# 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.

${\displaystyle V=Ah}$

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:

• ${\displaystyle SA=2lw+2lh+2wh}$
• 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: ${\displaystyle {\frac {1}{3}}Ah}$

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

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

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

## Cylinders

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

• 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: ${\displaystyle 2\pi r\ (r+h)}$

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

## Cones

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

• 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: ${\displaystyle \pi \ r(r+{\sqrt {r^{2}+h^{2}}})}$

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

## Spheres

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

• r = radius of sphere

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

• r = radius of the sphere