Many of the pyramids built in ancient Egypt are examples of golden pyramids because they incorporate the golden mean into their geometry. According to the University of Illinois at Urbana-Champagne, the golden mean is equal to the square root of five, added to one and divided by two, or approximately 0.618. This value is often represented with the Greek letter, "phi." In a golden pyramid, such as the Great Pyramid of Giza, the proportional height is equal to the square root of phi. The hypotenuse--from the exterior base to the apex--is phi. Lastly, each side of the base is proportionally two.
The golden pyramid is a special kind of square pyramid because its base is a perfect square. To calculate the volume of a square pyramid, multiply the area of the base times 1/3, times the height. To calculate the surface area, multiply the area of the base by 1/2, times the perimeter, times the hypotenuse. These formulas can be applied to any pyramid.
Volume of Pyramid = Area of Base * 1/3 * height
Surface Area of Pyramid = Area of Base * 1/2 * perimeter * hypotenuse
A triangular pyramid has a base with three sides, which can be regular, i.e., equilateral, or irregular in length. The triangular pyramid is the most basic form of a Platonic solid, or regular polyhedron. In the case of an equilateral triangular pyramid, it is known as a tetrahedron. According to Paul Kunkel, author of Whistler Alley, a website dedicated to mathematical curiosities, what makes a platonic solid special is that "all of their faces are congruent, regular polygons, with the same number of faces meeting at every vertex."
Many pyramids are classified by the shapes of their bases. As such, both triangular and square pyramids are examples of polygonal pyramids. These shapes can technically extend to infinity, but some common examples include pentagonal, hexagonal, heptagonal and octagonal pyramids, which are 5-, 6-, 7- and 8-sided, respectively. In these examples, the bases of the pyramids are regular. However, a pyramid can also have an irregular base, such as that of a rhombus or parallelogram.
In all the examples listed, the apexes of the pyramids exist above the centers of their bases. As such, these pyramids can be classified as right pyramids. In an oblique pyramid, the apex exists above a point other than the base's center. Regardless, the same equations for volume and surface area can be used for regular and oblique pyramids alike.