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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