Sexagesimal system

Index Angles Time Converting to another system Adding and Subtracting Exercises

Sexagesimal system

Matematicas 2o E.S.O.Alberto Pardo Milanes

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1 Angles

2 Time

3 Converting

4 Adding and Subtracting

5 Exercises

Alberto Pardo Milanes Sexagesimal system


Angles



Angles

What’s an angle? Measure an angle

Given two intersecting lines, the point ofintersection is called the vertex and theamount of rotation about the vertex re-quired to bring one line into correspondencewith the other is called the angle betweenthem.

There are several ways to measure the size of an angle. One way isto use units of degrees: In a complete circle there are three hundredand sixty degrees (one full rotation corresponds to 360 degrees).



Angles

Some angles

Half a full rotation is called a straight angle that is 180 de-grees, and a quarter of a full rotation is called a right angle(90 degrees). An angle less than 90 degrees is called an acuteangle, an angle greater than 90 degrees and less than 180degrees is called an obtuse angle, and an angle greater than180 degrees and less than 360 degrees is called a reflex angle.



Angles

Using the proctartor

Acute angle Right angle

Obtuse angle Straight angle



Angles

Express an angle

You can express angles to any precision using decimal numbers.Example: 4 degrees and a half=4.5 degrees.

There is another way to state the size of an angle, and that is bysubdividing a degree. The degree is divided into sixty parts calledminutes. These minutes are further divided into sixty parts calledseconds.In a full circle there are 360 degrees, each degree is split up into 60parts called minutes (each part being 1/60 of a degree), eachminute is split up into 60 parts called seconds (each part being1/60 of a minute).

Examples: • 1 degree is 60 minutes, so ,5 degrees=,5 · 60 = 30minutes, so 4.5 degrees is equal to 4 degrees and 30 minutes.• 14 degrees, 22 minutes and 55 seconds is usually written thisway: 14o 22′ 55′′

.Alberto Pardo Milanes Sexagesimal system


Time



To measure time we can use hours: In a day there are twenty-fourhours.You can express time to any precision using decimal numbers.There is another way to state the amount of time, subdividing anhour. The hour is divided into sixty parts called minutes. Theseminutes are further divided into sixty parts called seconds.In a day there are 24 hours, each hour is split up into 60 partscalled minutes (each part being 1/60 of an hour), each minute issplit up into 60 parts called seconds (each part being 1/60 of aminute).

Example: 18 hours,51 minutes and 42 seconds is usually writtenthis way: 18hr 51min 42sec.



Converting



Converting

Sexagesimal to decimal

To convert sexagesimal format to decimal format divide theseconds by three thousand six hundred, then add it to the minutesdivided by sixty, and finally add the degrees. To convert time dothe same.

Example: 22o 54′ 18′′ is written in the sexagesimal system. Toconvert it to decimal: 18/3600+54/60+22 = 0,005+0,9+22 =22,905o.



Converting

Decimal to sexagesimal

To convert decimal format to sexagesimal format the whole unitsof degrees will remain the same, then multiply the decimal by 60,the whole number becomes the minutes, take the remainingdecimal and multiply by 60, the resulting number becomes theseconds. Seconds can remain as a decimal. Take your three sets ofnumbers and put them together, using the symbols for degrees (o),minutes (′), and seconds (′′). To convert time do the same.

Example: 35,245o is written in the decimal system. To convert itto sexagesimal: 35 are the degrees, 0,245 ·60 = 14,7 so 14 are theminutes, 0,7 · 60 = 42 so 42 are the seconds. 35,245 degrees are35o 14′ 42′′.



Adding and Subtracting




To add or subtract angles or time, you must add or subtract theunits separately. But you may need to do some adjusting:the hours can not end up 24 or more,the degrees can not end up 360 or more,the minutes can not end up 60 or more,and the seconds can not end up 60 or more.Negative numbers can not be used.




Adding

Add the hours. If the hours are 24 or more, then you have tosubtract 24 from the hours and increase the number of days byone. Add the minutes. If the minutes are 60 or more, subtract 60from the minutes and increase the number of hours by one. Addthe seconds. If the seconds are 60 or more, subtract 60 from theseconds and increase the number of minutes by one. To add anglesdo the same method.




Adding

Example: To add 18hr 55min + 9hr 21min:

18 hr 55 min+ 9 hr 21 min

27 hr 76 min

1 day 3 hr 76 min

1 day 4 hr 16 minAdd the hours: 18 + 9 = 27. The hours are 24 or more, so wesubtract 24 from the hours (27 − 24 = 3hr) and increase thenumber of days by one. Add the minutes: 55 + 21 = 76. Theminutes are 60 or more, so subtract 60 from minutes (76− 60 =16min) and increase the number of hours by one (3 + 1 = 4hr).So 18hr 55min + 9hr 21min= 1day 4hr 16min.




Subtracting

Subtract the hours, if the hours are negative, add 24 to the hoursand subtract 1 from the days. Subtract the minutes, if the minutesare negative, add 60 to the minutes and subtract 1 from the hours.Subtract the seconds, if the seconds are negative, add 60 to theseconds and subtract 1 from the minutes.




Subtracting

Example: To subtract 12hr 30min - 7hr 55min:

12 hr 30 min- 7 hr 55 min

11 hr 90 min- 7 hr 55 min

4 hr 35 minSubtract the hours: 12− 7 = 5. Subtract the minutes: 30− 55 =−25. The minutes are negative, so subtract 1 from hours andincrease the number of minutes by sixty (30 + 60 = 90min). So12hr 30min - 7hr 55min ==11hr 90min - 7hr 55min= 4hr 35min.



Exercises



Exercises

Exercise 1

Find five objects in your classroom that have all kind of anglesangles but no right angles.



Exercises

Exercise 2

Using a set square is easy to draw a right angle, a 30o angle, a 45o

angle or a 60o angle. Use a set square to draw a 15o angle and a75o angle.

30o angle 45o angle 60o angle



Exercises

Exercise 3

Measure these angles:



Exercises

Exercise 4

Write down the following amounts:

• 12o 11′ 12′′:

• 145o 23′ 3′′:

• 3hr 25min 28sec:

• 16hr 46min 50sec:



Exercises

Exercise 5

Jane works six hours a day from Monday till Friday, and one hourless on Saturday. How many hours does she work a week?



Exercises

Exercise 6

At 13:45 AM, Peter went to have lunch at his friend Paul’s house.His mother told him to be back in two and a half hours. At whattime does Peter have to be home?



Exercises

Exercise 7

Thirty days have September, April, June, and November; All therest have thirty-one, Save February, with twenty-eight days clear,And twenty-nine each leap year. How many days are there in oneleap year? How many hours? How many minutes? How manyseconds?



Exercises

Exercise 8

Two angles are complementary if the sum of their measures is 90o.Find two complementary angles measuring exactly the same.



Exercises

Exercise 9

Two angles are supplementary if the sum of their measures is 180o.Find two supplementary angles being the measure of the big onethree times the measure of the small one.



Exercises

Exercise 10

Charles rode his bike on Monday and Thursday. He spend 1hr36min 48sec riding on Monday and 1hr 32min 25sec riding onThursday How much time did he ride over the two days in total?



Exercises

Exercise 11

The interior angles of a triangle always add up to 180o. If twoangles of a triangle measure 21o 24′ and 86o 49′. What is themeasure of the third angle?



Exercises

Exercise 12

Spencer got on the train to get to the airport at 11:23. He spent42 min in the train. It took him 25 min to get his plane ticket andHe spent 1 h 6 min waiting for the plane to take off. At what timedid the plane take off?



Exercises

Exercise 13

A tv show started at 20:25 and it ended at 23:30. How long didthe show last for?



Exercises

Exercise 14

School ended at 15:15. Al arrived home just in time to watch amovie. If the movie started at 16:05, how long did it take for Al toget home?



Exercises

Exercise 15

A bus runs from around half past six in the morning until around00:45 at night. How many hours does the drivers use the bus? Howmany minutes? If it rides the same route sixty times a day, howmuch time does the route need to be covered?


Education

Sexagesimal system