**Shiawassee Amateur Radio Association ~ W8QQQ **

**Whiskey 8 Quack Quack Quack**

**Baker College; Center for Technical Studies, Bldg 16; Room near 1632 (look for us); Owosso, MI 48867**

*Current Site Path: *SARA HomeSARA's Reference Page SARA's Mathematics Reference

Page Navigation:

Decimal Equivalents
| Important Constants
| Formulas for Area
| Sphere Formulas
| Trig Formulas for Triangles

# SARA's Science & Mathematics Reference Page

Shiawassee Amateur Radio Association presents Reference Information which we hope is useful to viewers.

We also have an electronics reference page ~ Electronics Reference Page.

## Decimal Equivalents (& Drill Sizes)

Fractions of an inch to decimals (same value conversions as drill sizes). Seems like I am always looking for these and it takes me a long time (minutes) to find the information, so I put it here for my use! Works for both purposes. Therefore I added in 'Number' & 'Letter' table so I know where the information is - ** ENJOY**.

### Fractions & ANSI Size Drill Bit Selections

Fraction | Decimal | Fraction | Decimal | Fraction | Decimal | Fraction | Decimal |

(in.) | (in.) | (in.) | (in.) | ||||

1/64 | 0.0156 | 17/64 | 0.2656 | 33/64 | 0.5156 | 49/64 | 0.7656 |

1/32 | 0.0313 | 9/32 | 0.2812 | 17/32 | 0.5312 | 25/32 | 0.7812 |

3/64 | 0.0469 | 19/64 | 0.2968 | 35/64 | 0.5469 | 51/64 | 0.7969 |

1/16 | 0.0625 | 5/16 | 0.3125 | 9/16 | 0.5625 | 13/16 | 0.8125 |

5/64 | 0.0781 | 21/64 | 0.3281 | 37/64 | 0.5781 | 53/64 | 0.8281 |

3/32 | 0.0937 | 11/32 | 0.3437 | 19/32 | 0.5937 | 27/32 | 0.8437 |

7/64 | 0.1093 | 23/64 | 0.3594 | 39/64 | 0.6094 | 55/64 | 0.8594 |

1/8 | 0.0125 | 3/8 | 0.3750 | 5/8 | 0.6250 | 7/8 | 0.8750 |

9/64 | 0.1406 | 25/64 | 0.3906 | 41/64 | 0.6406 | 57/64 | 0.8906 |

5/32 | 0.1562 | 13/32 | 0.4062 | 21/32 | 0.6562 | 29/32 | 0.9062 |

11/64 | 0.1719 | 27/64 | 0.4219 | 43/64 | 0.6719 | 59/64 | 0.9219 |

3/16 | 0.1875 | 7/16 | 0.4375 | 11/16 | 0.6875 | 15/16 | 0.9375 |

13/64 | 0.2031 | 29/64 | 0.4531 | 45/64 | 0.7031 | 61/64 | 0.9531 |

7/32 | 0.2187 | 15/32 | 0.4687 | 23/32 | 0.7187 | 31/32 | 0.9687 |

15/64 | 0.2344 | 31/64 | 0.4844 | 47/64 | 0.7344 | 63/64 | 0.9844 |

1/4 | 0.2500 | 1/2 | 0.5000 | 3/4 | 0.7500 | 1 | 1.0000 |

### Numbered and Letter Size ANSI Drill Bit Chart

Decimal | Decimal | Decimal | Decimal | ||||

Drill # | (in.) | Drill # | (in.) | Drill # | (in.) | Drill Letter | (in.) |

80 | 0.0135 | 53 | 0.0595 | 26 | 0.1470 | A | 0.2340 |

79 | 0.0145 | 52 | 0.0635 | 25 | 0.1495 | B | 0.2380 |

78 | 0.0160 | 51 | 0.0670 | 24 | 0.1520 | C | 0.2420 |

77 | 0.0180 | 50 | 0.0700 | 23 | 0.1540 | D | 0.2460 |

76 | 0.0200 | 49 | 0.0730 | 22 | 0.1570 | E | 0.2500 |

75 | 0.0210 | 48 | 0.0760 | 21 | 0.1590 | F | 0.2570 |

74 | 0.0210 | 47 | 0.0785 | 20 | 0.1610 | G | 0.2610 |

73 | 0.0240 | 46 | 0.0810 | 19 | 0.1660 | H | 0.2660 |

72 | 0.0250 | 45 | 0.0820 | 18 | 0.1695 | I | 0.2720 |

71 | 0.0260 | 44 | 0.0860 | 17 | 0.1730 | J | 0.2770 |

70 | 0.0280 | 43 | 0.0890 | 16 | 0.1770 | K | 0.2811 |

69 | 0.0292 | 42 | 0.0935 | 15 | 0.1800 | L | 0.2900 |

68 | 0.0310 | 41 | 0.0960 | 14 | 0.1820 | M | 0.2950 |

67 | 0.0320 | 40 | 0.0980 | 13 | 0.1850 | N | 0.3020 |

66 | 0.0330 | 39 | 0.0995 | 12 | 0.1890 | O | 0.3160 |

65 | 0.0350 | 38 | 0.1015 | 11 | 0.1910 | P | 0.3230 |

64 | 0.0360 | 37 | 0.1040 | 10 | 0.1935 | Q | 0.3320 |

63 | 0.0370 | 36 | 0.1065 | 9 | 0.1960 | R | 0.3390 |

62 | 0.0380 | 35 | 0.1100 | 8 | 0.1990 | S | 0.3480 |

61 | 0.0390 | 34 | 0.1110 | 7 | 0.2010 | T | 0.3580 |

60 | 0.0400 | 33 | 0.1130 | 6 | 0.2040 | U | 0.3680 |

59 | 0.0410 | 32 | 0.1160 | 5 | 0.2055 | V | 0.3770 |

58 | 0.0420 | 31 | 0.1200 | 4 | 0.2090 | W | 0.3860 |

57 | 0.0430 | 30 | 0.1285 | 3 | 0.2130 | X | 0.3970 |

56 | 0.0465 | 29 | 0.1360 | 2 | 0.2210 | Y | 0.4040 |

55 | 0.0520 | 28 | 0.1405 | 1 | 0.2280 | Z | 0.4130 |

54 | 0.0550 | 27 | 0.1440 |

## Some Important Constants

"Pi" or π is the ratio of Circumference to diameter for a circle.

Name | Symbol | Numeric Value |
---|---|---|

Pi | 3.14159265... | |

Pi^2 | 9.86960440... | |

Pi^.5 | 1.77245385... | |

1/Pi | 0.31830989... | |

Pi/180 |
0.01745329... | |

Pi/360 |
0.00872665... | |

e |
2.7182818... | |

Degrees in 1 Radian | 57.29577951... | |

Radians in 1 degree | 0.01745329... | |

Speed of Light | c |
299,792,458 m/sec or |

Speed of Light | c |
983,571,056.4 ft/sec or |

Speed of Light | c |
188,857.7297 mi/sec |

Newtonian Gravitational Constant | G |
6.67384 x 10^-11 m^3/(kg x sec^2) |

Plank constant | h |
6.62606957 x 10^-34 Jsec |

Coulomb's constant | 8.98755178 x 10^9 Nm^2/C^2 | |

Wavelength | lamda or | c is 'Speed of Light' and f is frequency in Hertz (cycles per second) |

Characteristic Impedance of vacuum | 376.73031346... Ohms |

## Web Math Programs

### Convert Units

'Convert for Windows' is a very useful program to convert from one value of measurement into another type of unit. It covers: Force; Light; Mass; Power;; Pressure; Speed (velocity); temperature; Time; Volume; Volume - Dry; Accelleration; Angle; Area; Density; Distance and Flow. Also, a 'custom' arae allows you to enter your own conversions (if required by you... example money conversions). Convert for Windows Freeware Program

### SMath Studio

'SMath Studio' is a free math studio for entering math equations and doing ploting, design, and other mathematical investigations in topics of your choosing. SMath Studio Tiny, powerful, free mathematical program with WYSIWYG editor and complete units of measurements support. It provides numerous computing features and rich user interface. The application also contains integrated mathematical reference book. A 'Built-in' Extensions Manager tool allows you to get access to hundreds official and third-party resources. Students and Engineers get a great tool for free!

## Formulas for Area

### Rectangle ~ Area

The most basic area formula is the formula for the area of a rectangle. Given a rectangle with length l and width w, the formula for the area is:

*A* = *l w* (rectangle)

That is, the area of the rectangle is the length multiplied by the width. As a special case, as *l* = *w* which is a square, the area of a square with side length s is given by the formula:

*A* = *s *^{2} (square)

### Triangle ~ Area

It follows that a triangle's area is shown to be one half the area of a rectangle (or parallelogram), namely: {b is base and h is height}

*A* = *1 (b * H) / 2 * (triangle)

### Dissection of parallelograms into triangles

Many simple formulas for area follow from the method of dissection. This involves cutting a shape into pieces, whose areas must sum to the area of the original shape.

For an example, any parallelogram can be subdivided into a trapezoid and a right triangle, as shown in the figure to the left. If the triangle is moved to the other side of the trapezoid, then the resulting figure becomes a basic rectangle. It follows that the area of any parallelogram has the same formula (thus same area) as a rectangle with same width and height.(area is just the product of the length and width).

*A* = *l w* (parallelogram)

However, the same parallelogram can be cut along one of the diagonals into two congruent triangles, as shown in the figure to the right. Each triangle has height, h, and width, b. You can use the triangle area formula to find the area of one triangle, and then multiply by 2 (two triangles) to get the parallelpgram's area.

It is easy to see that the area of each triangle is half the area of the parallelogram:

*A* = *1 (b * H) / 2 * (triangle)

Similar arguments and methods can be used to find area formulas for any trapezoid, as well as many more complicated polygons. If one tries the method of dissection and looking at results, usually the correct formula can be found. The polygon's area will be the sum of all the areas from each of the small divisions summed up. As you do these disections, triangles and the ability to work with them becomes important. This leads to the field of "Trigonometry" (see below).

### Circle ~ Area & Parts

*A* = *Pi r^2 * (Circle with Radius, r) or *A* = *Pi d^2 / 4* (Circle with diameter, d)

The formula for the area of a circle can be based on a similar method as used in parallelograms above. Given a circle of radius r (a disk), it is possible to partition the circle into sectors, as shown in the figure to the left. Each sector is approximately triangular in shape, and the sectors can be rearranged to form an approximate parallelogram. The height of this parallelogram is r, and the width is half the circumference of the circle, or (Pi)r. Thus, the total area of the circle is r x (Pi r), or Pi r^2, as 2r = d the formula can be rewritten as Pi d^2 / 4

The side [*s*] of an inscribed square becomes:

or

Length of an arc (small curved piece of the circle) becomes: [*L *]:

or

Area of a sector [*A s*]:

### List of Common Formulas for Area

Shape | Formula for Area | Variables |
---|---|---|

Regular Equilateral Triangle | s is the length of one side of the triangle. | |

Triangle 1 | s is half the perimeter, a, b and c are the length of each side. | |

Triangle 2 | a and b are any two sides, and C is the angle between them. | |

Triangle 3 | a and h are the base and altitude (measured perpendicular to the base). | |

Isosceles triangle | a is the length of one of the two equal sides and b is the length of the different side. | |

Rhombus/Kite | a and b are the lengths of the two diagonals of the rhombus or kite. | |

Parallelogram | b is the length of the base and h is the perpendicular height. | |

Trapezoid | a and b are the parallel sides and h the distance (height) between the parallels. | |

Regular hexagon | s is the length of one side of the hexagon. | |

Regular octagon | s is the length of one side of the octagon. | |

Regular polygon 1 | l is the side length and n is the number of sides. | |

Regular polygon 2 | p is the perimeter and n is the number of sides. | |

Regular polygon 3 | R is the radius of a circumscribed circle, r is the radius of an inscribed circle, and n is the number of sides. | |

Regular polygon 4 | n is the number of sides, b is the side length, a is the apothem, or the radius of an inscribed circle in the polygon, and p is the perimeter of the polygon. | |

Circle | r is the radius and d the diameter. | |

Circular sector | r and theta are the radius and angle (in radians), respectively and L is the length of the perimeter. | |

Ellipse | a and b are the semi-major and semi-minor axes. | |

Total surface area of a cylinder | r and h are the radius and height. | |

Lateral surface area of a cylinder | r and h are the radius and height. | |

Total surface area of a sphere | r and d are the radius and diameter. | |

Total surface area of a pyramid | B is the base area, p is the base perimeter and L is the slant height. | |

Square to circular area conversion | A is the area of the square in square units. | |

Circular to square area conversion | C is the area of the circle in circular units. |

The above calculations show how to find the areas of many common shapes. Hope they help you.

## Sphere Formulas

Where: *d* is diameter; *r* is radius; and 2*r = d*

Surface area [As] of a sphere:

Volume [V] of a sphere:

## Trigonometry Formulas for Triangles

Trigonometry is the mathematical branch dealing with relationships of sides and angles in triangles. It also deals with a "unit vector" rotating around a point of origin, which has an angle of rotation, a unit vector and the 'x axis' and 'y axis' values for the vector's end pont (a triangle with sides, 'x', 'y'; hypotenuse ~ 'c' and the rotational angle {theta}).

In triangle terms:

In 'unit circle' terms:

Also defined is the reciprocal of each function:

### Given a triangle with angles *A*, *B*, and *C* that are respectively opposite the sides *a*, *b*, and *c*.

NOTE: Some of the following equations can have two 'correct' solutions, so some **judgement is required !** Then the following results can be determined:

### "Law of Sines"

### "Law of Cosines"

Most useful if you have any two sides and the included angle.

and rearrange to two other forms:

### "Law of Tangents"

The law of tangents is most useful if you have two sides and an included angle or two angles and the included side.

### Trigonometry functions expressed in terms of the other basic trigonometry functions:

In terms of: | |||
---|---|---|---|

### Inverse Functions

Inverse functions are also called *"arc functions"*.

or

or

You and get to the electronics reference page ~ Electronics Reference Page from here.

The equations shown on this page are presented by using *"Latex"* and the equation writing capabilities from the Code Cogs Web Site. Images are then stored "locally" to allow proper HTML5 validation.

Page Navigation:

Decimal Equivalents
| Important Constants
| Formulas for Area
| Sphere Formulas
| Trig Formulas for Triangles

Web Site Contact: W8QQQ at arrl.net

Page Last Updated: 11-Aug-2017

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