15-4. Quadrature of plane figures
square
s=
d=
A=
A = area
A=s
2
=
A=d
2
/2=
s=0.7071d=
s=γA=
d=1.414s=
d=1.414γA=
Trapezoid
a=
b=
h=
A area
A=(a+b)*h/2
@=
Rectangle
a=
b=
d=
A=
A area
A=a*b=
A=a*(d
2
-a
2
)
0.5
=
A=b*(d
2
-b
2
)
0.5
=
d=(a
2
+b
2
)
0.5
a=(d
2
-b
2
)
0.5
=
a=Ab=
b=(d
2
-a
2
)
0.5
=
b=Aa=
Parallelogram
a=
b=
c=
H=
h=
A area
A=[(H+h)*a+bh+cH]/2
@=
As shown by the dotted line, there are two triangles, and the area of ??each is calculated. The area of ??the non-parallelogram may be calculated from the sum.
parallelogram
A=
a=
b=
Aarea
A=a*b=
a=Ab=
b=Aa=
[ Semi-consideration]
a Dimensions measured at right angles to side b
Regular hexagon
s=
R=
r=
Aarea
R=Radius of circumscribed circle,@r= Radius of inscribed circle
A=2.598s
2
=
A=2.598R
2
=
A=3.464r
2
=
R=s=
R=1.155r=
r=0.866s=
r=0.866R=
Right triangle
a=
b=
c=
A area
A=bc/2=
a=(b
2
+c
2
)
0.5
=
b=(a
2
-c
2
)
0.5
=
c=(a
2
-b
2
)
0.5
=
Regular octagon
R=
r=
s=
A area
R= Radius of circumscribed circle,@r= Radius of inscribed circle
A=4.828s
2
=
A=2.828R
2
=
A=3.314r
2
=
R=1.307s=
R=1.082r=
r=1.207s=
r=0.924R=
s=0.765R=
s=0.828r=
Acute triangle
a=
b=
c=
h=
A area
A=bh/2=
A=b/2{a
2
-[(a
2
+b
2
-c
2
)/(2b)]
2
}
0.5
@=
if@s=(a+b+c)/2@ If so
A=[s(s-a)(s-b)(s-c)]
0.5
@=
regular polygon
n=
s=
R=
r=
Aarea, n= Number of sides
a=360n
ΐ=180-a
A=nsr/2=
A=ns/2*(R
2
-s
2
/4)
0.5
@=
R=(r
2
+s
2
/4)
0.5
=
r=(R
2
-s
2
/4)
0.5
=
s=2(R
2
-r
2
)
0.5
=
Obtuse triangle
a=
b=
c=
h=
Aarea
A=bh/2=
A=b/2*{a
2
-[(c
2
-a
2
-b
2
)/(2b)]
2
}
0.5
@=
if@s=(a+b+c)/2@ If so
A=[s(s-a)(s-b)(s-c)]
0.5
@=
circle
d=
r=
n=
Aarea, c= Circumference
A=Ξr
2
=3.1416r
2
=
A=0.7854d
2
=
c=2Ξr=6.2832r=
c=3.1416d=
r=c6.2832=(A3.1416)
0.5
=0.564γA
@=
d=c3.1416=(A0.7854)
0.5
@=1.128γA
@=
Arc length for a central angle of 1 =0.008727d=
The length of the arc with respect to the central angle n =0.008727nd=
circle dividing
r=
Ώ=
l=
A=
AareaCl= Arc lengthCΏ=angle
l=r*Ώ*3.1416/180=0.01745rΏ
=
l=2A/r=
A=rl/2=
A=0.008727Ώr
2
=
Ώ=57.296l/r=
r=2A/l=
r=57.296l/Ώ=
Hyperbola
x=
y=
a=
b=
A areaBCD
A=xy/2-ab/2*log(x/a+y/b)
@=
Missing circle
r=
h=
c=
Ώ=
l=
A areaCl= Arc lengthCΏ= angle
c=2[h(2r-h)]
0.5
=
A=[rl-c(r-h)]/2=
r=(c
2
+4h
2
)/(8h)=
l=0.01745ar=
h=r-(4r
2
-c
2
)
0.5
/2=
Ώ=57.296l/r=
parabola
x=
y=
l = Arc length
@=p/2{[2x/p(1+2x/p)]
0.5
+hyp.log[(2x/p)
0.5
+(1+2x/p)
0.5
]}
Approximate formula when x is smaller than y
l =y[1+2/3*(x/y)
2
-2/5*(x/y)
4
]
@=
or
l = (y
2
+4/3*x
2
)
0.5
@=
Ring-shaped
R=
r=
D=
d=
A area
A=Ξ(R
2
-r
2
)=3.1416(R
2
-r
2
)
@=3.1416(R+r)(R-r)
@=
A=0.7854(D
2
-d
2
)
@=0.7854(D+d)(D-d)
@=
parabola
x=
y=
A area
A=2xy/3
@=
(That is, equal to 2/3 of the area of ??a rectangle with x as the base and y as the height)
Fan shape
R=
r=
D=
d=
Ώ=
AareaCΏ= angle
A=ΏΞ(R
2
-r
2
)/360
@=0.00873Ώ(R
2
-r
2
)
@=
A=ΏΞ(D
2
-d
2
)/(4*360)
@=0.00218Ώ(D
2
-d
2
)
@=
Parabolic intercept
BC=
FG=
A area
A=BFC
@= (Area of ??parallelogram BCDE)*2/3
If the height of the intercept measured at right angles to BC is FG
A=BFC=2/3*BC*FG
@=
Square edge
r=
c=
A area
A=r
2
-Ξr
2
/4=0.215r
2
@=
A=0.1075c
2
@=
cycloid
r=
d=
A area
l= Length of "Cycloid"
A=3Ξr
2
=9.4248r
2
= (Area of ??a circle)*3
@=
A=2.3562d
2
= (Area of ??a circle)*3
@=
l=8r=
l=4d=
ellipse
a=
b=
A areaCP= Around the ellipse
A=Ξab=3.1416ab=
Approximate formula for P
1. P=3.1416*[2*(a
2
+b
2
)]
0.5
@@ =
2. P=3.1416*[2*(a
2
+b
2
)-(a-b)
2
/22]
0.5
@@ =
References: "Standard Machine Design Chart Handbook Revised, Augmented 4th Edition" (co-authored by Fujio Oguri and Tatsuo Oguri) Numbers and Calculations of Numbers, page 1-1, refer to the quadrature of plane figures.
