What is area?What is area?

Think of a definition

How is it measured?

Should all of our area measurements be the same??

AreaArea

Area: the measure of the region enclosed by a plane figure.

Area is measured by counting the number of

square units that you can arrange to fill

the figure completely.

We may have used different units, which may result in

different answers.

Units of AreaUnits of Area

1 m

1 m

100 cm

100 cm

Area of 1m2

What is the area in cm2?

1m=100cm1m2= ? cm2

1m=100cm1m2= 10,000cm2

Dimensions of a figureDimensions of a figure

BaseAny side of a rectangle

or parallelogram could be

called a base.

HeightThe length of the side that

is perpendicular to the base

of a rectangle or the length of the altitude.

AltitudeAny segment from

one side of a parallelogram

perpendicular to a line through the

opposite side

Area Theorems 8.1Area Theorems 8.1

Rectangle Area Theorem: The area of a rectangle is given by the formula A=bh, where A is the Area, b is the length of the base, and h is the height of the rectangle.

Parallelogram Area Theorem: The area of a parallelogram is given by the formula A=bh, where A is the Area, b is the length of the base, and h is the height of the parallelogram.

Is area calculated the same way for other parallelograms?

TrianglesTriangles

What formula

does this give us for the area of a triangle?

Area of a triangle?

A triangle is half of a parallelogram, therefore its area is given by the formula;

Where b is the length of the base, and h is the height of the triangle

1

2A bh

b

h

Trapezoids What formula does this give us for the area of a trapezoid?

Area of a trapezoid?

A trapezoid is half of a parallelogram, therefore its area is given by the formula;

Where b1 and b2 are the lengths of the bases, and h is the height of the trapezoid.

1 2

1

2A b b h

b1

h

b2

Kites

What formula

does this give us for the area of

a kite?

Area of a kite?

A kite is half of a rectangle, therefore its area is given by the formula;

Where d1 and d2 are the diagonals of the kite.

1 2

1

2A d d

d1

d2

Group Area Problems

Work in a group to discuss a method for finding the area of the irregular shapes.

Find the area of each using a ruler (if group members get different answers, average them).

For each shape, write a sentence or two describing how you decided to measure the areas. Also, explain how accurate you think each measurement is.

Helios

It takes 65,000 solar cells, each 1.25 in. by 2.75 in., to power the Helios Prototype, pictured here.

How much surface area, in square feet, must be covered with the cells?

Helios pg 435 #3

One cell has an area of (2.75in*1.25in)=3.44in2

65,000 cells have an area of (65,000*3.44in2)=223437.5in2

1552ft2

Painting

You get a new job as a painter to make some extra money in the summer. Your boss tells you that you must paint 148 identical rooms. Each room needs a coat of base paint and finish paint. All four walls and the ceiling of each room must

be covered. The rooms are 14ft wide by 16ft long by 10ft high.

If one gallon of base paint covers an area of 500ft2 and one gallon of finish paint covers 250ft2, how many gallons of

each type of paint will you need to buy?

Painting #2

Area of one room the sum of 4 walls (opposing walls are the same size) and the ceiling.

148 rooms so total area to be painted is 148 times 824 square feet.

121,952 square feet!

2

2 2

2 16 10 2 14 10 16 14

824

A lh wh lw

A

A ft per room

2

2

2

2

121952244

500

121952488

250

ftGallonsbase

ft

ftGallons finish

ft

Flowerbed

A landscape architect is designing three trapezoidal flowerbeds to wrap around three sides of a hexagonal flagstone patio.

What is the area of the entire flowerbed?

The landscaper’s fee is $100 plus $5 per square foot. What will the flowerbed cost?

20 ft

7 ft

12 ft

Flowerbed #1

Each flowerbed has an area of:

Therefore, total area is 336ft2

336ft2 times $5 equals $1680, so you owe the Architect $1780

1 2

2

1

21

20 12 72

112

A b b h

A ft ft ft

A ft

8.4 The Area of Regular 8.4 The Area of Regular PolygonsPolygons

Draw 3 circles on a piece of paper using a compass.

Use these 3 circles to circumscribe 3 regular polygons so that the center of the

polygon is also the center of the circle. You may choose the amount of sides that your

polygons will have.Find a way to generalize the formula for

the area of a regular polygon.

If you are up to the challenge, find two ways!!

OctagonOctagon

As you have seen, All regular polygons can be split into an amount of

congruent triangles that equals the amount of sides

of the polygon

OctagonOctagon

Side of length “s” is the base of a triangle

Side of length “s” is the base of a triangle

Altitude of triangle is known as the apothem “a” of the

polygon (a perpendicular from the

center to a side.

Altitude of triangle is known as the apothem “a” of the

polygon (a perpendicular from the

center to a side.

OctagonOctagon

a

s

h

b

A 1

2san

A 1

2bh

n is the amount of triangles that have the same area. This is the same as the number of sides!

What is the perimeter (P) of a What is the perimeter (P) of a regular polygon in terms of n regular polygon in terms of n

& s ???& s ???

A 1

2san

P sn

A 1

2aP

Regular Polygon Area Regular Polygon Area TheoremTheorem

The area of a regular polygon is given by the formula or , where A is the area, P is the perimeter, a is the apothem, s is the length of a side, and n is the number of sides.

A 1

2san

A 1

2aP

8.5 Areas of Circles8.5 Areas of Circles

Draw 1 large circle on a piece of paper using a compass.

Cut out your circle, fold it in half 4 times, and then unfold it and cut along creases

forming 16 pieces.Arrange them into something that resembles

a parallelogram

What would happen if you cut the pieces even smaller?

How does this help us with a formula to find the area of a circle?

Areas of CirclesAreas of Circles

CircumferenceCircumference

RadiusRadius

One half of the original

circumference is the base of

the parallelogram

One half of the original

circumference is the base of

the parallelogram

The tinier we make our piece of the circle the

smoother our parallelogram becomes!

The tinier we make our piece of the circle the

smoother our parallelogram becomes!

Areas of CirclesAreas of Circles

b=C/2

h=r

C

r

A bh

A C

2r

Areas of CirclesAreas of Circles

b

h

C

r

A C

2r

A 2r

2r

A r2

8.6 Any Way You Slice 8.6 Any Way You Slice ItIt

Circle Terms: Sector of a Circle- The region between two radii and an arc of the

circle.

Segment of a Circle- The region between a chord and an arc of the circle.

Annulus- The region between two concentric circles (circles that share the same center).

See if you can find the See if you can find the area of these sections!area of these sections!

Circle Terms: Sector of a Circle- The region between two radii and an arc of the

circle.

Segment of a Circle- The region between a chord and an arc of the circle.

Annulus- The region between two concentric circles (circles that share the same center).

Area of a SectorArea of a Sector

r

Amount of circle being used times the area of the

whole circle

arc measure

360

area of sector

area of circle

arc measure

360area of circle area of sector

Area of a SectorArea of a Sector

r=5cm

Asec t

360Acirc

θ=96°

Asec t

360r2

Asec t 96360

5cm 2

Asec t 21cm2

Area of a Segment of a Area of a Segment of a CircleCircle

r

Aseg Asec t Atri

Aseg 360

r2 1

2bh

area of segment area of sec tor area of triangle

The base of the triangle is the length of a chord.

The radius is the other two sides of the isosceles triangle.

Area of a Segment of a Area of a Segment of a CircleCircle

Aseg Asec t Atri

Aseg 360

r2 1

2bh

Aseg 90360

3cm 2 1

23cm 3cm

θ=90°

r=b=h=3cm

Aseg 2.57cm2

Area of AnnulusArea of Annulus

r

R

Aann AR Ar

Aann R2 r2

Area of AnnulusArea of Annulus

r=1m

R=3m

Aann AR Ar

Aann R2 r2

Aann 3m 2 1m 2

Aann 8cm2 25cm

Pick’s Formula for AreaPick’s Formula for Area

Exploration on pages 446-448

Figure out Pick’s formula for area estimation

Use Pick’s formula to find the area of your hand using graph paper.

Help estimate the area of a pi symbol

Cut out your hand and add it to the class poster Decorate your hand and label it with your name and

area

Pick’s formula

Consider the vertices on your graph papers to be

points. Draw these shapes on your paper and confirm that their

areas are each 12 blocks.

Create a table like the one below. Create a table like the one below. Complete it for the 4 shapes of equal Complete it for the 4 shapes of equal

area that you made.area that you made.

D E F G

# of boundary points (b)

# of interior points (i)

D E F G

# of boundary points (b) 14 8 10 12

# of interior points (i) 6 9 8 7

Study this table for a pattern. Plot the points on a small graph to find equation

relating the two.

Now investigate what happens Now investigate what happens if you hold the interior points if you hold the interior points

constant.constant.Draw several polygons between you and

your partners that have zero interior points. Find the area of those shapes. Fill out another table like the one below. Also

, do this for shapes with exactly one interior point.

Example area of ½ block

i=0 H I J K

# of boundary points (b)

Area

When you hold interior points constant, what When you hold interior points constant, what happens to the area with each boundary happens to the area with each boundary

point that is added?point that is added?

The area increases by ½ !!!

Now investigate what happens Now investigate what happens if you hold the boundary points if you hold the boundary points

constant.constant.For example, make different triangles

that hit no other points other than corners.

b=3 L M N O

# of interior points (i)

Area

When you hold boundary points constant, what happens to the

area each time one interior point is added?

When you hold boundary points constant, When you hold boundary points constant, what happens to the area with each interior what happens to the area with each interior

point that is added?point that is added?

The area increases by 1 !!!

Try a few random polygons and see if Try a few random polygons and see if you can develop a formula for area in you can develop a formula for area in terms of boundary points and interior terms of boundary points and interior

points.points.

Pick’s Formula for Estimating the Area of a Polygon

A 1

2b i 1

8.7 Surface Area8.7 Surface Area

Surface Area: the sum of all of the areas of the faces or surfaces that enclose the solid.

Bases & Lateral Faces:In a prism, the bases are two congruent polygons and the lateral faces are rectangles or other parallelograms

In a pyramid, the base can be any polygon and the lateral faces are triangles.

Steps for finding Surface Steps for finding Surface AreaArea

1)Draw and label each face of the solid as if you had cut the solid apart along its edges and laid it flat. Label the dimensions.

2) Calculate the area of each face. If some faces are identical, you only need to find the area of one.

3) Find the total area of all of the faces.

Example- Rectangular Example- Rectangular PrismPrism

3ft

6ft

8ft

8ft

6ft

6ft

8ft

3ft 3ft

€

SA = 2(8ft)(6ft) + 2(8ft)(3ft) + 2(6ft)(3ft) =180ft 2

Example- CylinderExample- Cylinder

12in

10in

r=5inr=5in

h=12in

Top Bottom

Lateral Surface

b=C=2πr

€

SA = bh + 2 πr2( )

€

SA = 2πr( )h + 2 πr2( )

€

SA = 2π5in( )12in + 2 π(5in)2( ) ≈ 534in2

PyramidsPyramids

Surface area is the area of the base plus the area of triangular faces.

Slant height (l): The height of each triangular face of a pyramid

Example-PyramidExample-Pyramid

Example-PyramidExample-Pyramid

€

SA = bh = n1

2ls

⎛

⎝ ⎜

⎞

⎠ ⎟

€

SA =1

2san

s

a

s s ss

s s s s

b

l=h

+

Example-PyramidExample-Pyramid

s

a

s s ss

s s s s

b

l=h

€

SA = n1

2ls

⎛

⎝ ⎜

⎞

⎠ ⎟+

1

2san

€

SA =1

2ns l + a( )

€

SA =1

2P l + a( )

Example- ConeExample- Cone

l

r

l

2πrπr

πr

l

Example-ConeExample-Cone

r=5cm

l=10cm

€

SA = πr2 + bh

SA = πr2 + πr l( )

€

SA = π 5cm( )2

+ π 5cm( ) 10cm( )

SA = 75πcm2 ≈ 235.6cm2

Practice Problems from Practice Problems from text-pg 466 #’s 1-12text-pg 466 #’s 1-12

SpheresSpheres

€

SA = 4πr2