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: $$\frac{1}{3} A h$$ The surface area of a Pyramid can be found by the following formula:$$A = A_b + \frac{ps}{2}$$
 * A = area of base, h = height from base to apex
 * $$A$$ = 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: $$\pi r^2 \cdot h$$ 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 = radius of circular face, h = distance between faces
 * $$r\,$$ 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: $$\frac{4}{3} \pi r^3$$ The surface area of a Sphere can be found by the following formula: $$4 \pi\ r^2$$
 * r = radius of sphere
 * r = radius of the sphere