刘徽从半径1尺圆的内接正6边形开始，逐次分割为12边形，24边形，48边形，96边形。反复使用勾股定理求得各多边形的边长，又用刘氏多边形面积公式求多边形面积。
分割6边形为12边形
刘徽割圆术原理
令圆直径为2尺，折半得半径1尺。圆内接正6边形的边长也是1尺。[8]
如图：
半径OA=r=1尺=10寸
6边形单边长AB=M=10寸
从圆心O作AB的垂直平分线OC，将AB平分为二，
AP=BP=M/2，AP+BP=AB
垂直平分线OC和圆周相交于C，
作直线AC
AC就是12边形的一边，OAP是一个直角三角形
弦=半径=r=10寸
勾=AP=M/2=5寸
股OP 可用勾股定理求得：令弦长=X，股长=G， 句长=M/2，则：
G
2
=
r
2
−
(
M
2
)
2
=
100
−
25
=
75
{\displaystyle {}G^{2}=r^{2}-\left({\frac {M}{2}}\right)^{2}=100-25=75}
平方寸
G
=
r
2
−
M
2
4
{\displaystyle {}G={\sqrt {r^{2}-{\frac {M^{2}}{4}}}}}
因为1寸 =100000忽
1平方寸 =10000000000平方忽
G
=
750000000000
=
866025
2
5
{\displaystyle {}G={\sqrt {750000000000}}=866025{2 \over 5}}
忽[9]APC是一个小直角三角形
令小弦AC长度为m，令小句PC长度为j
j
=
r
−
G
=
1000000
−
866025
2
5
=
133974
3
5
{\displaystyle {}j=r-G=1000000-866025{2 \over 5}=133974{3 \over 5}}
忽用勾股定理求m：
m
2
=
(
M
2
)
2
+
j
2
{\displaystyle {}m^{2}=\left({\frac {M}{2}}\right)^{2}+j^{2}}
=
(
500000
)
2
+
(
133974.6
)
2
=
267949193445
{\displaystyle {}(500000)^{2}+(133974.6)^{2}=267949193445}
平方忽12边形的一边长度
=
m
=
267949193445
=
517638.09
{\displaystyle =m={\sqrt {267949193445}}=517638.09}
忽
12边形的一边长度的一半
=
m
2
=
517638.09
2
=
258819.045
{\displaystyle ={m \over 2}={517638.09 \over 2}=258819.045}
忽分割12边形为24边形
将上一轮的多边形边长m作为新一轮割圆的开始，
作替换M=m=12边形的一边长度
=
517638.09
{\displaystyle =517638.09}
忽
继续将此多边形的一边平分，周而复始，重复使用[10]：
G
=
r
2
−
M
2
4
{\displaystyle {}G={\sqrt {r^{2}-{\frac {M^{2}}{4}}}}}
j
=
r
−
G
{\displaystyle {}j=r-G}
m
2
=
(
M
2
)
2
+
j
2
{\displaystyle {}m^{2}=\left({\frac {M}{2}}\right)^{2}+j^{2}}
=
M
2
4
+
j
2
{\displaystyle {}={\frac {M^{2}}{4}}+j^{2}}
由上M^2已有现成数值
M
2
=
267949193445
{\displaystyle {}M^{2}=267949193445}
M
2
4
=
267949193445
4
=
66987298361
{\displaystyle {\frac {M^{2}}{4}}={267949193445 \over 4}=66987298361}
G
=
r
2
−
M
2
4
=
1000000000000
−
66987298361
=
965925
4
5
{\displaystyle {}G={\sqrt {r^{2}-{\frac {M^{2}}{4}}}}={\sqrt {1000000000000-66987298361}}=965925{4 \over 5}}
j
=
r
−
G
=
1000000
−
965925
4
5
=
34074
1
5
{\displaystyle {}j=r-G=1000000-965925{4 \over 5}=34074{1 \over 5}}
m
2
=
(
M
2
)
2
+
j
2
=
66987298361
+
(
34074
1
5
)
2
=
68148349466
{\displaystyle {}m^{2}=({\frac {M}{2}})^{2}+j^{2}=66987298361+(34074{1 \over 5})^{2}=68148349466}
24边形一边长度
m
=
68148349466
=
261052
2
5
{\displaystyle m={\sqrt {68148349466}}=261052{2 \over 5}}
分割24边形为48边形
将第二轮的多边形边长m作为第三轮割圆的起点[11]，
作替换
M
=
m
=
261052
2
5
{\displaystyle M=m=261052{2 \over 5}}
M
2
=
m
2
=
68148349466
{\displaystyle {}M^{2}=m^{2}=68148349466}
M
2
4
=
68148349466
4
=
17037087366
{\displaystyle {\frac {M^{2}}{4}}={68148349466 \over 4}=17037087366}
G
=
r
2
−
M
2
4
=
1000000000000
−
17037087366
=
991444
4
5
{\displaystyle {}G={\sqrt {r^{2}-{\frac {M^{2}}{4}}}}={\sqrt {1000000000000-17037087366}}=991444{4 \over 5}}
j
=
r
−
G
=
1000000
−
991444
4
5
=
8555
1
5
{\displaystyle {}j=r-G=1000000-991444{4 \over 5}=8555{1 \over 5}}
m
2
=
(
M
2
)
2
+
j
2
=
68148349466
4
+
(
8555
1
5
)
2
=
17110278813
{\displaystyle {}m^{2}=({\frac {M}{2}})^{2}+j^{2}={68148349466 \over 4}+(8555{1 \over 5})^{2}=17110278813}
开平方，得48边形一面
m
=
17110278813
=
130806
{\displaystyle m={\sqrt {17110278813}}=130806}
忽根据刘徽多边形面积公式：
96边形的面积= 48边形的半周长×半径=
m
×
48
2
×
r
{\displaystyle m\times {\frac {48}{2}}\times r}
,所以96边形的面积
A
96
=
130806
×
48
2
×
1000000
{\displaystyle A_{96}=130806\times {\frac {48}{2}}\times 1000000}
=
130806
×
24
×
1000000
=
31393440000000
{\displaystyle ={130806\times 24\times 1000000}=31393440000000}
平方忽
A
96
=
31393440000000
10000000000
=
313
584
625
{\displaystyle A_{96}={\frac {31393440000000}{10000000000}}=313{584 \over 625}}
平方寸分割48边形为96边形
将第三轮的多边形边长m作为第四轮割圆的起点[12]
作替换
M
=
m
=
130806
{\displaystyle M=m=130806}
忽
M
2
=
m
2
=
17110278813
{\displaystyle M^{2}=m^{2}=17110278813}
M
2
4
=
17110278813
4
=
4277569703
{\displaystyle {\frac {M^{2}}{4}}={17110278813 \over 4}=4277569703}
G
=
r
2
−
M
2
4
=
1000000000000
−
4277569703
=
997858
9
10
{\displaystyle {}G={\sqrt {r^{2}-{\frac {M^{2}}{4}}}}={\sqrt {1000000000000-4277569703}}=997858{9 \over 10}}
j
=
r
−
G
=
1000000
−
997858
9
10
=
2141
1
10
{\displaystyle {}j=r-G=1000000-997858{9 \over 10}=2141{1 \over 10}}
m
2
=
(
M
2
)
2
+
j
2
=
17110278813
4
+
(
2141
1
10
)
2
=
4282154012
{\displaystyle {}m^{2}=({\frac {M}{2}})^{2}+j^{2}={17110278813 \over 4}+(2141{1 \over 10})^{2}=4282154012}
开方得
96边形的一边
m
=
4282154012
=
65438
{\displaystyle m={\sqrt {4282154012}}=65438}
忽根据刘徽多边形面积公式：
192边形的面积
A
192
=
{\displaystyle A_{192}=}
96边形的半周长×半径=
m
×
96
2
×
r
{\displaystyle m\times {\frac {96}{2}}\times r}
所以192边形的面积
A
192
=
65438
×
96
2
×
1000000
{\displaystyle A_{192}=65438\times {\frac {96}{2}}\times 1000000}
平方忽
=
65438
×
48
×
1000000
=
3141024000000
{\displaystyle ={65438\times 48\times 1000000}=3141024000000}
平方忽
A
192
=
3141024000000
10000000000
=
314
64
625
{\displaystyle A_{192}={\frac {3141024000000}{10000000000}}=314{64 \over 625}}
平方寸