}^2 \][/tex]

Now, we can use the formula for the volume of the pyramid:

[tex]\[ V = \frac{1}{3} \times 64 \, \text{cm}^2 \times 6 \, \text{cm} \][/tex]

[tex]\[ V = \frac{1}{3} \times 384 \, \text{cm}^3 \][/tex]

[tex]\[ V = 128 \, \text{cm}^3 \][/tex]

Thus, the volume of the pyramid is \( 128 \, \text{cm}^3 \).

Therefore, the correct answer is:

[tex]\[ \boxed{128 \, \text{cm}^3} \][/tex]