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)
1. Volume of the triangular prism:
Volume = Base area x Height
Base area = (1/2) x base x height
Base area = (1/2) x 6 cm x 4 cm
Base area = 12 cm^2
Volume = 12 cm^2 x 9 cm
Volume = 108 cm^3
Therefore, the volume of the triangular prism is 108 cubic centimeters.
2. Volume of the triangular pyramid:
Volume = (1/3) x Base area x Height
Base area = 12 cm^2 (from above)
Height = 9 cm
Volume = (1/3) x 12 cm^2 x 9 cm
Volume = 36 cm^3
Therefore, the volume of the triangular pyramid is 36 cubic centimeters.
To compare the volume of the prism to the volume of the pyramid, we need to calculate the volumes of both shapes using the given information.
Volume of the triangular prism:
Volume = Base area x Height
Base area = 10 inches^2
Height = 7 inches
Volume = 10 inches^2 x 7 inches
Volume = 70 cubic inches
Volume of the triangular pyramid:
Volume = (1/3) x Base area x Height
Base area = 10 inches^2
Height = 7 inches
Volume = (1/3) x 10 inches^2 x 7 inches
Volume = 23.33 cubic inches
Therefore, the volume of the prism is 70 cubic inches, while the volume of the pyramid is 23.33 cubic inches.
Therefore, the volume of the prism is greater than the volume of the pyramid.
To find the volume of the pyramid, we first need to calculate the area of the base triangle using the side lengths given.
By Heron's formula, the area of a triangle with side lengths 21, 17, and 10 is given by:
s = (21+17+10)/2 = 24
Area = √(24(24-21)(24-17)(24-10)) = 84
Now, for the pyramid:
Volume = (1/3) x Base area x Height
Height is given to be the same as the prism, which gives a volume of 1,092 cubic units.
However, since the volume of a pyramid is 1/3 of the volume of a corresponding prism with the same base and height, the volume of the pyramid is:
1/3 * 1,092 = 364 cubic units
Therefore, the volume of the pyramid is 364 cubic units.
You're welcome! I'm glad I could help. If you have any more questions or need further assistance, feel free to ask.
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? (1 point)
Use the image to answer the question.
An illustration shows a triangle with sides measuring 21, 17, and 10. A perpendicular line, from the side measuring 21 to the opposite angle, measures 8. A right angle symbol is shown to the left of the perpendicular line.
A prism and a pyramid both have this triangular base. If both shapes have the same height and the volume of the prism is 1,092 cubic units, what is the volume of the pyramid? (1 point)