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### Involute of a Circle

```
Date: 01/22/99 at 10:44:30
From: EDWARD MALONE
Subject: Formula for the involute of a circle

I know how to construct the involute of a circle, but I can't find a
formula for it.
```

```
Date: 01/22/99 at 11:53:45
From: Doctor Anthony
Subject: Re: Formula for the involute of a circle

A string is wound around a circle and then unwound while being held
taut. The curve traced out by the point P at the end of the string is
called the involute of the circle. If the circle has radius r, the
circle centre is at the origin O, the string is tangent to the circle
at Q, and the initial position of P is (r,0), then the parameter,
theta, is chosen as angle between a radius line to Q and the x-axis.
The parametric equations of the involute are:

x = r[cos(theta) + (theta)sin(theta)]
y = r[sin(theta) - (theta)cos(theta)]

- Doctor Anthony, The Math Forum
http://mathforum.org/dr.math/
```

```
Date: 08/08/2001 at 11:19:37
From: Scott McConnehey
Subject: Involute Form on Splines and Gears

Dear Dr. Math:

I'm not a mathematician but normally I can use existing formulas and
equations effectively. I saw your formula for the involute of a circle
and tried to plug in some values but came up with what I viewed as
unrealistic numbers. In the formula X=r(cos(theta)+(theta)sin(theta)),
does (theta)sin(theta) mean angle times the sin of the angle? For
instance, if theta is 45 degrees, do I figure this as 45 degrees time
the sin of 45 degrees?

When I take this approach, I seem to get unrealistic numbers. In this
case, I know that the value for "X" can't be 32.5269 using a 1-inch
radius. Could you please tell me what I'm doing wrong or point me in the
right direction if there's a resource I can access that would make this a
little clearer for me?

Let me explain that I work in a navy calibration lab and I need to
derive a method for verifying the involute form on spline teeth.
Anything you can do to help would be greatly appreciated.

Thanks,
Scott.
```

```
Date: 08/08/2001 at 12:36:29
From: Doctor Rick
Subject: Re: Involute Form on Splines and Gears

Hi, Scott.

The angle theta in Dr. Anthony's response is assumed to be in radians.
The formula is derived in part from the length of the arc subtended
by a central angle theta on a circle of radius r. This arc length is
r*theta, but only if the angle is in radians. There are 2*pi radians in a
full circle (360 degrees), so that the "arc length" of a full circle is
2*pi*r, the circumference of the circle.

To convert degrees to radians, multiply by pi/180. Thus, 45 degrees
equals

45 * 3.1416 / 180 = 0.7854 radian

- Doctor Rick, The Math Forum
http://mathforum.org/dr.math/
```
Associated Topics:
High School Conic Sections/Circles
High School Geometry

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