# Geometry/Chapter 8

## PerimeterEdit

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:

$P=4 l$ 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

$P=n l$ 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).

## AreaEdit

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

• For a triangle:

$A={b h \over 2}$ 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:

$A=b h$ 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:

$A={1 \over 2} |x_1 y_2 - x_2 y_1 + x_2 y_3 - x_3 y_2 + x_3 y_4 - x_4 y_3 + ... + x_n y_1 - x_1 y_n|$ where $(x_i, y_i)$ 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.

## VolumeEdit

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

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

• For a sphere

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

• For a cone or pyramid

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

$V = A_{base} * h$ 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 AreaEdit

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

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

• Sphere: $SA = 4\pi r^2$

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Last modified on 20 November 2013, at 01:24