1. Chapter 11: Areas of Polygons and Circles Sections 11.1 to 11.4

2.  Warm-Up: Find the missing side lengths (perimeter & area depend on these dimensions!) Section 11.1 - Areas of Parallelograms objective: we will find the perimeters and areas of parallelograms, and determine whether points on a plane define a parallelogram. Let’s start with…  How do we find the area of a rectangle? 12 y z 10 x b h A = bh

3. Dividing a rectangle gives… Looking at a rectangle we can determine the area of a triangle… how much of the rectangle? HALF So then, h b however, the height must be _____________________ to the base! How do we find the height (h) of this triangle given a side? b We can also call this height the __________ of the. h Trigonometry, Pythag. Thm. or Special triangles

4.  Draw the altitudes of a parallelogram: Focus back on a parallelogram h b  Moving a shaped section over still gives: A = bh  (remember not to use the slant length of the for the height!) Examples: 1) 7 ft 10.2 ft 2) 5 cm 6 cm 35.7ft 2

5.  average of bases Section 11.2 - Areas of Triangles, Trapezoids, and Rhombi objective: we will find the areas of trapezoids and rhombi.  Triangles:  Remember… b h h b2b2 b1b1  Trapezoids: also then, b h h Make 2 triangles: factor out ½ and h to get, or: leg

6.  Looking at a rhombus: Areas of Rhombi  Finding the perimeter of any polygon, we… find all the lengths of the sides and add them up Can we make triangles? h d1d1 b OR, = bh

7.  10) Examples to try (pg. 598) Find the area and perimeter of each parallelogram (round to nearest tenth). 16) 14) 12) 4 m 10 in. 15 in. 5.4 ft. 4.2 ft. 12 cm. 4 cm. 15 cm. 7 cm. 8 cm. 3 cm. Ans.: 13.7m, 8m 2 50”,106.1 in 2 19.2’,22.7ft 2 202cm 2

8. Pg. 598 #22 J(-1, -4), K(4, -4), L(6, 6), M(1, 6) area Determine whether this is a square, rectangle, or a parallelogram, then find the area. Ans.: parallelogram; 50 sq. units

9. Examples to try (pg. 606) Find the area of each figure & round to the nearest tenth. 18) rhombus 17 cm 16) 12 cm 8.5 yd 14.2 yd 8.5 yd 17 cm 12 cm Ans.: 96.5yd 2 ; 408cm 2 20) 6 in 21 in 4 in 18 in Ans.: 99 in 2

10. 28) Find the area of rhombus JKLM given the coordinates of the vertices. J(-1, -4), K(2, 2), L(5, -4), M(2, -10) 30) Trapezoid ABCD has an area of 750 m 2. Find the height of ABCD. 24) Find the area of trapezoid PQRT given the coordinates of the vertices. P(-3, 8), Q(6, 8), R(6, 2), T(1, 2) 35 m 25 m A D C B Ans.: 25 mAns.: 36 units 2 Ans.: 42 units 2

11. A regular octagon with a perimeter of 96 m has sides of what length? Section 11.3 – Area of Reg. Polygons & Circles objective: we will learn how to find the areas of circles and regular polygons  Warm-Up Question: Areas of REGULAR polygons  Let’s break the shape into triangles again. (= sides ; = angles) (P = perimeter; n = number of sides) s a

12. Example (step by step) Pg. 611 #1 Find the area of a REGULAR pentagon with a perimeter of 40 cm. s a 110 cm 2 To find the area we need the apothem (a). 1. Draw a smaller triangle and label. 2. Find the angle. 3. Label side opposite the angle with half of a side length. 4. Use trigonometry to find apothem. a

13. 2. What is the area of a decagon with an apothem length of 2”? 1. What is the area of an equilateral triangle with sides equal to ½ mile?.108 mi 2 Examples 2

14. Circles Diameter: distance from a point on the perimeter through the center to another point on the perimeter Center Radius: distance from center to perimeter Circumference: length around outside of circle (perimeter) 1.) Find the area of a circle with a radius of 4.5 cm. 63.6 cm 2 2.) A circle has a circumference of 10. Find its area. 78.5 units 2

15. An irregular (or composite) figure is a figure that cannot be classified into the specific shapes we’ve studied : (triangle, parallelogram, trapezoid, rhombus, rectangle, circle, etc.) Section 11.4 - Areas of Irregular Figures objective: we will learn how to find the areas of irregular figures (on and off the coordinate plane) But, can be separated into the shapes we know! Use auxiliary lines to divide up the shape into non-overlapping parts Find the areas of each Add them together Let’s practice some….

16. 19 6 22 6 6 Example p. 617 ex #1 Find the area. 3 3 area of irreg. figure = area of - area of + area of semi- 112.5 sq. units 6 6 type of triangle?

17. Examples p. 618 ex #3; p.613 # 6 T(1, 10) R(-5, 0) U(3, 7) S(-3, 7) V(3, 0) 58 sq. units Find the area of figure RSTUV. Find the area of the shaded part of the equilateral triangle. 3