Helpful Tools for Calculus, Chapter 0: Numbers, Notations and Notes
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A basic understanding of two- and three-dimensional shapes is particularly helpful for applications in Calculus. This chapter will help to form a basic background on shapes, their perimeters and areas, information about their constructions and more.
2D Shapes
Rectangles
Rectangles are a four-sided polygon. The sides can be denoted by any variable you wish, however the most commonly used are b/h for base/height or l/w for length/width. You shouldn’t refer to them simply as ‘sides’ because the sides are usually different in length.
Area = base ∙ height
Perimeter = b + b + h + h =2(b+h)
Angles add to 360 degrees
Triangles
It is easiest to think of triangles as half of a rectangle. We can easily find the area of a triangle by comparing it to its rectangle like the shape above. We then divide the shape in two pieces by drawing a vertical line from the tip of the triangle.
We can see that each side of the box is divided in half by the new (right) Triangles. So, we can conclude the area is half of the rectangle.
Area = 1/2 bh
Perimeter = a + b + c Where the sides are denoted a,b,c and generally c is the hypotenuse, though not always. To solve for any side, 
where C is the angle opposite of side c.
Angles add to 180 degrees
We also know that the angles and the sides have related ratios (through trigonometry).
Law of Sines: 
There are some special right triangles that may be helpful to know:
Trapezoids
The pictures above describe transformations made to an isosceles trapezoid (meaning that it is symmetrical, or that the two angles are equal). In other trapezoids we can do something similar.
We should always find ways to re-write the dimensions of a shape based on what we know. So, representing a trapezoid as a square and two triangles is a fantastic idea. The area of the triangles can be evaluated as above, and the square should be easy.
Area = 
The total value of the angles in a trapezoid add to 360.
Circles
For circles, things are a little bit different, because of π.
Circumference (perimeter) = π times Diameter = 2 π radius
Area = π radius2
A portion of a circle (like a pizza slice) can be defined so that the Area of the sector = ½ θ r2
3D Shapes
Rectangular and Other Prisms
These should be pretty simple: It’s a box.
Volume = Base x Height x Depth
= Length x Width x Height
To do Surface Area, think of a 6-sided die. There are six sides, and opposite sides are equal.
Surface Area = 2 (BH + BD + HD)
Imagining the ‘net’ of an object is also helpful. Below is the net of a rectangular prism :
For non-rectangular prisms, like a trapezoidal prism, we take the area of the face (the main shape it is named after) and multiply that by the depth for the area.
Surface area is usually four rectangles and the two faces.
Cylinder
It is easiest to think of a cylinder as a circle that has depth.
Its volume is given by V = πr2h where h is the depth, or height.
The surface area is a bit trickier. Imagine you have a tube of cinnamon rolls. You smack it on the table and it unrolls. If you laid it flat it would be in the shape of a rectangle !
Surface area = Circumference x Height + top + bottom= 2πrh +2πr2 = 2πr(r+h)
Sphere
This is a sphere with radius r.
What is shaded is a single slice of the sphere z units from the center.
This slice has a radius of x and is in the shape of a circle.
Using the Pythagorean theorem, we can say that 
Therefore, the area of the shaded disk is 
If we did this an infinite number of times to get all of the disks in the sphere, and then we added them all up, we would get the volume of a sphere.
I won’t go through the details here, but this will be continued later in Chapter 8.
Volume = 4/3 πr3
The surface area is also quite difficult to prove and uses similar ideas and another called the Archimedes’ Hat-Box Theorem. Give it a quick Google if you have the time.
Surface Area = 4πr2
Regular Polygons
To expand our knowledge of geometry, we can study what are generally called ‘regular polygons.’ These are shapes that are equiangular and equilateral. This means that all angles are equal and all sides are equal. So basically, these are the easy shapes.
Interior Angles
To find the angle of a given n-gon, we can do start with a simple shape : the triangle.
In a pentagon (5 sides), we can draw 3 triangles. Draw from the top, like I have shown here.
Since each triangle hold 180o, we can say the pentagon has 540o.
In a hexagon (6 sides), we can draw 4 triangles.
Use the same method, and we show it has 720o
A nonagon (9 sides) holds 7 triangles.
You can draw them in yourself.
We can see a n-gon (n sides) has 2 less triangles than sides.
Thus, the formulas for total angles = 180 (n-2)
We can then identify that each angle (there are n of them) is (180 (n-2))/n degrees = (π (n-2))/n radians
Area and Perimeter
The perimeter should be easy. Since all the sides have the same length, s, we can say that P = n times s.
Area, however, is a bit tricky. We shall use a different triangle method. The triangles will be constructed from the center of the shape like below
Using this method, we have one triangle per side. So the area of the shape is (Area of Triangle) times n. The area of the triangle is a bit tricky though. To find the area of the triangles we need to find what’s called the apothem. The apothem is the (shortest) length from the center of a polygon to one of its sides. This forms right triangles in the triangles we already created and thus we can identify the height of the triangle using trigonometry since we know the angles.
Height of triangle (apothem) = a = ½ s tan θ
Using the formula for θ, we ‘simplify’ this to: ½ s tan ((π (n-2))/n) and there are n of the triangles.
Thus, the total area of a regular polygon is given by :
A = ½ sn tan ((π (n-2))/n) which is equal to ½ P tan ((π (n-2))/n), where P is the perimeter = sn
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