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?
Since the prism and pyramid both have the same height, their volumes are proportional to their bases.
The area of the triangle can be calculated using the Heron's formula:
Area = √(s*(s-a)*(s-b)*(s-c)), where s is the semi-perimeter of the triangle and a, b, c are the sides.
In this case, s = (21+17+10)/2 = 24
Area = √(24*(24-21)*(24-17)*(24-10)) = √(24*3*7*14) = √(7056) = 84.
So, the area of the base of both shapes is 84 square units.
Since the volume of a prism is given by the formula base area * height, the volume of the prism is 84 * h = 1092, where h is the height.
h = 1092 / 84 = 13 units.
Since the volume of the pyramid is 1/3 * base area * height, the volume of the pyramid is:
1/3 * 84 * 13 = 364 cubic units.
Therefore, the volume of the pyramid is 364 cubic units.