properties of area elements

Properties of Area ElementsNotation:

C = centroidr = radiusx = x centroid locationy= y centroid locationx', y' = centroidal axesA = areaP = perimeterIx = area moment of inertia about the x axisIy = area moment of inertia about the y axisIx' = area moment of inertia about the x' axisIy ' = area moment of inertia about the y' axis

 Circle

r x'

y, y'

C

x

rP

rA

ry

x

2

0

2

441

''

445

rII

rI

yx

x

Semicircular Area

r x'

y, y'

C

x

)2(

3

4

0

221

rP

rA

ry

x 

42

'

481

'

72

649rI

rIII

x

yyx

Quarter circular Area

r x'

y'

C x

y  

)2(

3

4

21

241

rP

rA

ryx

42

''

4161

144

649rII

rII

yx

yx

Circular Sector Area

r

x, x'

y'

C

y

)1(2

03

sin2

2

rP

rA

y

rx

9

)(sin4))2sin((

))2sin((

))2sin((

24

214

41

'

214

41

214

41

'

rrI

rI

rII

y

y

xx

Circular Segment Area    

r x

C

y, y'

x'

)sin(2

)cossin(

)cossin(3

sin2

0

2

3

rP

rA

ry

x

)cossin3

cossin23(

)cossin

cossin2(

34121

'

3441

rII

rI

yy

x

Ellipse

a

x, x'

y, y'

C

b

abA

y

x

0

0 

341

341

abI

baI

y

x

Rectangle

h x'

y'

C

b

y

x

bhP

bhA

hy

bx

2

21

21

3121

'

3121

'

331

331

bhI

bhI

bhI

bhI

y

x

y

x

Trapezoid

h x'

y'

C

b

a

x

y

)(

2

3

21 bahA

ba

bahy

)(36

)4(

)3(223

'

3121

ba

babahI

bahI

x

x

Triangle

h x'

y'

C

b x

y c

hbA

hy

cbx

21

31

31 )(

)(

)(

22361

'

3361

'

22121

3121

cbcbhbI

hbI

cbcbhbI

hbI

y

x

y

x

Right Triangle

h x'

y'

C

b x

y  

22

21

31

31

hbhbP

hbA

hy

bx

hbI

hbI

hbI

hbI

y

x

y

x

3361

'

3361

'

3121

3121

Quarter-circle spandrel

r

x'

y'

C x

y  

rrP

rrA

ry

rx

21

2412

2

)4(3

)310(

)4(3

2

4

4

163

1

16

51

rI

rI

y

x

Parabolic semisegment

2

2

1a

xby

x '

y '

C

x

y

a

b

abA

by

ax

32

52

83

3

3

15

2105

16

baI

abI

y

x

Parabolic spandrel

22x

a

by

x '

y '

C

x

y

a

b

abA

by

ax

31

103

43

5

213

3

baI

abI

y

x