All examples come from RegentsPrep.org
Practice Problems:

Prisms are threedimensional closed surfaces. A prism has two parallel faces, called bases, that are congruent polygons. The lateral faces are rectangles in a right prism, or parallelograms in an oblique prism. In a right prism, the joining edges and faces are perpendicular to the base faces. 

Prisms are also called polyhedra since their faces are polygons.
A regular prism is a cube. 




We will be working with regular pyramids unless otherwise stated. 
Pyramids are threedimensional closed surfaces. The one base of the pyramid is a polygon and the lateral faces are always triangles with a common vertex. The vertex of a pyramid (the point, or apex) is not in the same plane as the base. Pyramids are also called polyhedra since their faces are polygons. The most common pyramids are regular pyramids. A regular pyramid has a regular polygon for a base and its height meets the base at its center. The slant height is the height (altitude) of each lateral face. 
In a regular pyramid, the lateral edges are congruent. 

Pyramids are named for the shape of their base.
Triangular pyramid Square pyramid


We will be working with right circular cylinders unless 
Cylinders are threedimensional closed surfaces. In general use, the term cylinder refers to a right circular cylinder with its ends closed to form two circular surfaces, that lie in parallel planes. Cylinders are not called polyhedra since their faces are not polygons. In many ways, however, a cylinder is similar to a prism. A cylinder has parallel congruent bases, as does a prism, but the cylinder’s bases are circles rather than polygons. 
The volume of a cylinder can be calculated in the same manner as the volume of a prism: the volume is the product of the base area times the height of the cylinder,


The surface area (of a closed cylinder) is a combination of the lateral area and the area of each of the bases. When disassembled, the surface of a cylinder becomes two circular bases and a rectangular surface (lateral surface), as seen in the net at the left. Note that the length of the rectangular surface is the same as the circumference of the base. Remember that the area of a rectangle is length times width. The lateral area (rectangle) = height × circumference of the base. 
When working with surface areas of cylinders, read the questions carefully.
Will the surface area include both of the bases? 
Will the surface area include only one of the bases? 
Will the surface area include neither of the bases? The lateral area only. 

Cones are threedimensional closed surfaces. In general use, the term cone refers to a right circular cone with its end closed to form a circular base surface. The vertex of the cone (the point) is not in the same plane as the base. Cones are not called polyhedra since their faces are not polygons. In many ways, however, a cone is similar to a pyramid. A cone’s base is simply a circle rather than a polygon as seen in the pyramid. 
The volume of a cone can be calculated in the same manner as the volume of a pyramid: the volume is onethird the product of the base area times the height of the cone, 
In a right circular cone, the slant height, s, can be found using the Pythagorean Theorem: 
The surface area (of a closed cone) is a combination of the lateral area and the area of the base. When cut along the slant side and laid flat, the surface of a cone becomes one circular base and the sector of a circle (lateral surface), as shown in the net at the left. Note that the length of the arc in the sector is the same as the circumference of the small circular base.



The lateral area (sector) =

When working with surface areas of cones, read the questions carefully.
Will the surface area include the base? 
Will the surface area not include the base? 

Spheres are threedimensional closed surfaces. A sphere is a set of points in threedimensional space equidistant from a point called the center. The radius of the sphere is the distance from the center to the points on the sphere. Spheres are not polyhedra. Of all shapes, a sphere has the smallest surface area for its volume. 
The volume of a sphere is fourthirds times pi times the radius cubed.

Note: A cross section of a geometric solid is the intersection of a plane and the solid.


A great circle is the largest circle that can be drawn on a sphere. Such a circle will be found when the crosssectional plane passes through the center of the sphere.
The equator is an examples of a great circle. Meridians (passing through the North and South poles) are also great circles. The shortest distance between two points on a sphere is along the arc of the great circle joining the points.
The shortest distance between points on any surface is called ageodesic. In a plane, a straight line is a geodesic. On a sphere, a great circle is a geodesic. 