There are also 25 quadrangular pyramids, and the other one has a bottom.
A quadrangular pyramid with any two sides of a regular pentagonal prism as the bottom and any one of the other six points as the vertex has 6 vertices, and the bottom is 2=5*4/2= 10, so there are 6 * * * 10 quadrangular pyramids in total.
A quadrangular pyramid consisting of two parallel sides on two bottom surfaces, where any two points on the upper bottom surface are parallel to the straight line formed by two points on the lower bottom surface, the combination of two out of five is 5*4/2= 10, and the other six points form a quadrangular pyramid of 10*6=60.
To sum up, the quadrangular pyramid has 25+25+60+60= 170.