# Inscribed Polygons in Circles Worksheets

An inscribed angle is formed by having its vertex found from within in touching a circle. The rays of the angles form chords within the circle. If two angles that are inscribed within a circle share starting points, but their vertex is in then center of the circle then the second angle is twice that of the inscribed angle. Using these foundational thoughts, when given information about one of the shapes you can learn a great deal about the other. Students often ask why we learn this and do all this, what seems like unnecessary math. This is a daily, minute-by-minute problem for the super computers of companies like Amazon, Walmart Online. They are constantly evaluating the best type of packaging to put your items in as they ship them to you. These worksheets and lessons with teach you how to determine the proper dimensions of various inscribed polygons.

### Aligned Standard: High School Geometry - HSG-CO.D.13

- Hexagons Step-by-step Lesson - How big of a hexagon can you fit into that circle?
- Guided Lesson - Not only do we shove shapes into circles, but we also ask you to jam a circle into a triangle.
- Guided Lesson Explanation - I'm always wondering if this reminds people of the round peg into the square hole.
- Practice Worksheet - At first look, these problems should fly by, but they take a good chunk of time.
- Matching Worksheet - I'm on a roll here with Geometry, another easy to sort through answer set.

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

### Homework Sheets

I love to see the faces of kids when they see these problems for the first time. The skill looks unattainable, at first.

- Homework 1 - Mark a point anywhere on the circle. This will be the first vertex of the hexagon.
- Homework 2 - Start by constructing a circle. Construct a diagonal through this circle. Extend the diagonal beyond the circumference of the circle. Construct a perpendicular line to this line through the center.
- Homework 3 - Increase the compasses width, just a bit. Place the compass point where each arc crosses the side and draw two crossing arcs inside the triangle.

### Practice Worksheets

Bisectors and circle geometry should have been mastered before students move on to these.

- Practice 1 - Construct the largest regular hexagon that will fit in the circle below.
- Practice 2 - Construct a regular pentagon that will fit in the circle below.
- Practice 3 - Draw the largest possible circle inscribed in this equilateral triangle.

### Math Skill Quizzes

These quizzes are more artistic than anything else we have seen in this content area.

- Quiz 1 - Construct a regular square that will fit in the circle below.
- Quiz 2 - Get yourself a ruler and draw a line from the vertex of the triangle to the point where the two arcs cross.
- Quiz 3 - Construct a dodecagon that will fit in the circle below.

### What are Inscribed Shapes?

Do you like eggs? Then you surely know what they look like. Delicious and yummy with an outer white part and a yellow circular yolk fitting in the center. Do you know the geometrical term for figures like eggs? They are called inscribed shapes. Let's see what the term means. Any planar shape that fits into another shape is called an inscribed shape. In the case of the egg, the outside shape is the eggshell and the inside (yolk and egg white) would be the inscribed shapes.

To understand better, take a piece of paper and draw whichever shape you like. Triangle, square, rectangle, or any other shape. Next, draw another shape inside the shape you first drew such that it perfectly fits inside it. Now take a look. Can you see how one shaping is holding another inside of it? This image in front of you is now called an inscribed shape.

Using various geometric theorems if we have some information about either of the shapes, we can learn a great deal about the other shapes. While you may not quickly see how this could ever be used in the real world, think about packaging or boxing something. Isnâ€™t that box from Amazon just placing inscribed shapes within it? Would it not be great to save having to use unnecessary larger boxes or packaging as a result of knowing what is about to be placed inside it? Think of all the waste you may have saved.