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

Navigation

• Geometry Main Page
• Motivation
• Introduction
• Geometry/Chapter 1 Definitions and Reasoning (Introduction)
  • Geometry/Chapter 1/Lesson 1 Introduction
  • Geometry/Chapter 1/Lesson 2 Reasoning
  • Geometry/Chapter 1/Lesson 3 Undefined Terms
  • Geometry/Chapter 1/Lesson 4 Axioms/Postulates
  • Geometry/Chapter 1/Lesson 5 Theorems
  • Geometry/Chapter 1/Vocabulary Vocabulary
• Geometry/Chapter 2 Proofs
• Geometry/Chapter 3 Logical Arguments
• Geometry/Chapter 4 Congruence and Similarity
• 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 ${\displaystyle \mathbb {R} ,\mathbb {R} ^{2},\mathbb {R} ^{3}}$
• 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