# Geometry/Chapter 8

## Perimeter[edit]

The perimeter of a particular shape is the total length of its sides.

- For a triangle:

The perimeter is equal to the length of side a, , plus the length of side b, , plus the length of side c, .

- For a square:

The perimeter is equal to 4 times the length (l) of a side.

- For a rectangle:

The perimeter is equal to 2 times the sum of the base plus the height.

- For regular polygons

The perimeter is equal to the number of sides (n) times the length (l) of a side.

Circles do not have sides made of line segments like polygons do but they do have a perimeter known as a circumference. The circumference is equal to 2 times pi times the radius (r).

## Area[edit]

Area of a shape is how much space is inside the perimeter.

- For a triangle:

The area is equal to the product of the base (b) times the height (h) divided by 2.

- For a square:

The area is equal to the length (l) of a side squared.

- For a rectangle:

The area is equal to the length of the base (b) times the base of the height (h).

- For a circle:

The area is equal to pi times the radius (r) squared.

- For polygons with irregular shapes a sum of smaller areas can be used. The smaller area must completely compose the polygon. Useful smaller areas can be squares, triangles, or rectangles.

- There is another method to calculate the area of a polygon located in an 2D coordinate system:

where is the ith vertex of the polygon, they have to be given in correct order, clockwise and counter clockwise is both ok. The polygon NEED NOT to be convex.

## Volume[edit]

Volume is the amount of space an object occupies. Only shapes with 3 dimensions have a volume. This is because a 2 dimensional object has no thickness, and, therefore, takes-up no space.

- For a cube:

The volume is equal to the length of a side (l) cubed.

- For a rectangular prism

The volume is equal to the base (b) times the width (w) times the height (h).

- For a sphere

The volume is equal to four-thirds pi times the radius cubed.

- For a cone or pyramid

The volume is one-third the area of the base times the height.

- For a cylinder with a base of
**any**shape (as long as the cross sectional area is constant),

where h is the height (not slant height) of the cylinder and is the area of the base. For example, the volume of a **circular cylinder** is

## Surface Area[edit]

For most shapes you can find the surface area by adding up the area of all its sides. For example,

- (closed) Box with dimensions w, l, and h:
- Closed cube:
- Closed Cylinder with base area A and base perimeter P:
- For a circular cylinder,

Spheres are special because they have no sides but using calculus it's possible to show that:

- Sphere:

## Exercises[edit]

## Links[edit]

- 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
- 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