Answer :
To find the volume of a regular triangular pyramid with the given dimensions, follow these steps:
### 1. Calculate the Area of the Triangular Base
To find the area of the triangular base, use the formula for the area of a triangle:
[tex]\[ A_{\text{base}} = \frac{1}{2} \times \text{base} \times \text{height} \][/tex]
Here, the base \( b \) of the triangular base is 10 cm and the altitude \( h_b \) of this triangle is approximately 8.7 cm. Substituting these values into the formula gives:
[tex]\[ A_{\text{base}} = \frac{1}{2} \times 10 \, \text{cm} \times 8.7 \, \text{cm} \][/tex]
[tex]\[ A_{\text{base}} = \frac{1}{2} \times 87 \, \text{cm}^2 \][/tex]
[tex]\[ A_{\text{base}} = 43.5 \, \text{cm}^2 \][/tex]
So, the area of the triangular base \( A_{\text{base}} \) is \( 43.5 \, \text{cm}^2 \).
### 2. Calculate the Volume of the Pyramid
The formula for the volume \( V \) of a pyramid is:
[tex]\[ V = \frac{1}{3} \times A_{\text{base}} \times h \][/tex]
Here, \( A_{\text{base}} \) is the area of the triangular base which we have already calculated as \( 43.5 \, \text{cm}^2 \), and \( h \) is the height of the pyramid, which is 12 cm. Substituting these values into the formula gives:
[tex]\[ V = \frac{1}{3} \times 43.5 \, \text{cm}^2 \times 12 \, \text{cm} \][/tex]
[tex]\[ V = \frac{1}{3} \times 522 \, \text{cm}^3 \][/tex]
[tex]\[ V = 174 \, \text{cm}^3 \][/tex]
Therefore, the volume of the regular triangular pyramid is [tex]\( 174 \, \text{cm}^3 \)[/tex].
### 1. Calculate the Area of the Triangular Base
To find the area of the triangular base, use the formula for the area of a triangle:
[tex]\[ A_{\text{base}} = \frac{1}{2} \times \text{base} \times \text{height} \][/tex]
Here, the base \( b \) of the triangular base is 10 cm and the altitude \( h_b \) of this triangle is approximately 8.7 cm. Substituting these values into the formula gives:
[tex]\[ A_{\text{base}} = \frac{1}{2} \times 10 \, \text{cm} \times 8.7 \, \text{cm} \][/tex]
[tex]\[ A_{\text{base}} = \frac{1}{2} \times 87 \, \text{cm}^2 \][/tex]
[tex]\[ A_{\text{base}} = 43.5 \, \text{cm}^2 \][/tex]
So, the area of the triangular base \( A_{\text{base}} \) is \( 43.5 \, \text{cm}^2 \).
### 2. Calculate the Volume of the Pyramid
The formula for the volume \( V \) of a pyramid is:
[tex]\[ V = \frac{1}{3} \times A_{\text{base}} \times h \][/tex]
Here, \( A_{\text{base}} \) is the area of the triangular base which we have already calculated as \( 43.5 \, \text{cm}^2 \), and \( h \) is the height of the pyramid, which is 12 cm. Substituting these values into the formula gives:
[tex]\[ V = \frac{1}{3} \times 43.5 \, \text{cm}^2 \times 12 \, \text{cm} \][/tex]
[tex]\[ V = \frac{1}{3} \times 522 \, \text{cm}^3 \][/tex]
[tex]\[ V = 174 \, \text{cm}^3 \][/tex]
Therefore, the volume of the regular triangular pyramid is [tex]\( 174 \, \text{cm}^3 \)[/tex].
