# Geometry/Chapter 8
## Perimeter

The perimeter of a particular shape is the total length of its sides.
* For a triangle:
$P=l_{a}+l_{b}+l_{c}$ The perimeter is equal to the length of side a, $l_{a}$ , plus the length of side b, $l_{b}$ , plus the length of side c, $l_{c}$ .
* For a square:
$P=4l$ The perimeter is equal to 4 times the length (l) of a side.
* For a rectangle:
$P=2(b+h)$ The perimeter is equal to 2 times the sum of the base plus the height.
* For regular polygons
$P=nl$ 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. $C=2\pi r$ The circumference is equal to 2 times pi times the radius (r).

## Area

Area of a shape is how much space is inside the perimeter.
* For a triangle:
$A={\frac {bh}{2}}$ The area is equal to the product of the base (b) times the height (h) divided by 2.
* For a square:
$A=l^{2}$ The area is equal to the length (l) of a side squared.
* For a rectangle:
$A=bh$ The area is equal to the length of the base (b) times the base of the height (h).
* For a circle:
$A=\pi r^{2}$ 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:
$A={\frac {{\big |}x_{1}y_{2}-x_{2}y_{1}+x_{2}y_{3}-x_{3}y_{2}+x_{3}y_{4}-x_{4}y_{3}+\cdots +x_{n}y_{1}-x_{1}y_{n}{\big |}}{2}}$ where $(x_{k},y_{k})$ 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

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:
$V=l^{3}$ The volume is equal to the length of a side (l) cubed.
* For a rectangular prism
$V=bwh$ The volume is equal to the base (b) times the width (w) times the height (h).
* For a sphere
$V={\frac {4\pi }{3}}r^{3}$ The volume is equal to four-thirds pi times the radius cubed.
* For a cone or pyramid
$V={\frac {Bh}{3}}$ 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),
$V=A_{base}\cdot h$ where h is the height (not slant height) of the cylinder and $A_{base}$ is the area of the base. For example, the volume of a circular cylinder is $\pi r^{2}h$

## Surface Area

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: $SA=2(lw+lh+wh)$
* Closed cube: $SA=6s^{2}$
* Closed Cylinder with base area A and base perimeter P: $SA=Ph+2A$
    For a circular cylinder, $SA=2\pi r(r+h)$
Spheres are special because they have no sides but using calculus it's possible to show that:
* Sphere: $SA=4\pi r^{2}$

## Exercises

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

## Links

Geometry

### Navigation

* Geometry Main Page
* Motivation
* Introduction
* Geometry/Chapter 1 - HS 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 $\mathbb {R} ,\mathbb {R} ^{2},\mathbb {R} ^{3}$
* 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
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