# Geometry/Chapter 9

## Prisms

An n-sided prism is a polyhedron made of an n-sided polygonal base, a translated copy, and n faces joining corresponding sides. Thus these joining faces are parallelograms. All cross-sections parallel to the base faces are the same. A prism is a subclass of the prismatoids.

The volume of a prism is the product of the area of the base and the distance between the two base faces, or height. In the case of a non-right prism, the height is the perpendicular distance.

In the following formula, V=volume, A=base area, and h=height.

$V=Ah$

The surface area of a prism is the sum of the base area and its face, and the sum of each side area, which for a rectangular prism is equal to:

• $SA=2lw+2lh+2wh$
• where l = length of the base, w = width of the base, h = height

## Pyramids

The volume of a Pyramid can be found by the following formula: ${\frac {1}{3}}Ah$

• A = area of base, h = height from base to apex

The surface area of a Pyramid can be found by the following formula:$A=A_{b}+{\frac {ps}{2}}$

• $A$  = Surface area, $A_{b}$  = Area of the Base, $p$  = Perimeter of the base, $s$  = slant height.

## Cylinders

The volume of a Cylinder can be found by the following formula: $\pi r^{2}\cdot h$

• r = radius of circular face, h = distance between faces

The surface area of a Cylinder including the top and base faces can be found by the following formula: $2\pi r\ (r+h)$

• $r\,$  is the radius of the circular base, and $h\,$  is the height

## Cones

The volume of a Cone can be found by the following formula: ${\frac {1}{3}}\pi r^{2}h$

• r = radius of circle at base, h = distance from base to tip

The surface area of a Cone including its base can be found by the following formula: $\pi \ r(r+{\sqrt {r^{2}+h^{2}}})$

• $r\,$  is the radius of the circular base, and $h\,$  is the height.

## Spheres

The volume of a Sphere can be found by the following formula: ${\frac {4}{3}}\pi r^{3}$

• r = radius of sphere

The surface area of a Sphere can be found by the following formula: $4\pi \ r^{2}$

• r = radius of the sphere