C = 2πR. What does this mean? What is its usefulness? C = The distance around any circle. But where did such a formula come from?
Start by drawing a circle. Within the middle of the circle, draw a dot. Now, draw a line that passes through the dot. This line should divide the circle into two halves and not stretch beyond the outside of the circle. Let’s call the distance around the circle ( C = circumference ). The line that dissects the circle is the diameter ( D ).
C divided by D will be noted as ( C / D ). If you draw any circle of any size, the ratio of ( C / D ) will always be the same. As it turns out, the ratio of ( C / D ) is approximately 3.14. The number 3.14 is commonly abbreviated using a symbol pronounced as ” pie ” = π . So ( C / D ) = π.
Let’s take another look at ( C / D ) = π. If you look at the picture you’ve drawn, you will notice that the dot at the center of the circle divides the line ( D ) into two segments. If you take one of these segments, it is referred to as being the radius of a circle ( r ). Multiplying the radius ( r ) by two will give you the diameter. In other words, D = 2r ( 2 times r ). Therefore, ( C / D ) = ( C / 2r ) = π = 3.14.
Now, a quick recap of the variables listed above. C = Circumference = Total distance around a circle. D = Diameter of the circle = line that dissects a circle into two halves. Take half of the diameter ( D ) and you have the radius ( r ) of a circle.
IF ( C / 2r ) = ( Distance around a circle / 2r ), using the rules of algebra, we can multiply ( C / 2r ) by 2r to obtain the distance around a circle. Remember that ( C / 2r ) = π. So ( π ) times ( 2r ) = The distance around a circle. This can be written as ( π )( 2r ), or you may write the more familiar term……………..2πR = C.
NOTE: The capital R in 2πR is = r.
It does not matter what order the multiplication occurs. 1 times 2 times 3 = 1 times 3 times 2. So ( π )( 2r ) above is more commonly written as 2πr.
Now, let’s dig a little deeper!
Look at the circle you’ve drawn. Assign a value of r = 1 to your radius. This is commonly done in math ( google ” unit circle ” if you wish, but for the time being, that is not necessary ). 1 could equal 1 inch, 1 meter, 1 mile, etc. For the time being, just remember that the metric system is used in science, and anything that isn’t a meter must be converted to meters using methods I’ve described in a prior entry. Okay, let’s assume that the radius ( r ) of our circle = 1 m.
If we drew a circle in some sand, it could have a radius ( r ) equal to 1 m. Now, starting at the position where the radius touches the outer part of the circle, we could take 1 m of rope, and we could carefully superimpose the 1 m rope onto a segment of the circle. Since this 1 m length is not straight, and it has the curvature of the circle, we can call it 1 radian.
EXTREMELY IMPORTANT: 1 radian is the distance around a circle that is equal in magnitude to the radius. It would be helpful now to attempt to draw a radian on a circle.
Draw a new circle. Put a dot in the middle of the circle. Draw a radius ( r ) extending from the dot. As your pin reaches the outer surface of the circle ( without lifting your pin off the paper ), trace out a distance around the circle’s edge that is equal in length to ( r ). This distance = 1 radian.
HOW MANY RADIANS ARE NEEDED TO COMPLETELY WRAP AROUND A CIRCLE?
Notice that a radian ( theta ) is a fraction of the entire circumference ( distance around ) a circle. If we divide $1.00 by 1 quarter ( $0.25 ), we get the number 4. Likewise, dividing the entire distance around a circle ( circumference ( C ) = 2πR ) by 1 radian ( which is a fraction of the whole ), we will see how many radians are needed to complete any circle.
Since C = 2πR, and 1 radian ( theta ) is equal in length to the radius, ( 2πR / r ) = 2π. It takes 2π radians to travel around a circle. Since π = 3.14, 2π radians = 6.28 radians. IF YOU MEASURE THE RADIUS OF ANY CIRCLE, CONVERT THAT LENGTH TO METERS ( IF NEED BE ), MULTIPLY THAT LENGTH BY 6.28 to get the circle’s circumference ( C ).
This is why 360 degrees = 2π radians.
Q: You are given a length of rope and a meter stick. Someone will pay you to create a circular track whose distance equals 1 mile. What would you do?
A: 1 mile = 1.60 kilometers, approximately. Let’s now use a couple of conversion fractions to convert 1 mile to meters ( since we are using a meter stick to measure things ).
( 1 mi )( 1.60 km / 1mi )( 1 m / ( 1 * 10^-3 km ) ) = 160 m, so 160 m = C. And remember that C = 2πR. Also remember that 2π = 6.28 .
C = 2πR = ( 6.28 )( R ) = 160 m
R = ( 160 m / 2π ) = ( 160 m / 6.28 ) = 25.5 m
Therefore, you’d take a length of rope = 25.5 m. Attach that rope to a pole that is the center of the circle that will carve out 1 mile. Stretch out the rope that has been attached to the pole, and walk around in a circle. The path you carve out will equal 1 mile.
RENAISSANCE PHYSICS 