Perimeter section of a circle?

Question

Lets say I have a pipe that has a diameter of 20 inches.
I the have a piece of flat bar that is 8 inches.
If I hold the 8 inch flat bar next to the pipe.
How do I determine how many inches of perimeter are equal to the 8 inch straight bar?

Answer 1

Your question is not well posed, but I'm going to restate it in the way that I'm guessing you're thinking.

Suppose you have a 20 inch diameter circle. Take two points on the circle and draw a straight line segment between them that is precisely 8 inches in length (this is termed a "secant", or a "chord"). What is now the arc-length of the segment of circle between those two points?

If that is your question, then here is how I work the problem:

You have defined an equilateral triangle, two edges of which are the radii of the circle, 10 inches each, and one which is the chord of 8 inches. We can now determine the ANGLE between the two 10 inch edges using basic trigonometry. The easiest way for me to think about this, offhand is to divide the equilateral triangle in half, making two right triangles. The hypotenuse is 10 inches, the opposite leg is 4 inches, and therefore the sin function of the angle is equal to 4/10. the angle is therefore (approximately) 23.6 degrees. Since we "trigged" a right triangle that was half of the original equilateral triangle, we need to double the result, yielding 47.2 degrees.

The angle between the two radii that subtend the arc who's chord is 8 inches (on a 20 inch circle) is 47.2 degrees.

47.2 degrees represents 47.2/360 of the circle (or 0.13 of the circle). Therefore the arc length of the segment in question is given by 0.13 x the circumfrence.

The circumfrence is pi times the diameter or 3.14 x 20 = 62.8 inches, therefore the arc length is given by 0.13 x 62.8 =
8.2 inches

The answer here isn't terribly surprising. The 20 inch diameter circle is quite large compared to the short segment of an 8 inch straight line. The "gentle curvature" of the circle compared to the line means that the arc length of the circular segment isn't going to be much longer than the straight line. If you took it to the extreme and compared the circular segment from a circle who's DIAMETER was about 8 inches, then the arc length would become nearly half the circle, 3.14 x 8 / 2 =12.5 inches. On the other hand, if the diameter of the circle got really really big (like the diameter of the earth for example) then you'd not be able to tell the difference between the straight 8 inch rod and the curve of the circle at all. The answer would approach exactly 8 inches.

I hope that answered your question!

Answer 2

The perimeter of the circular pipe is given by 2pi*r

For the pipe P = 2pi*10 = 62.831inches

I am not sure what you asking to find but the 8 inch bar would wrap around the pipe 62.831/8 times or 7.85 (3sf) times.

OR the bar is 8/62.831 of the perimeter. That is the bar is 0.127 (3sf) of the perimeter.

Answer 3

2 π R is the general formula for working out the perimeter of a circle. Where pi is 3.142 (to 3 decimal places) and R is the radius of the circle. The radius is the length of a line from circumference to the centre of the circle, it is half the diameter. The area of a circe is πR². Therefore if the radius is unknown the radius is the squareroot of the (area divided by 3.142.) Then you can use the other formula to work out the circumference.

Answer 4

Ce circumference of the circle is 20* pi = 20*3.1416=62.38 inches

The bar corresponds to 8/62.38 =0.128 of the circle