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:

$A=\frac{|x_1{y}_2-x_2{y}_1+x_2{y}_3-x_3{y}_2+x_3{y}_4-x_4{y}_3+\cdots +x_n{y}_1-x_1{y}_n|}{2}$ 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

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