Creating and Exploring Circles

Transcription

1 Creating and Exploring Circles 1. Close your compass, take a plain sheet of paper and use the compass point to make a tiny hole (point) in what you consider to be the very centre of the paper. The centre was believed by the Ancient Greeks to be the most honourable place the keeper of Zeus. Is your point exactly in the centre? How could you find out? How close were you? 2. Stand the compass upright in the hole you made and carefully open the compass to the radius of the circle you will draw. This symbolises letting in light according to many creation myths. The Sanskrit sound Ohm, or I Am represents the centre, radius and circumference of a circle. Can you remember what parts of a circle the radius and circumference are? How about diameter? 3. Draw your circle, trying to do it in one smooth movement, leaning the compass in the direction your pencil is travelling. As your pencil is cycling around the point, consider the cycles in nature, in time and even in your body. 4. Admire your finished circle. A point inside a circle is the Egyptian, Chinese and Mayan glyph for light. Why do you think so many cultures have seen the circle as sacred? How many places in nature can you think of where circles appear naturally?

2 Hunting for Perhaps you learned about using a strange number called pi in school. I wonder whether they showed you how mysterious it was. Let s do an experiment: Draw some different sized circles with the compass. Carefully draw the diameter of each circle, making sure it goes through the centre point and just touches each side. Measure the diameter of each circle exactly and make a note of it. Now (this is harder) measure the circumference of the circle as accurately as you can. What could you use to do this? Note these measurements beside the circle. Which are longer, the diameters or the circumferences? Using a calculator if you need to, divide the circumference of each circle by the diameter. So what do you notice? Is something rather strange happening? Note that your measurements probably weren t exact, as it s really hard to measure the circumference accurately. You should be finding something significant, though. In fact, you should be finding something pretty close to π. When the Ancient Greeks discovered this, they were very excited and threw a party! Why? Because they wouldn t have to bother trying to measure the circumference of a circle any more (which was fiddly and boring). They could work it out by using π. Have a go: Draw a new circle, a different size to all the others. Measure the diameter easy then multiply that by π. That will give you your circle s circumference. Check it out. (Oh, I forgot to mention nobody knows exactly the value of π. It s roughly 3.14, so use that for now. We ll try and do a bit better next time.)

3 The longest division sum in the history of the world! Right, you ready for this? We are going to try and find the value of π. As a fraction the closest anyone has come is 355/113. But a pretty close estimate is 22/7 So let s try 22 divided by 7. Sounds easy enough, doesn t it? Have a go. Remember to put a decimal point after the 22 and to add zeros after it, to keep dividing if you need to. (IF!!!) 7 22 Maybe we should have turned the page sideways! So when does it stop??? Er, never. It doesn t have any helpful repeating patterns in the digits! Here are some of the decimal places that have been found: π= To help remember these digits, people like to make up sentences or rhymes, called mnemonics. "May I have a large container of coffee?" is one for the first eight digits. You work out the numbers by counting the letters in each word. Some clever person even came up with this one for the first 31 digits: "Now I will a rhyme construct, By chosen words the young instruct. Cunningly devised endeavour, Con it and remember ever. Widths in circle here you see, Sketched out in strange obscurity." Try making up your own mnemonic for the digits of pi.

4 As you ve discovered, using π is a really useful way to find the circumference of a circle. When you stop to think about it though, there s another problem with circles. How do you find a circle s AREA??? Squares and rectangles are easy. So are triangles if you remember that trick about making a rectangle around them and halving the area of that. Most other 2D shapes can be divided up into triangles and rectangles but circles can t. Maybe a grid would help We measure area in squares, so see if you can work out how many squares there are in each circle. Decide what you re going to do about the odd bits of squares.

5 Make a note of your estimated areas. Remember the little squares aren t square centimetres or anything, so just call them squares. Is there an easier way? Believe it or not, there s more magic! Those Greeks were a clever lot. This is what they discovered: 1. Find the radius of each circle using the little squares as units of measurement (may be easier to count the diameter and halve it as there s no centre dot). 2. Square the radius. (Squares have to come into it somewhere if we re finding area.) 3. Now multiply it by π (3.14) Do you get a similar number to your estimate? That is the area of your circle! Most people call the formula A = π r 2 which means Area = π x the radius squared. So now you have an easy way to find the area of a circle: