# Chapter 8: Circles

**At Grade**Created by: CK-12

Here you will start by connecting circles and similarity and justifying why all circles are similar. You will use the fact that all circles are similar to derive the area and circumference formulas for circles. Then, you will investigate different angles, shapes, and lines that are related to circles including central angles, inscribed angles, inscribed shapes, tangent lines, and secant lines. Finally, you will look at portions of circles and calculate the length of arcs and the area of sectors.

- 8.1.
## Circles and Similarity

- 8.2.
## Area and Circumference of Circles

- 8.3.
## Central Angles and Chords

- 8.4.
## Inscribed Angles

- 8.5.
## Inscribed and Circumscribed Circles of Triangles

- 8.6.
## Quadrilaterals Inscribed in Circles

- 8.7.
## Tangent Lines to Circles

- 8.8.
## Secant Lines to Circles

- 8.9.
## Arc Length

- 8.10.
## Sector Area

### Chapter Summary

You saw that you can use a similarity transformation to show that all circles are similar. You learned that you can use the fact that circles are similar to derive the area and circumference formulas for circles. You learned about central and inscribed angles and that central angles are the same measure as the arcs they intercept while inscribed angles are half the measure of the arcs they intercept. You learned that a circle is *inscribed in a shape* if it is *inside* the shape and tangent to all sides of the shape, while a circle is *circumscribed about a shape* if it is outside the shape and passes through all vertices of the shape.

You learned the difference between tangent and secant lines and the relationships between the arcs, angles, and segments created by these lines. Finally, you considered portions of circles. You learned that arc length is another way to describe the size of an arc. You saw that the concept of arc length leads to radians, a new way of measuring angles. You also learned how to find the area of a sector of a circle based on its radius and the measure of its central angle.

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