Use the image to answer the question.
An illustration shows an equilateral triangle with each side measuring 6 centimeters. The perpendicular height is shown as a dashed line from the base to the apex and and is 4 centimeters. A right angle symbol is shown to the right of the perpendicular line.
Use the model for the base of a triangular prism and triangular pyramid. If the heights are both 9 cm, what is the volume of each shape?
To find the volume of both the triangular prism and the triangular pyramid, we first need to find the area of the base triangle.
Since the base is an equilateral triangle with side length 6 cm, we can find the area using the formula:
Area = (sqrt(3) / 4) * side^2
Area = (sqrt(3) / 4) * 6^2
Area = (sqrt(3) / 4) * 36
Area = (sqrt(3) * 36) / 4
Area = (6sqrt(3)) cm^2
Now, we can find the volume of the triangular prism using the formula:
Volume = base area * height
Volume = (6sqrt(3)) * 9
Volume = 54sqrt(3) cm^3
Next, we can find the volume of the triangular pyramid using the formula:
Volume = (1/3) * base area * height
Volume = (1/3) * (6sqrt(3)) * 9
Volume = (18sqrt(3)) cm^3
So, the volume of the triangular prism is 54sqrt(3) cubic centimeters and the volume of the triangular pyramid is 18sqrt(3) cubic centimeters.