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: $\frac{1}{3} A h$

• 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

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