# Unit 6Measuring Circles, Angles, and Shapes

## Lesson 1

### Learning Focus

Find the perimeter and area of regular polygons.

### Lesson Summary

In this lesson, we developed a strategy for finding the perimeter and area of a regular polygon. We first inscribed the regular polygon in a circle so we could draw upon our knowledge of central angles, radii, and chords. By decomposing the regular polygon into right triangles, we could use trigonometric ratios to find the lengths that we needed to calculate perimeter and area.

## Lesson 2

### Learning Focus

Relate the circumference and area of circles to the perimeter and area of regular polygons.

### Lesson Summary

Previously, we have been given formulas for the circumference and area of a circle and have used them in application problems. In this lesson, we learned where these formulas came from and gained a better sense of why is such an important number, and also how it occurs naturally in the context of relating regular polygons to the circles in which they are inscribed.

## Lesson 3

### Learning Focus

Find formulas for arc length and area of a sector of a circle.

### Lesson Summary

In this lesson, we found a relationship between arc length and the area of a sector of a circle bounded by the arc and the two radii of the circle drawn to the endpoints of the arc.

## Lesson 4

### Learning Focus

Develop a new unit for measuring angles.

### Lesson Summary

In this lesson, we learned about a new unit for measuring angles called a radian. We developed this new unit of angle measurement by examining the ratio of arc length to radius for various central angles and for various distances from the center of the circle.

## Lesson 5

### Lesson Summary

In this lesson, we learned how to approximate the size of an angle measured in radians, and we learned the radian measure for some familiar angles measured in degrees, such as 90° and 180°. We also learned how to convert between degree and radian measures.

## Lesson 6

### Learning Focus

Use similarity to scale area and volume.

### Lesson Summary

In this lesson, we learned how area and volume scale if we scale the linear measures in a 3-D shape. Consequently, if we scale up a 3-D figure, we do not need to recalculate its surface area and volume using formulas for volume and area.

## Lesson 7

### Learning Focus

Derive and use formulas for right prisms and pyramids.

### Lesson Summary

In this lesson, we derived a formula for the volume of right prisms with non-rectangular bases, and a formula for the volume of pyramids. These formulas were derived by decomposing rectangular prisms in various ways and considering how the sum of the volumes of the resulting prisms or pyramids had to add up to the volume of the original rectangular prism.

## Lesson 8

### Learning Focus

Apply Cavalieri’s principle to 2-D and 3-D figures.

### Lesson Summary

In this lesson, we learned about Cavalieri’s principle, which can be used as a tool for deriving the volume formulas for oblique prisms, pyramids, cones, and spheres.

## Lesson 9

### Learning Focus

Find the sum of a geometric sequence.

### Lesson Summary

In this lesson, we learned how to find the sum of the terms in a geometric sequence without needing to add each term in the sequence one at a time.