Circles 1

One of the earliest identifiable shapes

The circle must have been one of the earliest shapes identified by humans as it is the most easily identifiable shape and there are many examples in nature and around us.

The heavenly bodies closest to use - Sun, Moon & the planets are circular. Raindrops are spherical. Most seeds, fruits & nuts are spherical. Rainbows are actually circular, though it is difficult to see a complete one!

Ancient geometers discovered a lot of interesting properties that circles had!

It is the figure easiest to draw. A circle can be drawn with 2 sticks and a length of string. One stick fixed to the ground acts as the centre. One end of the string can be tied to the centre stick and the other tied to the other end of the string. By holding the 2ndstick taut a circle can be drawn on the ground.

Using the same centre, the length of the string can be increased or decreased and many other circles can be drawn. It can be seen that all these circles are parallel to one another.

Any line drawn through the centre, will cut the circle at 2 opposite points and its length would always be same for a particular circle. This is the Diameter of the circle. The length of the string is the Radius of the circle.

It was also seen that the Diameter of a circle is twice the Radius of the circle.

The outer curve which bounds the circle is called the Circumference.

Since the radius of the circle is same at any point on the circumference, when a circle rolls on the ground, the centre is always at a fixed distance to the ground. This is the reason, wheels of vehicles are circular.

There is archaeological evidence that humans were making circular wheels in Sumeria as far back as 5000 years.

By drawing many circles as part of daily routines, it was discovered that the circumference of a circle was a fixed multiple of the length of the diameter. Many cultures made efforts to find the value of this "fixed multiple". The Greeks called this multiple as π (Pi). One of the earliest measures for this multiple was 3. As the need for more and more accurate values of π was required, better approximations were found out. It was only about 300 years back that it was realized that π was a very fascinating number (Please refer to chapter 32.2 on π for more details).

Many interesting facts were discovered by Greek geometers about the circle.

Angles in the Segment & Angle Subtended at the Centre

If you plot two points A & B, anywhere on the circumference and imagine a line joining them, we can see that it divides the circle into 2 parts, each of which is called a segment. If you imagine a third point C in any one of the segments, angle ACB was called the angle in that segment.

The interesting discovery was that whatever the location of C, between A & B, the angle ACB remains same! Angles in the same segment are equal.

Similar to C, another point D can be imagined on the other segment. It is obvious that whatever the location of D, angle ADB remains same.

If we assume that the centre of the circle to be O, the angle AOB is called the” angle subtended at the centre”. It was also found that the angle subtended at the centre for any segment was twice the angle in the segment.

The surprise was that angles ACB & ADB were Supplementary angles. Their sum was 180 degrees.

There are some additional surprises.

Angle in a semi-circle

In any segment, if A & B coincide, the angle ACB will be 0. As A & B move away from each other, ACB will increase. When A & B are farthest from each other from each other, ACB was found to be a right angle. At this point AB also form the diameter of the circle. Hence it was discovered that angle in a semicircle is a right angle!

If the 2 points move in the same direction, they will start coming together and ACB will keep on increasing. When A & B are very close, ACB will be very close to 180 degrees.