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

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• Geometry/Chapter 5 Triangle: Congruence and Similiarity
• Geometry/Chapter 6 Triangle: Inequality Theorem
• Geometry/Chapter 7 Parallel Lines, Quadrilaterals, and Circles
• Geometry/Chapter 8 Perimeters, Areas, Volumes
• Geometry/Chapter 9 Prisms, Pyramids, Spheres
• Geometry/Chapter 10 Polygons
• Geometry/Chapter 11 $\mathbb {R} ,\mathbb {R} ^{2},\mathbb {R} ^{3}$
• Geometry/Chapter 12 Angles: Interior and Exterior
• Geometry/Chapter 13 Angles: Complementary, Supplementary, Vertical
• Geometry/Chapter 14 Pythagorean Theorem: Proof
• Geometry/Chapter 15 Pythagorean Theorem: Distance and Triangles
• Geometry/Chapter 16 Constructions
• Geometry/Chapter 17 Coordinate Geometry
• Geometry/Chapter 18 Trigonometry
• Geometry/Chapter 19 Trigonometry: Solving Triangles
• Geometry/Chapter 20 Special Right Triangles
• Geometry/Chapter 21 Chords, Secants, Tangents, Inscribed Angles, Circumscribed Angles
• Geometry/Chapter 22 Rigid Motion
• Geometry/Appendix A Formulae
• Geometry/Appendix B Answers to problems
• Appendix C. Geometry/Postulates & Definitions
• Appendix D. Geometry/The SMSG Postulates for Euclidean Geometry