# 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. Area is the amount of space inside the perimeter

## 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]

1. If a cylinder has a base area of 10cm and height of 12cm.what is its volume? 932

## 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