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]
[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]
