unit 3: perimeter and area of plane figures - myfirstwiki-mar

Transcription

M.Mar Agüera de Pablo-B lanco
IES Caura. Coria del Río
Bilingual programme
UNIT3. PERIMETER AND AREA OF
PLANE FIGURES
What’s the unit about?
 Perimeter and area of triangles. (Perímetro y área de
triángulos).
 Perimeter and area of quadrilaterals.
( Perímetro y área de cuadriláteros).
 Areas of regular polygons. (areas de pentágonos,
hexágonos,…..regulares).
 The circumference and area of a circle. (Longitud de
la circunferencia y área del círculo).
 Other measures related to circles. (Otras medidas
relativas al círculo).
Key words:
Area
Perimeter
Base and height
Apothem of a regular polygon
Central angle of a regular polygon
Larger diagonal
Shorter diagonal
Larger base
Shorter base
Circumference
Circle
Centre of a circle
Radius of a circle
Diameter
Chord
Sesment
Tangent
Arc
Sector
1
Bilingual programme
3.1
PERIMETER AND AREA OF
PLANE FIGURES.
Definitions:
 The perimeter of a polygon is the sum of the length
of its sides.
 The area of a polygon is the number of square units
contained in its interior.
Exercises:
a) Work out the Perimeters of both figures:
 P=1+2+1+1+1+4+1+1+1+2+1+6=
 P=1+1+1+1+1+1+1+2+1+2+1+7=
b) Work out the areas of both figures:
 A=1+4+7=
 A=4+2+6=
The measurements of perimeter use the next units.
The measurements of area use the next units.
2
Bilingual programme
Exercises:
1. Complete:
Km
Hm
Dam
47
m
cm
mm
124
21.5
6.459
2. Complete:
m
dm
4.73
3.91
3. Put in order the next lengths, from the smallest to
largest: (Use <)
43 hm,
25.7 dam,
157 mm,
23 cm,
397.23 m.
2 dm.
4. Put in order the next lengths, from largest to smallest:
(Use >)
a) 31.5 m,
b) 0.07 km,
243 cm,
4257 mm.
2.54 hm,
3
32 m.
Bilingual programme
5.
Find the perimeter of the next figures:
6. Find the area of the figure, using the next different
units:
7. Express in the indicated unit the next measurements:
a)
b)
c)
d)
e)
f)
in m 2
700dam2 in m 2
150mm2 in cm 2
0.03cm2 in m 2
7mm 2 in dm 2
3cm 2 in km 2
6hm 2
4
Bilingual programme
3.2
AREA OF QUADRILATERALS
A) AREA OF THE SQUARE:
l
The area of the square is:
All l2
(equals)
(times)
(The square of its side)
l
( l squared)
Exploring the area of a square
if its side measures 2.5cm.
B) AREA OF THE RECTANGLE:
The area of the rectangle is:
A  b h
(The product of its base and
h
height)
b
(is equal to) (base times height)
Exploring the area of a
rectangle with base equals 6cm
and height equals 2.25cm.
5
Bilingual programme
C) AREA OF THE RHOMBOID:
The area of the rhomboid
is:
h
hh
A  b  h (The product of its base
and height)
b
rhomboid with base equals
8.2cm and height equals.
D) AREA OF THE RHOMBUS:
D
The area of the rhombus is:
A
Dd
(One half of the product
2
of its diagonals)
(larger diagonal) (shorter diagonal)
Exploring the area of a rhombus
with its larger diagonal equals 7cm
and its shorter diagonal equals
4.75cm.
6
d
Bilingual programme
E) AREA OF THE TRAPEZIUM:
b
The area of the trapezium is:
(Bb)h
A
(One half of the product
2
of the sum of its bases and its height)
(larger base)
h
B
(shorter base)
Exploring the area of a trapezium
with B=6cm, b=3.5cm and h=2.9cm.
Exercises:
 Find the areas of the next parallelograms: (b is the base
and h is the height)
a)
b  7.3cm
b)
b 12.1m
h 15cm
h  34dm
7
Bilingual programme
 Find the area of the next figures:
 The roof of my house is rectangular. It is 7.2 meters wide
and 14.8 meters long. Find the area of the roof.
 The perimeter of a rectangle is 70cm. and the length of its
base is 15cm. Find the height and the area of this
rectangle.
 Find the perimeter of a rectangle. It is 4 cm height and its
area is equal to 56 cm 2 .
8
Bilingual programme
 The area of a square is equal to 36 cm 2 . Which is the length
of its side?
 Find the area of the next trapeziums ( measured in cm):
 Un terreno tiene forma de rombo. Si las diagonales miden
180m y 90m, ¿cuál debería ser el lado de un cuadrado de
igual área?
9
Bilingual programme
3.3
AREA OF TRIANGLES
A) AREA OF THE TRIANGLE:
The area of the triangle is:
A
b h
(One half of the
2
h
product of its base and height)
b
triangle:
(It is half of the area of a parallelogram)
Exercises:
 Find the area of the next triangles:
 Find the area of the next triangles(measured in cm)
10
Bilingual programme
3.4
AREA OF A REGULAR
POLYGON.
A) AREA OF REGULAR POLYGONS:
The area of a regular pentagon is:
la 5
la P
a
 

A5Areatriangle= 5
2
2
2
(One half of the product of its perimeter and its
apothem)
The area of a regular hexagon is:
l

a6

l

aP

a
triangle

6
  
A6Area
2 2 2
(Six times the area of the triangle)
The area of a regular octagon is:
l

a8

l

aP

a
triangle

8
  
A8Area
2 2 2
Exploring the
area
of a regular
polygon.
11
Bilingual programme
Exercises:
 Find the area of the following regular polygons:
 A regular hexagon with
l  12cm
and
a 10cm.
 A regular pentagon with
l  6cm
and
a4.13
cm
.
 An equilateral triangle with
l  7cm
and
a  2cm.
 The mirror of my bathroom is octagonal-shaped.
Can you find its area?
12
Bilingual programme
3.5
AREA OF COMPOUND SHAPES
A) THE AREA OF COMPOUND SHAPES:
 A compound shape is any shape made up from
two or more basic shapes.
 You find the total area by addition or
subtraction of the areas of its parts.
 YOU MUST REMEMBER:
Two polygons are congruent if they have the same
area.
Exercises:
1. Find the area of the next figure. The lengths of its
sides are expressed in cm.
2. The figure represented a houseplant. Can you find the
area of this house?
13
Bilingual programme
3. In a rectangular-shaped plot of 40 meters long and 25
meters wide, we have built a chalet with rectangular
base of 17 meters long and 11 meters wide. Find the
area of the garden.
4. Find the area of the next figures:
5. Find the area of the next shading regions:
14
Bilingual programme
3.6
CIRCUMFERENCES AND
CIRCLES.
A) REMEMBER the following specific terms:
Todos los puntos de una
circunferencia se
encuentran a la misma
distancia de otro punto
llamado centro.
The center is the only
point which is the same
distance from every
point on the
circumference.
El radio es la distancia
del centro a cualquiera
de los puntos de la
circunferencia.
The circumference is
the line which defines
the edge of the circle.
Cuerda es el segmento
que une dos puntos de la
circunferencia.
Diámetro es la cuerda
que pasa por el centro.
The radius is any
straight line from the
center to the
circumference.
A chord is the segment
which joins any two
points of the
circumference.
The diameter is the
chord which passes
through the centre.
15
Bilingual programme
Arco es cada una de las
partes en que una cuerda
divide a la
circunferencia.
Any chord divides a
circumference in two
different parts whose
name is arc.
Si la cuerda es un
The diameter divides
diámetro, el arco se llama the circumference in
semicircunferencia.
two equal parts whose
name is
Dos circunferencias se
semicircumference.
llaman concéntricas si
tienen el mismo centro.
The circumference and
all the intirior points
La circunferencia y los
form the circle.
puntos interiores forman
el círculo.
B) RECINTOS EN EL CIRCULO
Parte de
círculo
limitada
por dos
radios.
Parte de
círculo
limitada por
una cuerda
y su arco.
Parte de
círculo
limitada
por dos
cuerdas
paralelas.
Parte de
círculo
limitada por
dos
circunferencias
concéntricas
.
16
Parte de
círculo
limitada por
dos
circunferencias
concéntricas
y dos radios.
Bilingual programme
C) POSICIONES RELATIVAS DE RECTA Y
CIRCUNFERENCIA
La recta es secante a la
circunferencia.
La recta es tangente a
la circunferencia.
La recta es exterior a
la circunferencia.
Se cortan en dos
puntos.
Se cortan en un punto.
No se cortan.
D) POSICIONES RELATIVAS DE DOS
CIRCUNFERENCIAS
Las
circunferencias
son
secantes.
Se
cortan en
dos
puntos.
Las
circunferencias
son tangentes
exteriores.
Se cortan en un
punto y
Las
circunferencias son
tangentes
interiores.
Se cortan en
un punto y
d
(C
,C
R
r
1
2)
d
(C
,C
R

r
1
2)
17
Las
circunferencias son
exteriores.
Las
circunferencias son
interiores
No se cortan y
No se cortan y
d
(C
,C
R
r d
(C
,C
R

r
1
2)
1
2)
Bilingual programme
Exercises:
1. Which is the name of the next coloured regions?
2. A circumference with radius 3cm. Decide about the
relative position of one line which is placed to:
a. 5cm of the centre of the circumference.
b. 2cm of the centre of the circumference.
c. 7cm of the centre of the circumference.
3. Two circumferences with radius 3 and 5cm respectively.
Decide about their relative position if the distance
between their centres is:
a. 2cm
b. 4cm
4. In Spanish. Una casa de campo está en el centro de un
terreno circular de 72 metros de radio. Se quiere
construir una carretera que pase a 35 metros de la
puerta del terreno. ¿A qué distancia de la casa pasará la
carretera?
18
Bilingual programme
E)ÁNGULOS EN LA CIRCUNFERENCIA.
Ángulo central: Es el que tiene
Ángulo inscrito: Es el que tiene
su vértice en el centro de la
circunferencia
su vértice en un punto de la
circunferencia y sus lados son
cuerdas.
Exercises:
1. Calculate the measurement of the central angle â in the
next cases:
2. Work out the measurement of the inscribed angle â in
the next cases:
19
Bilingual programme
F) CIRCUMFERENCE LENGTHS.
CIRCUMFERENCE LENGTH (L)
Theorem:
For all circles, the ratio of the
Definition:
The circumference of a circle is the circumference to the diameter is
the same.
distance around the circle.
This ratio is known as  . It is
approximately equal to 3.1416
L

2 r

L2  r

L Circumference length
2 r 
diameter (the diameter is twice the
radius)
  pi =3.1416
ARC LENGTH (AB)
Definition:
An arc length is a portion of the
circumference of a circle.
Theorem:
In a circle, the ratio of the length
of an arc to the circumference is
equal to the ratio of the measure of
the arc to 360
20
Bilingual programme
Exercises:
3. Find:
a. The circumference of a circle with radius 6
centimeters.
b. The radius of a circle with circumference 12 inches.
4. Find the circumference of circles with radius:
a. 31.42 m
b. 215.3 mm
c. 62.74 km
5. In a circumference with radius 3 cm, work out the arc
length that its central angle measures:
a) 60
b) 45
c) 180
6. All the circumferences of the figure have their radius
equal to 2 cm. Find the length of the coloured line:
7. The next figure is made up of 4 semicircumferences.
Find its perimeter.
21
Bilingual programme
G) AREAS OF CIRCLES AND SECTORS
AREA OF A CIRCLE (A)
Theorem:
The area of a circle is  times the square of the radius.
A    r2
Proof:
If we divide the circle in congruent parts and we place them in
this way, we obtain approximately a parallelogram.

2
A

b

h


r
r


r
22
Bilingual programme
Exercises:
1. The diameter of a circular carpet is equal to 1.2 meters.
Find:
a. The area of the carpet.
b. Its price, knowing that one squared meter costs
360€.
2. Find the area of the next sectors. They are measured in
meters:
3. In Spanish. Una tienda de campaña circular para 6 personas
tiene un diámetro de 4.5 metros. ¿Cuántos metros
cuadrados le corresponden a cada persona?
4. Find the area of the shaded region, knowing that the
radius of the circle is equal to 10 cm.
23
Bilingual programme
5. Our coins are circular-shaped. In the next table, we show
you the diameters of some of them. Find the
circumference and the area of each one:
6. Find the area of the coloured figure:
7. What is the area of a circle with diameter 28 feet?
8. What is the radius of a circle with area 40 square inches?
9. Find the area of the shaded regions:
10.
Find the area of the shaded region:
24
Bilingual programme
11. Find the area of the shaded region:
12.
Find the area of the shaded region:
13.
Find the area of the shaded regions:
25

unit 3: perimeter and area of plane figures - myfirstwiki-mar

Transcription

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