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?
(1 point)
To find the volume of the triangular prism and triangular pyramid, we need to first calculate the area of the base triangle.
The area of an equilateral triangle can be found using the formula: A = (s^2 * sqrt(3))/4, where s is the length of one side.
Given that each side of the equilateral triangle is 6 cm, the area of the base triangle is:
A = (6^2 * sqrt(3))/4 = (36 * sqrt(3))/4 = 9sqrt(3) cm^2
Next, we can calculate the volume of the triangular prism:
Volume of a prism = base area * height = 9sqrt(3) cm^2 * 9 cm = 81sqrt(3) cm^3
Finally, we can calculate the volume of the triangular pyramid:
Volume of a pyramid = (1/3) * base area * height = (1/3) * 9sqrt(3) cm^2 * 9 cm = 27sqrt(3) cm^3
Therefore, the volume of the triangular prism is 81sqrt(3) cubic centimeters, and the volume of the triangular pyramid is 27sqrt(3) cubic centimeters.
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?
(1 point)
To find the volume of the triangular prism and triangular pyramid, we first need to determine the area of the equilateral triangle base.
The area of an equilateral triangle can be calculated using the formula: Area = (√3/4) * side length^2
Given that each side of the equilateral triangle is 6 cm, the area of the base triangle is:
Area = (√3/4) * 6^2 = (√3/4) * 36 = 9√3 cm^2
The volume of a prism is calculated as the area of the base times the height. Since the height is 9 cm, the volume of the triangular prism is:
Volume = 9√3 cm^2 * 9 cm = 81√3 cm^3
The volume of a pyramid is calculated as one-third of the base area times the height. So, the volume of the triangular pyramid in this case will be:
Volume = (1/3) * 9√3 cm^2 * 9 cm = 27√3 cm^3
Therefore, the volume of the triangular prism is 81√3 cubic centimeters, and the volume of the triangular pyramid is 27√3 cubic centimeters.
Use the image to answer the question.
An illustration shows a triangular prism and a triangular pyramid. The edges that are not visible are marked as dashed lines. The triangular prism has its triangular face as the base. The area of the triangular face is labeled as upper B equals 10 inches squared. The length is 7 inches. The triangular pyramid has the triangular face as its base with the area labeled upper B equals 10 inches squared. The perpendicular height of the pyramid is 7 inches.
How does the volume of the prism compare to the volume of the pyramid?
The volume of a prism is given by the formula: Volume = Base Area * Height
The volume of a pyramid is given by the formula: Volume = (Base Area * Height) / 3
Given that the base area (upper B) of both the prism and the pyramid is 10 square inches, and the height of both shapes is 7 inches, we can calculate the volumes:
Volume of the prism = 10 sq in * 7 in = 70 cubic inches
Volume of the pyramid = (10 sq in * 7 in) / 3 = 23.33... cubic inches
Therefore, the volume of the prism is 70 cubic inches and the volume of the pyramid is approximately 23.33 cubic inches.
Comparing the volumes, we can see that the volume of the prism is greater than the volume of the pyramid.