Geometry

1. Geometry Day 78

2. Today’s Agenda • Area • Rectangles • Parallelograms • Triangles • Trapezoids • Kites/Rhombi • Circles/Sectors • Irregular Figures • Regular Polygons

3. Area • Area is a measurement that describes the amount of space a figure occupies in a plane. • Area is a two-dimensional measurement. It is measured in square units. • Area Addition Postulate – The area of a region is the sum of the areas of its non-overlapping parts.

4. Area • Area problems will often refer to the base and height of a figure. Typically (but not always), any side of a figure can act as a base. • The height must always be perpendicular to the base! The height will typically not be a side of a figure.

5. Area of a Rectangle • The area of a rectangle is the length of its base times the length of its height. • A = bh HEIGHT BASE

6. Examples • Find the areas of the following rectangles: 5 4 12 ½

7. Area of a Parallelogram • The area of a parallelogram is the length of its base times the length of its height. • A = bh • Why? • Any parallelogram can be redrawn as a rectangle without losing area. BASE HEIGHT

8. Examples • Find the areas of the following parallelograms: 10 7 12 5 8 6

9. Area of a Triangle • The area of a triangle is one-half of the length of its base times the length of its height. • A = ½bh • Why? • Any triangle can be doubled to make a parallelogram. HEIGHT BASE

10. Examples • Find the areas of the following triangles: 8 7 13 12 5

11. Area of a Trapezoid • Remember for a trapezoid, there are two parallel sides, and they are both bases. • The area of a trapezoid is the length of its height times one-half of the sum of the lengths of the bases. • A = ½(b1 + b2)h • Why? • Red Triangle = ½ b1h • Blue Triangle = ½ b2h • Any trapezoid can be divided into 2 triangles. BASE 1 HEIGHT BASE 2

12. Examples • Find the areas of the following trapezoids: 10 15 15 7 10 12 20

13. Area of a Kite/Rhombus • The area of a kite is related to its diagonals. • Every kite can be divided into two congruent triangles. • The base of each triangleis one of the diagonals.The height is half of theother one. • A = 2(½•½d1d2) • A = ½d1d2 d1 d2

14. Area of a Rhombus • Remember that a rhombus is a type of kite, so the same formula applies. • A = ½d1d2 • A rhombus is also a parallelogram, so its formula can apply as well. • A = bh

15. Area of a Circle/Sector • Recall the area of a circle: • A = πr2 • Page 782 shows how a circle can be dissected and rearranged to resemble a parallelogram, and how the above formula can be derived. • Recall that the area of a sector is a proportion of the area of the whole circle: • or

16. Area of Irregular Figures • A composite figure can separated into regions that are basic figures. • Add auxiliary lines to divide the figure into smaller sub-figures. • Look to form rectangles, triangles, trapezoids, circles, and sectors. • Find the area of each sub-shape. • Add the sub-areas together to find the area of the whole figure. • Sometimes you may have to subtract pieces

17. EXAMPLE 10 9  3 = 27 3 8  3 = 24 4 1 10  12 = 120 3 9 Total Area = 27 + 24 + 120 = 171 Sq. Units

18. Example 2  4 = 8 12 8  8 = 64 2 ½  4  8 = 16 4 6 12 Total Area = 8 + 64 + 16 = 88 Sq. Units

19. Another Way To Solve… ½  4  8 = 16 12 12  8 = 96 2 4  6 = 24 4 6 12 Total Area = 16 + 96 – 24 = 88 Sq. Units

20. Area of a Regular Polygon • Because a regular polygon has unique properties, you only need a little bit of information to find the area. • The basic idea is to dissect the figure as we did before. However, with a regular polygon, we can divide it into congruent isosceles triangles.

21. Area of a Regular Polygon • What is the relationship to the number of sides of the polygon and the number of triangles you can draw from the center? • So to find the area of the polygon, we find the area of one of these triangles, and multiply by the number of sides.

22. A few definintions • The segment that connects the center of a regular polygon to one of its vertices is called the radius. • This is also a radius of the polygon’s circumscribed circle.

23. A few definintions • The segment that connects the center of a regular polygon to the midpoint of one of its sides is the apothem. • The apothem will be perpendicular to that side. • This is also a radius of the polygon’s inscribed circle.

24. Apothem and area • The apothem also is the height of one of the congruent triangles we drew when dividing the figure up. • So, if we know the height and base of the triangle, we can find its area, and then we multiply by the number of triangles.

25. Apothem and area • To put it in terms of the polygon, if we know the length of a side (s) and the apothem (a), and the number of sides (n), then the area would be: A = (½as)n • What would be another way to express s•n? A = ½ap

26. Example • Find the area of the following regular octagon: 12 cm 14.5 cm

27. Example • What if we don’t know the apothem? • Is there a way we can calculate it? • TRIG!!! • Find the area. 22 cm 14.5 cm

28. Other Triangle Formulas • Equilateral Triangle • An equilateral triangle with side s can be divided into two 30-60-90 triangles. • Using the special right triangle ratios, we can represent the height in terms of s. • Substituting into the formulaA = ½bh… s s ½ s s

29. Other Triangle Formulas • SAS Triangle • If we know two sides and an included angle of any triangle, we can use trig to find the area. • Drawing the altitude creates a right triangle, of which we know the hypotenuse and angle. • Substituting into A = ½bh: b h C a

30. Other Triangle Formulas • Heron’s Formula (SSS) • There’s a formula for calculating the area of a triangle if you know the three sides. • s in the above formula represents thesemi-perimeter, which half of theperimeter c b a

31. Assignments • Homework 46 • Workbook, pp. 140, 142 • Homework 47 • Workbook, pp. 144, 145