Doin’ Circlework!

Circlework” is a common country activity – put utes, dirt and young farmers together, and quickly it will devolve into who can kick up the most dirt while spinning their utes in a circle. Every B&S has one, and the prize for best circlework is always hard fought.  Unfortunately, you don’t have your license yet, so the only circle work you will be doing in the near future is in mathematics.

Unlikely as it may seem, circles are actually quite interesting and show up in many places – architecture, art, technology, nature, basically everywhere. In fact, people love circles so much, that when lost they automatically walk in circles… spooky!

But what we are interested in is angles in circles – first the simple things like measuring angles in degrees and in radians. What are radians, you say? Well, they are a sensible way of measuring the size of angles that doesn’t rely on babylonian astrology.

Waitonesec – babylonian astrology? Yep – that’s right. The original decision to divide the angle at the centre of the circle into 360 parts (degrees) was by King Nebuchadnezzar. Converting from radians is easy if you remember that there are 2 pi radians in a full revolution and also 360 degrees. Radians measure the distance around the outside of the circle (circumference), Degrees measure the amount of rotation at the centre. Here are some pictures that explain the concept and how to convert between the different systems. If you want to read some more about, try this page.

         

But measuring the angles is just the beginning – Angles in circles are so fascinating, they were researched even by the ancient Greeks. One of the first ever Mathematical texts was all about geometry – Euclid’s Elements. If you want to see what they were like, here they are.

What you need to know about now are the “Circle Theorems” that show you the relationship between angles and sections of the circle. Here are two pages that explain what they different theorems are (1, 2). You may have previous played with a geoboard (rubber bands and nails) – you can explore the theorems with this simulation.

If you prefer something more modern – or you want to play with your ClassPad, here are two pages that teach you how to use your geometry tab to explore circular theorems (1, 2)

Finally, here are is a video about converting radians to degrees and vice versa:

And here is the set of videos I expect you to watch – there are 8, and you must watch them all.

I’d like you all to comment about examples of how circle theorems might be used in the real world. Here’s an example I found

See you in class!

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34 Comments on “Doin’ Circlework!”

  1. Prathu Khairnar Says:

    hey sir i need help with excersise 4.6, question 13, b -find th are of the playground.

    i checked the back of the book and the answer is 133.05m^2
    the answer i keep getting is 134.15m^2 for some reason.

    • Bernice Bagano Says:

      Hi Sir,

      I have the same question, I keep getting either 138.25m^2 or 138.35m^2.

    • Ellayne Garcia Says:

      Hi Bernice and Prathu,

      For that question too, I keep getting the wrong answer first, but I got it when I tried to look for an alternative way of dividing the whole composite shape. Maybe you can try if this will work as there are usually many possible ways of separating composite shapes.

      • CyberChalky Says:

        It is a very challenging problem; It’s good to see you all supporting each other to try and find the solution. I have noticed, and it does make a difference when I think about your end of semester reports.

        As to this particular problem, I’d start with a rectangle (8.3 * 16), and subtract the lower triangle (0.5 * 16 + 2.5), then add the sector (0.5 * 5^2 * (2.1 – sin2.1)) and then subtract the overlap triangle (0.5 * 4 * 3).

        Make sure that your calculator is in radians mode!

  2. prathamesh Says:

    did u substract the the *overlapping trianlge*?

  3. Ellayne Garcia Says:

    Hi Sir,

    I need help with Circle Theorems 4.7, question 4b. I already got the diagram correctly labelled, but I still can’t get how to solve the angle formed at the centre.

    • CyberChalky Says:

      Hi Elayne,

      Two things. First, right angles with the tangents formed by the fence line. This allows you to construct two right angled triangles (with the radius (2m), the fenceline (4m) and a line from the centre of the circle to fence corner (you can find the length with pythagoras, but it isn’t necessary). Second, Use this triangle and trigonometry (tangent ratio you learned in year 9), you can find the central angle as 126.9 degrees.

  4. andrea duval Says:

    I have checked out the blog like you said Sir. I know I wasn’t meant to do, but how would I work out 22 (D) n (E)? It’s similar to the goat attached to the fence problem.

  5. Hao Le Says:

    Sir, is the checklist due on Monday or on Friday?

  6. Jack Huynh Says:

    Good material for methods 😛

  7. milica Says:

    hey sir for the test tomorrow will we need to know the perimeter for measuring the arc length for sectors??

  8. prathamesh Says:

    u are not going to believe this! I made a mistake in figureing out the area of the overlapping triangle :O *silly mistakes, shoots myself*

  9. Hao Le Says:

    Sir, with the Maths Online. Some worksheets don’t require us to enter an answer, just a “grade”. Do we need to print off the worksheet to show to you?

  10. Zahidah Zain Says:

    Hi Mr Grichting,
    I need help with 4.7 in the Maths Textbook. I really don’t understand it. I can’t tell the difference between the theorems.

    Would you be able to put up a post on circle theorems and how to calculate the pronumerals given?
    Thank you

  11. Ellayne Garcia Says:

    Hi Mr. G,

    About the tricky question on x = y; 1 = 2, I have a guess that the answer should the answer be undefined.

    x = y
    xy = y^2
    xy – x^2 = y^2 – x^2
    x(y – x) = (y + x)(y – x)
    Up to this point, the equation is correct. However, when we divided both sides by (y – x), I think we made a mistake.

    x(y – x) = (y + x)(y – x)
    ———— —————
    (y – x) (y – x)
    The result was x = y + x

    However, y = x.
    Therefore y – x is equals to zero.
    Dividing a number by zero should be undefined.
    Therefore, the answer should not be x = y – x, but undefined.

    Is that correct sir?

    • CyberChalky Says:

      Absolutely correct process Elayne – Good job.
      The answer isn’t undefined – there was a step that you cannot do (dividing by zero), so it should not be done, and thus the proof is invalid.

  12. Prathu Khairnar Says:

    Clear example of AI, the computer is learning ur moves and u cant win after round 30

    http://www.nytimes.com/interactive/science/rock-paper-scissors.html

  13. Damien Meyepa Says:

    Hey sir,
    I would like to know how much we have to do on Chaptes 1 and 2 during the holidays.
    Thanks Sir

    • CyberChalky Says:

      There will be a book check up on the blog by wednesday; You are not

        required

      to do *any* of chapters 1 & 2, but you should do all of the MathsOnline and ensure your bound reference is complete.

  14. Chantelle Says:

    Hey sir,
    I don’t think I will be able to finish the mathsonline tasks by the 29th. I get back on the 1st, so is it possible to extend the due date to the 7th?

  15. Zahidah Zain Says:

    Sir,
    I got a heart attack when i saw this…
    I went on maths online to do the new tasks you gave…
    and I look at my past work
    and it says I didn’t complete it when I know I did.
    I am 100038290039208% sure I did it..
    What do I do?
    😥
    But maybe I’m looking at it wrong..
    It says I got 87% though… but it says I only passed one of them. But I am very certain I got mostly 100% on all the lessons you gave us on the holidays..

    Would you be able to check for me and tell me if the records show that I’ve completed them? (Because I have.)
    Thank you
    From the sad Zahidah
    🙂


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