F-Areas_and_Volumes

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AREAS AND VOLUMES 
 
ipri
 
 Departament of Mathematics 1 
 
A
R
E
A
S
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F
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L
A
N
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IG
U
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S
 
 NAME FORM AREA
TRIANGLES 
(3-sided polygons) 
Triangle 
b
h
 
𝐴
𝑏 ∙ ℎ
2
 
Q
U
A
D
R
IL
A
T
E
R
A
L
S
 
(4
-s
id
ed
 p
ol
yg
on
s)
 
Q
U
A
D
R
IL
A
T
E
R
A
L
S
 
(W
it
h 
tw
o 
in
 tw
o 
pa
ra
ll
el
 s
id
es
) Square l
l
𝐴 𝑙 ∙ 𝑙 
Rectangle 
b
a
 
𝐴 𝑏 ∙ 𝑎 
Rhombus D
d
 
𝐴
𝐷 ∙ 𝑑
2
 
Rhomboid or 
parallelogram 
b
h
 
𝐴 𝑏 ∙ ℎ 
T
R
A
P
E
Z
O
ID
S
 
(W
it
h 
tw
o 
pa
ra
ll
el
 s
id
es
) Right trapezoid 
b
B
h
 
𝐴
𝐵 𝑏 ∙ ℎ
2
 Isoceles trapezoid h
B
b
 
Scalene trapezoid 
B
b
h
 
T
R
A
P
E
Z
O
ID
S
 
Trapezoid 
 
It is divided into 
two triangles and 
their areas added 
N
 –
 S
ID
E
D
 
P
O
L
Y
G
O
N
 
Regular polygon 
a
 
𝐴
𝑝 ∙ 𝑎
2
 
p = perimeter 
Irregular polygon 
 
It decomposes into 
triangles and their 
areas are added 
 
AREAS AND VOLUMES 
 
Areas and Volumes 2 
A
R
E
A
S
 
 
C
IR
C
U
L
A
R
 F
IG
U
R
E
S
 
Circumference 
r
 
𝐿 2 ∙ 𝜋 ∙ 𝑟 
Circle 𝐴 𝜋 ∙ 𝑟 
Circular sector 
 
rºn
 
 
𝐴
𝜋 ∙ 𝑟 ∙ 𝑛º
360º
 
 
nº = number of degrees 
Circular crown r
R
 
 
𝐴 𝜋𝑅 𝜋𝑟 
Circular trapezoid 
 
r
R
ºn
 
 
𝐴
𝜋 𝑅 𝑟 𝑛º
360º
Circular segment 
rºn
 
sec tor isosceles
triangle
A A A  
 
A
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F
 
G
E
O
M
E
T
R
IC
 S
O
L
ID
 
 NAME FORM AREAS VOLUMES
P
O
L
Y
H
E
D
R
A
 
(G
eo
m
et
ri
c 
so
li
ds
 li
m
it
ed
 b
y 
po
ly
go
ns
) 
PRISM h
Ba
𝐴 𝑝 ∙ ℎ 
𝑝 = perimeter base 
 
𝐴
𝑝 ∙ 𝑎
2
 
𝑎 = base apothem 
 
𝐴 𝐴 2𝐴 
𝑉 𝐴 ∙ ℎ 
PYRAMID h
la
 
. 2
B l
TRIANG
l a
A

 
la = lateral apothem 
 
𝐴
𝑝 ∙ 𝑎
2
 
 
𝐴 𝐴 2𝐴 
𝑉
𝐴 ∙ ℎ
3
 
S
O
L
ID
 
O
F
 
R
E
V
O
L
U
T
IO
N
 
(S
ol
id
s 
w
hi
ch
 
ar
e 
ob
ta
in
ed
 b
y 
tu
rn
in
g 
a 
pl
an
e 
fi
gu
re
)
CILINDER h
r
𝐴 2𝜋𝑟 ∙ ℎ 
h = height 
 
𝐴 𝜋 ∙ 𝑟 
 
𝐴 𝐴 2𝐴 
𝑉 𝐴 ∙ ℎ 
AREAS AND VOLUMES 
 
ipri
 
 Departament of Mathematics 3 
 
CONE h
g
r
𝐴 𝜋 ∙ 𝑟 ∙ 𝑔 
g = generatrix or slant 
height 
 
𝐴 𝜋 ∙ 𝑟 
 
𝐴 𝐴 𝐴 
𝑉
𝐴 ∙ ℎ
3
 
SPHERE R
 
𝐴 4𝜋𝑅 𝑉
4
3
𝜋𝑅 
 
 
 
 
A
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G
E
O
M
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IC
 S
O
L
ID
 
 NAME FORM AREAS VOLUMES
F
R
U
S
T
R
U
M
S
 
(S
ol
id
s 
th
at
 a
re
 o
bt
ai
ne
d 
fr
om
 o
th
er
s,
 
w
he
n 
cu
t b
y 
a 
pl
an
e 
pa
ra
ll
el
 to
 th
e 
ba
se
 )
 
FRUSTUM 
PYRAMID 
or 
TRUNCATED 
PYRAMID 
h ap
 
𝐴
𝑃 𝑝 𝑎𝑝
2
 
P = perimeter of the largest 
base 
p = perimeter of the 
smaller base 
ap = frustum apothem 
 
T L B bA A A A   
𝐴 = larger base area 
𝐴 = smaller base area 
 
3
B b B bA A A A h
V
  

FRUSTRUM 
CONE 
or 
TRUNCATED 
CONE 
r
h
g
R
 LA R r g  
 
 
2 2 
TA g R r
R r

 
  
 
 
 2 2
3
h R r Rr
V
  
 
S
P
H
E
R
IC
A
L
 S
O
L
ID
S
 
(S
ol
id
 o
bt
ai
ne
d 
fr
om
 a
 s
ol
id
 a
nd
 c
ut
 in
to
 
on
e 
or
 m
or
e 
pl
an
es
) 
SPHERICAL 
SEGMENT 
h
R
r
 
 
2A r h   
2 2 23 3
6
h h R r
V
  
 
SPHERICAL 
CAP 
 
h
R
 
2A R h   
2 3
3
h R h
V
 
 
SPHERICAL 
WEDGE or 
UNGULA or 
SPHERICAL 
LUNE 
 
ºnr
 
 
2 º4
360º
n
A r  34 º
3 360º
n
V r  
 
If we don’t want to memorize the formulas for finding the volume of frustums, we use similar triangles 
and the Thales’ Theorem. 
To find the area and volume of a spherical wedge we can use a rule of three.