# Perimeter of Polygons with Inscribed Circles Worksheets

An inscribed polygon is a many-sided object that is orientated in such a way that all of the vertices are found on a circle. Because of this unique situation there also a number of features that come to light when this takes place. The shape would therefore appear to be inside the circle. It can also be said that the circle is circumscribed about the polygon. Both shapes are positioned in such a fashion that the circumcenter of the polygon is also the center of the circle. In order for it to fit, if a parallelogram becomes inscribed it must be a rectangle. Also if a quadrilateral becomes inscribed in a circle, the opposite angles that are formed are supplementary. 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.

### Aligned Standard: High School Geometry - HSG-C.A.3

- 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.

- Answer Keys - These are for all the unlocked materials above.

### 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.

- Quiz 1 - See if you can print this at a slightly higher resolution.
- Quiz 2 - Finding and determining the halfway point is helpful here.
- Quiz 3 - I wish we could spend even more time here with this.

### How to Find the Perimeter of Polygons with Inscribed Circles

The common polygon that is found when dealing with this type of situation is a triangle with circle that is inscribed. The perimeter is the sum of all the lengths of the sides of the polygon. To begin to break this down take in all the information that is provided for you. After that infer as much as you can about this system.

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 360 degrees, n number of sides divides the circle with 360/n.

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] or 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).