## Perimeter of Polygons with Inscribed Circles

#### Aligned To Common Core Standard:

**Circles** - HSG-C.A.3

How to find the Perimeter of Polygons with Inscribed Circles?
The method given below solves the problem for any polygon inscribed in a circle, given that each side touches the circle. Therefore, the distance between the middle point of the polygon and the circle's surface is equal to the radius. Not to mention that a polygon, with n number of sides, divides the circle into n parts. In other words, from a total 360o, n number of sides divides the circle with 360/n
In this example, let's consider an octagon with 8 sides in a circle.
Let us also consider the law of cosines, which applies to any triangle. The value of a and b is equal to r, while the law of cosines is as follows. The value of Θ is 360/π
c^{2}= a^{2}+b^{2}-2ab cosΘ Let us derive this equation in the context of our example: c^{2}= r^{2}+r^{2} - 2(r)(r) cos [360/π]
c^{2}=r^{2} (2 - 2 cos [360/π)]
Take square root on both sides to get a single value of c. c=r √ ( 2 - 2 cos [360/π]) This equation above is used to find out the perimeter of any polygon. However, you need to multiply the answer with the number of sides (n). These worksheets and lessons help students learn a great deal of information about the incircle of a regular polygon and how to use that information your advantage.

### Printable Worksheets And Lessons

- Finding Missing Sides Step-by-step
Lesson - Every time I see this with triangles I ask myself,
"Couldn't we just use the Pythagorean theorem?"

- Guided Lesson
- I finish this one out with triangles this helps kids gain confidence.

- Guided Lesson
Explanation - Sorry for the six pages to print, but you will
find it seriously helpful I you don't have time to spend going through
it.

- Practice Worksheet
- More crazy triangles to help solve.

- Matching Worksheet
- Match the lengths to triangle sides that are in the diagrams.

- Perimeter of Polygons & Circumference
of Circles Worksheet Five Pack - These are like gym exercises;
consistently painful if you don't pay attention to detail.

#### Homework Sheets

There has got to be engineer somewhere right now who's only job is to solve problems like this. It probably pays well.

- Homework 1 - HI and HG are tangent to the inscribed circle from H so, HI is congruent to HG.
- Homework 2 - YX and YZ are tangent to the inscribed circle from Y. so, YX is congruent to YZ. YX = YZ = 1.
- Homework 3 - If you can find one side, you can find them all.

#### Practice Worksheets

We work more on these midpoint based problems.

- Practice 1 - What is BC?
- Practice 2 - We know that HI and HJ. We have use additive property of length to write an equation and find IJ.
- Practice 3 - We have to use the additive property of length to write an equation and find missing lengths.

#### Math Skill Quizzes

The problems here are a good size so that everyone can see them clearly.