 Study Guide: Geometry->Chapter 8
Chapter 8
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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?

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