# Geometry/Chapter 8

## Perimeter

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

• For a triangle:

${\displaystyle P=l_{a}+l_{b}+l_{c}}$ The perimeter is equal to the length of side a, ${\displaystyle l_{a}}$, plus the length of side b, ${\displaystyle l_{b}}$, plus the length of side c, ${\displaystyle l_{c}}$.

• For a square:

${\displaystyle P=4l}$ The perimeter is equal to 4 times the length (l) of a side.

• For a rectangle:

${\displaystyle P=2(b+h)}$ The perimeter is equal to 2 times the sum of the base plus the height.

• For regular polygons

${\displaystyle P=nl}$ 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. ${\displaystyle C=2*\pi *r}$ 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:

${\displaystyle A={bh \over 2}}$ The area is equal to the product of the base (b) times the height (h) divided by 2.

• For a square:

${\displaystyle A=l^{2}}$ The area is equal to the length (l) of a side squared.

• For a rectangle:

${\displaystyle A=bh}$ The area is equal to the length of the base (b) times the base of the height (h).

• For a circle:

${\displaystyle A=\pi r^{2}}$ 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:

${\displaystyle A={1 \over 2}|x_{1}y_{2}-x_{2}y_{1}+x_{2}y_{3}-x_{3}y_{2}+x_{3}y_{4}-x_{4}y_{3}+...+x_{n}y_{1}-x_{1}y_{n}|}$ where ${\displaystyle (x_{i},y_{i})}$ 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:

${\displaystyle V=l^{3}}$ The volume is equal to the length of a side (l) cubed.

• For a rectangular prism

${\displaystyle V=bwh}$ The volume is equal to the base (b) times the width (w) times the height (h).

• For a sphere

${\displaystyle V={4 \over 3}\pi r^{3}}$ The volume is equal to four-thirds pi times the radius cubed.

• For a cone or pyramid

${\displaystyle V={1 \over 3}Bh}$ 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),

${\displaystyle V=A_{base}*h}$ where h is the height (not slant height) of the cylinder and ${\displaystyle A_{base}}$ is the area of the base. For example, the volume of a circular cylinder is ${\displaystyle \pi r^{2}h}$

## 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: ${\displaystyle SA=2lw+2lh+2wh}$
• Closed cube: ${\displaystyle SA=6s^{2}}$
• Closed Cylinder with base area A and base perimeter P: ${\displaystyle SA=P*h+2A}$
For a circular cylinder, ${\displaystyle SA=2\pi rh+2\pi r^{2}}$

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

• Sphere: ${\displaystyle SA=4\pi r^{2}}$