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### Circle Inscribed in Sector

```
Date: 05/20/99 at 17:38:29
From: Chris Wipke
Subject: Geometry

The arc of a sector has degree measure 60 degrees. The radius of the
sector is 12 inches. Find the area of the circle that can be inscribed
in the sector.

I have drawn a diagram of a sector whose arc measure is 60 degrees,
and have constructed a line perpendicular to the lower segment forming
the arc. I do not know what to do from here, or if I am right so far.
Please provide me the way to figure out the problem, along with the
```

```
Date: 05/21/99 at 13:04:25
From: Doctor Peterson
Subject: Re: Geometry

Hi, Chris. Interesting problem!

Here's how I understand your picture:

**
/  ****
/  +++++**
+++       +++
+            *+
+       +       +
/+               +*
/  +             + *
/60  +++       +++   *
*---------+++++-------*
12

I would draw in a couple lines: a radius from the center of the
inscribed circle to its point of tangency with a radius of the sector,
and a radius of the sector passing through the center of the
inscribed circle.

A
**
/  ****
/  +++++**  C
+++       +++
+      D  /r *+
+       +       +
/+    /  |       +*
/  +/     |r     + *
/ /  +++   |   +++   *
*---------+++++-------*
O            E          B

Now because the circle is tangent to the arc, OC is a radius of the
arc and has length 12. OD, the hypotenuse of a right triangle, has
length 12-r. But triangle ODE is a 30-60-90 triangle, so we know this
hypotenuse is exactly twice r. This will give you an equation you can
solve for r.

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

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