Lesson 8.1 Meaning of Area pp. 310-315

Objectives: 1. To define and illustrate area concepts and postulates. 2. To prove and apply the area formula for rectangular regions. 3. To apply the Area Addition Postulate to regions reducible to rectangular regions.

What is area?

What is the difference between area and perimeter?

What kind of units are used to measure area?

Definition The area of a region is the number of square units needed to cover it completely.

Postulate 8.1 Area Postulate. Every region has an area given by a unique positive real number.

Postulate 8.2 Congruent Regions Postulate. Congruent regions have the same area.

Postulate 8.3 Area of Square Postulate. The area of a square is the square of the length of one side: A = s2.

EXAMPLE 1 Find the area of square ABCD. A = s2 A = (4)2 A = 16 sq. units

Find the area of a rectangle that measures 4 ft. wide by 7 ft. long. 5 7 6 8 9 11 10 12 13 15 14 16 17 19 18 20 21 23 22 24 25 27 26 28 3 2 1

Postulate 8.4 Area Addition Postulate. If the interiors of two regions do not intersect, then the area of the union is the sum of their areas.

Theorem 8.1 The area of a rectangle is the product of its base and height: A = bh.

Practice: Find the area of a rectangle with b = 6 and h = 8.

Practice: Find the area of a rectangle with b = 2 and h = 5. 1. 10 2. 7 3. 10 4. 2 2 + 5

Practice: Find the area of a rectangle with b = x + 2 and h = x - 2. 4. x2 - 4x - 4

EXAMPLE 2 Find the area of the given polygonal region. 10 A=2(8) = 16 A = 8(2) = 16 2 16 4 +16 52 4 4 A=4(1)=4 1 8 3 8 A = 8(2) = 16 2 10

Find the area of the region in the figure. 10m A = 9(10) 90 A = 3(7) -21 69 m² 9m 7m 5m 2m

Find the area of the region in the figure. 10m A=3(2) =6 45 A=5(9) =45 A=2(9) =18 6 +18 9m 7m 69 m² 5m 2m

Find the area of the region in the figure. 4 4 A=4•4 =16 A=4•4 =16 4 2 2 A=2•4 =8 A=2•4 =8 12 4 12 7 A=4•11 =44 11

Find the area of the region in the figure. =16 A=2•4 =8 A=4•11 =44 16 8 +44 92

Find the area of the region in the figure. 4 4 4 132 2 2 A=11(12) =132 12 4 12 7 11

Find the area of the region in the figure. 4 4 A=3(4) =12 4 132 -12 -28 2 2 A=7(4) =28 92 12 4 12 7 11

Find the area. 12 120 A=12(10) =120 10 4 6 3 6 3

Find the area. 12 120 -18 10 4 A=3(6) =18 6 3 6 3

Find the area. 12 120 -18 -12 10 90 A=3(4) =12 4 6 3 6 3

12 10 4 6 3 6 3 Find the area. 12 A=12(4) =48 12 48 +18 A=3(4) =12 90 =18 6 3 A=6(2) =12 6 3

Homework pp. 314-315

►A. Exercises Complete the following tables. Square with side s s A 1. 15 yd. 3. 12 cm 5. 32.49 sq. m

►A. Exercises Complete the following tables. Rectangles b h A 7. 8 ft. 9 ft. 9. 2.4 cm 10.9 cm 11. 32 yd. 175 sq. yd.

►A. Exercises Find the area of each polygonal region. 13. 2 7 3 4 12

►A. Exercises Find the area of each polygonal region. 15. 9 13 3

►B. Exercises 19. How many 8-inch square tiles would be needed to cover the floor of a room that is 12 by 15 feet?

►B. Exercises Find the area of each rectangle. 21. b = 5 h = 7

►B. Exercises Find the area of each rectangle. 23. b = x + 7 h = x - 7

►C. Exercises 25. The inner square has its vertices at the midpoint of the sides of the outer square. Prove that the area of the outer square is double the area of the inner square.

►C. Exercises 25. b Ainner = a2 Aouter = 4b2 a a2 = b2 + b2 = 2b2 Aouter = 2Ainner

26. Rectangular region of exercise 19 ■ Cumulative Review Find the perimeter of each region. 26. Rectangular region of exercise 19 How many 8-inch square tiles would be needed to cover the floor of a room that is 12 by 15 feet?

27. Polygonal region of exercise 14 ■ Cumulative Review Find the perimeter of each region. 27. Polygonal region of exercise 14 3 12 8 5 19 7 27

28. Circular region with diameter of 3 in. ■ Cumulative Review Find the perimeter of each region. 28. Circular region with diameter of 3 in.

29. Give bounds for the measure of angle x. ■ Cumulative Review 29. Give bounds for the measure of angle x. 120° x

■ Cumulative Review 30. Give bounds for s. 5 6 s