Day 78

Today’s AgendaArea

RectanglesParallelogramsTrianglesTrapezoidsKites/RhombiCircles/SectorsIrregular FiguresRegular Polygons

AreaArea

• AreaArea is a measurement that describes is a measurement that describes the amount of space a figure occupies in the amount of space a figure occupies in a plane.a plane.

• Area is a two-dimensional measurement. Area is a two-dimensional measurement. It is measured in square units. It is measured in square units.

• Area Addition PostulateArea Addition Postulate – The area of a – The area of a region is the sum of the areas of its non-region is the sum of the areas of its non-overlapping parts.overlapping parts.

AreaArea

• Area problems will often refer to the Area problems will often refer to the basebase and and heightheight of a figure. Typically of a figure. Typically (but not always), any side of a figure (but not always), any side of a figure can act as a base.can act as a base.

• The height must always be The height must always be perpendicular to the base!perpendicular to the base! The The height will typically height will typically notnot be a side of a be a side of a figure.figure.

Area of a RectangleArea of a Rectangle

• The area of a rectangle is the length The area of a rectangle is the length of its base times the length of its of its base times the length of its height.height.• A = bhA = bh

BASEBASE

HEIGHHEIGHTT

ExamplesExamples

• Find the areas of the following Find the areas of the following rectangles:rectangles:

1212

5544

½½

• The area of a parallelogram is the length The area of a parallelogram is the length of its base times the length of its height.of its base times the length of its height.• A = bhA = bh

• Why?Why?

• Any parallelogram can be redrawn as a Any parallelogram can be redrawn as a rectangle without losing area.rectangle without losing area.

Area of a ParallelogramArea of a Parallelogram

BASEBASE

HEIGHHEIGHTT

ExamplesExamples

• Find the areas of the following Find the areas of the following parallelograms:parallelograms:

1212

66

5577

1010

88

Area of a TriangleArea of a Triangle• The area of a triangle is one-half of the The area of a triangle is one-half of the

length of its base times the length of its length of its base times the length of its height.height.• A = A = ½bh½bh

• Why?Why?

• Any triangle can be doubled to make a Any triangle can be doubled to make a parallelogram.parallelogram.

BASEBASE

HEIGHHEIGHTT

ExamplesExamples

• Find the areas of the following Find the areas of the following triangles:triangles:

1313

55

77

1212

88

Area of a TrapezoidArea of a Trapezoid• Remember for a trapezoid, there are two Remember for a trapezoid, there are two

parallel sides, and they are both parallel sides, and they are both basesbases..• The area of a trapezoid is the length of its The area of a trapezoid is the length of its

height times one-half of the sum of the height times one-half of the sum of the lengths of the bases.lengths of the bases.• A = A = ½(b½(b11 + b + b22)h)h

• Why?Why?

• Red Triangle = ½ bRed Triangle = ½ b11hh

• Blue Triangle = ½ bBlue Triangle = ½ b22hh• Any trapezoid can be Any trapezoid can be

divided into 2 triangles.divided into 2 triangles.

HEIGHHEIGHTT

BASE 1BASE 1

BASE 2BASE 2

ExamplesExamples

• Find the areas of the following Find the areas of the following trapezoids:trapezoids:1010

77

1212

1010

2020

1515

1515

Area of a Kite/RhombusArea of a Kite/Rhombus

• The area of a kite is related to its diagonals.The area of a kite is related to its diagonals.

• Every kite can be divided into two congruent Every kite can be divided into two congruent triangles.triangles.

• The base of each triangleThe base of each triangleis one of the diagonals.is one of the diagonals.The height is half of theThe height is half of theother one.other one.

• A = 2(½A = 2(½••½d½d11dd22))

• A = ½dA = ½d11dd22

dd11

dd22

Area of a RhombusArea of a Rhombus

• Remember that a rhombus is a type Remember that a rhombus is a type of kite, so the same formula applies.of kite, so the same formula applies.• A = ½dA = ½d11dd22

• A rhombus is also a parallelogram, so A rhombus is also a parallelogram, so its formula can apply as well.its formula can apply as well.• A = bhA = bh

Area of a Circle/SectorArea of a Circle/Sector

• Recall the area of a circle:Recall the area of a circle:• A = A = ππrr22

• Page 782 shows how a circle can be Page 782 shows how a circle can be dissected and rearranged to resemble a dissected and rearranged to resemble a parallelogram, and how the above parallelogram, and how the above formula can be derived.formula can be derived.

• Recall that the area of a sector is a Recall that the area of a sector is a proportion of the area of the whole circle:proportion of the area of the whole circle:

• oror3602

x

r

A

2

360r

xA

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

EXAMPLE

3

3

9

1

4

109 3 = 278 3 = 2410 12 = 120

Total Area =

27 + 24 + 120 =

171 Sq. Units

Example

2

4

6

12

12

2 4 = 8

8 8 = 64½ 4 8 =

16

Total Area =

8 + 64 + 16 =

88 Sq. Units

Another Way To Solve…

2

4

6

12

12

12 8 = 96

½ 4 8 = 16

Total Area =

16 + 96 – 24 =

88 Sq. Units

4 6 = 24

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.

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.

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.

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.

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.

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

Find the area of the following regular octagon:

12 cm

14.5 cm

What if we don’t know the apothem? Is there a way we can calculate it?

22 cm

14.5 cm

•TRIG!!!•Find the area.

Other Triangle FormulasEquilateral 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

s

½ s

s2

3ssA

2

3

2

1

2

4

3sA

Other Triangle FormulasSAS 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:

C

b

a

h

b

hC sin

Cbh sin

CabA sin2

1

Other Triangle FormulasHeron’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

b c

a

))()(( csbsassA

2

cbas

AssignmentsHomework 46

Workbook, pp. 140, 142Homework 47

Workbook, pp. 144, 145