Geometry

GeometryGeometry

Exploring Area and PerimeterExploring Area and Perimeter

What You'll LearnWhat You'll LearnWhy You Should Learn ItWhy You Should Learn It

 How to find the perimeter of a polygon How to find the perimeter of a polygon

 How to find the area of a square and How to find the area of a square and rectangle rectangle

You can use perimeters and areas to You can use perimeters and areas to solve real-life problems, such as finding solve real-life problems, such as finding the amount of fertilizer to cover a lawnthe amount of fertilizer to cover a lawn

PerimeterPerimeter

 The The perimeterperimeter of a polygon is the sum of the of a polygon is the sum of the length of its sides length of its sides – Linear Measurement (cm, m, in, ft, mi, etc.)Linear Measurement (cm, m, in, ft, mi, etc.)

Example 1Example 1

 Use a compass and Use a compass and straightedge to construct straightedge to construct a segment whose length a segment whose length is equal to the perimeter is equal to the perimeter of PQRSof PQRS

Example 1Example 1

 Draw a long segment Draw a long segment

 Use your compass to Use your compass to measure each segment measure each segment

P Q R S P

PQ + QR + RS + SP = Perimeter of PQRS

Areas of Squares & RectanglesAreas of Squares & Rectangles

Area of a polygon is the number of square Area of a polygon is the number of square units contained in its interior units contained in its interior   

Areas are measured in square units such Areas are measured in square units such as mas m22 or ft or ft22

3

3

1 2 3

4 5 6

7 8 9

9 units2

All formulas for areas of polygons All formulas for areas of polygons based on these 3 postulatesbased on these 3 postulates

Postulate 22Postulate 22 - - Area of a Square Area of a Square PostulatePostulate: The area of a square is the : The area of a square is the square of the length of its side, or A = ssquare of the length of its side, or A = s2 2

Postulate 23Postulate 23 – – Area Congruence Area Congruence PostulatePostulate: If two polygons are congruent, : If two polygons are congruent, then they have the same area then they have the same area Postulate 24Postulate 24 – Area Addition Postulate: – Area Addition Postulate: The area of a region is the sum of the areas The area of a region is the sum of the areas of all its non-overlapping partsof all its non-overlapping parts

Theorem 11.1Theorem 11.1

 Area of a Rectangle Theorem: Area of a Rectangle Theorem:

 The area of a rectangle is the product of The area of a rectangle is the product of its base and height, or A = bh its base and height, or A = bh – The base and height meet at right anglesThe base and height meet at right angles

h

b

Example 2Example 2

 Each of the three rectangles has a height of 6 Each of the three rectangles has a height of 6 units and a base of 4 units. Each is composed of units and a base of 4 units. Each is composed of congruent polygons. Find the area of each congruent polygons. Find the area of each polygonpolygon

Example 2Example 2

 Each of the three rectangles has a height of 6 Each of the three rectangles has a height of 6 units and a base of 4 units. Each is composed of units and a base of 4 units. Each is composed of congruent polygons. Find the area of each congruent polygons. Find the area of each polygonpolygon

11

Twenty-four 1 unit by 1 unit squares

1 Square Unit

Twelve 1 unit by 2 unit squares

12

2 Square Units

Six congruent hexagons 24÷6 =4

4 Square Units

Example 3Example 3Comparing Perimeters and Areas of SquaresComparing Perimeters and Areas of Squares

 Find a pattern for the perimeters and a pattern Find a pattern for the perimeters and a pattern for the areas of the following squares. Graph for the areas of the following squares. Graph your results and find functions that give the your results and find functions that give the perimeter and area in terms of the side lengthsperimeter and area in terms of the side lengths

Example 3Example 3

 Begin by making a table that shows the Begin by making a table that shows the perimeter and area of each squareperimeter and area of each square

Side length, s

1 2 3 4 5

Perimeter, P

Area, A

 Begin by making a table that shows the Begin by making a table that shows the perimeter and area of each squareperimeter and area of each square

Side length, s

1 2 3 4 5

Perimeter, P

4 8 12 16 20

Area, A 1 4 9 16 25

Example 3Example 3

P = 4s A = s2