# Chapter 10: Perimeter and Area

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

## Introduction

Now that we have explored triangles, quadrilaterals, polygons, and circles, we are going to learn how to find the perimeter and area of each. First we will derive each formula and then apply them to different types of polygons and circles. In addition, we will explore the properties of similar polygons, their perimeters and their areas.

- 10.1.
## Area and Perimeter of Rectangles

- 10.2.
## Area of a Parallelogram

- 10.3.
## Area and Perimeter of Triangles

- 10.4.
## Area of Composite Shapes

- 10.5.
## Area and Perimeter of Trapezoids

- 10.6.
## Area and Perimeter of Rhombuses and Kites

- 10.7.
## Area and Perimeter of Similar Polygons

- 10.8.
## Circumference

- 10.9.
## Arc Length

- 10.10.
## Area of a Circle

- 10.11.
## Area of Sectors and Segments

- 10.12.
## Area of Regular Polygons

### Chapter Summary

## Summary

This chapter covers perimeter and area of all the basic geometric figures. Perimeter and area are compared and calculated for rectangles, parallelograms, triangles, and then for composite shapes of those figures. The chapter then branches into perimeter and area for other special geometric figures, namely trapezoids, rhombuses, and kites, as well as similar polygons. The chapter wraps up with the circumference of circles and arc length followed by the area of a circle and the area of sectors and segments.

### Chapter Keywords

- Perimeter
- Area of a Rectangle: \begin{align*}A=bh\end{align*}
- Perimeter of a Rectangle \begin{align*}P=2b+2h\end{align*}
- Perimeter of a Square: \begin{align*}P=4s\end{align*}
- Area of a Square: \begin{align*}A=s^2\end{align*}
- Congruent Areas Postulate
- Area Addition Postulate
- Area of a Parallelogram: \begin{align*}A=bh\end{align*}.
- Area of a Triangle: \begin{align*}A= \frac{1}{2} bh\end{align*} or \begin{align*}A=\frac{bh}{2}\end{align*}
- Area of a Trapezoid: \begin{align*}A=\frac{1}{2} h(b_1+b_2)\end{align*}
- Area of a Rhombus: \begin{align*}A=\frac{1}{2} d_1 d_2\end{align*}
- Area of a Kite: \begin{align*}A=\frac{1}{2} d_1 d_2\end{align*}
- Area of Similar Polygons Theorem
- \begin{align*}\pi\end{align*}
- Circumference: \begin{align*}C=\pi d\end{align*} or \begin{align*}C=2 \pi r\end{align*}
- Arc Length
- Arc Length Formula: length of \begin{align*}\widehat{AB}=\frac{m \widehat{AB}}{360^\circ} \cdot \pi d\end{align*} or \begin{align*}\frac{m \widehat{AB}}{360^\circ} \cdot 2 \pi r\end{align*}
- Area of a Circle: \begin{align*}A=\pi r^2\end{align*}
- Sector of a Circle
- Area of a Sector: \begin{align*}A=\frac{m \widehat{AB}}{360^\circ} \cdot \pi r^2\end{align*}
- Segment of a Circle
- Perimeter of a Regular Polygon: \begin{align*}P=ns\end{align*}
- Apothem
- Area of a Regular Polygon: \begin{align*}A=\frac{1}{2} asn\end{align*} or \begin{align*}A=\frac{1}{2} aP\end{align*}

### Chapter Review

Find the area and perimeter of the following figures. Round your answers to the nearest hundredth.

- square
- rectangle
- rhombus
- regular pentagon
- parallelogram
- regular dodecagon

Find the area of the following figures. Leave your answers in simplest radical form.

- triangle
- kite
- isosceles trapezoid
- Find the area and circumference of a circle with radius 17.
- Find the area and circumference of a circle with diameter 30.
- Two similar rectangles have a scale factor \begin{align*}\frac{4}{3}\end{align*}. If the area of the larger rectangle is \begin{align*}96 \ units^2\end{align*}, find the area of the smaller rectangle.

Find the area of the following figures. Round your answers to the nearest hundredth.

- find the shaded area (figure is a rhombus)

### Texas Instruments Resources

*In the CK-12 Texas Instruments Geometry FlexBook, there are graphing calculator activities designed to supplement the objectives for some of the lessons in this chapter. See http://www.ck12.org/flexr/chapter/9695.*