Jiuzhang Suanshu Shanggong 9

## **Problem**

答曰：二千一百一十二尺。

術曰：周自相乘，以高乘之，十二而一。

Answer: 2112 [cubic] *chi*.

Method: Multiply the circumference by itself, then multiply by the height, divide by 12.

1 Literally translates to “round watchtower”.

## **Derivation**

The *yuan bao dao* (圓堡壔, “round watchtower”) is a cylinder. The cylinder is a prism with a circular base (Figure 1). To derive the formula for calculating the volume of a cylinder, begin by placing a cylinder within a rectangular prism. The rectangular prism must be of equal height, and have a square base whose width is equal in length to the diameter of the cylinder’s circular base.Let the height of the cylinder be

h

{\displaystyle h}

and the width of the square base of the rectangular prism be

a

{\displaystyle a}

. When the cylinder is placed inside the rectangular prism, the horizontal cross-sections of the two solids are circles inscribed in a square of equal width (Figure 2). The ratio of the area of a circle to the area of a square is

π
:
4

{\displaystyle \pi : 4 }

. Therefore, the volume of the cylinder to the rectangular prism with a square base is also

π
:
4

{\displaystyle \pi : 4 }

.

V

c
y
l
i
n
d
e
r

=

π
4

V

p
r
i
s
m

=

π
4

a
2

h
.

{\displaystyle V\_{cylinder} = \frac{\pi}{4} V\_{prism}= \frac{\pi}{4}a^2h. }

Since the circumference of the cylinder is

# C
π
a

{\displaystyle C = \pi a }

, the formula for volume of the cylinder expressed in terms of its circumference and height is

V

c
y
l
i
n
d
e
r

=

π
4

(

C
π

)

2

# h

1

4
π

C
2

h
.

{\displaystyle V\_{cylinder} = \frac{\pi}{4} \left(\frac{C}{\pi} \right)^2 h = \frac{1}{4\pi} C^2 h. }

The *Jiuzhang Suanshu* used the approximation

# π
3

{\displaystyle \pi = 3 }

 for its original solution method, resulting in the formula

V

c
y
l
i
n
d
e
r

=

1
12

C
2

h
.

{\displaystyle V\_{cylinder} = \frac{1}{12}C^2 h. }

 More refined formulas for the volume of the cylinder can be found the commentaries of Liu Hui and Li Chunfeng. Liu Hui used the rational approximation

# π

157
50

{\displaystyle \pi = \frac{157}{50} }

 and obtained the formula

V

c
y
l
i
n
d
e
r

=

25
314

C
2

h

{\displaystyle V\_{cylinder} = \frac{25}{314}C^2 h }

; Li Chunfeng used the rational approximation

# π

22
7

{\displaystyle \pi = \frac{22}{7} }

 and obtained the formula

V

c
y
l
i
n
d
e
r

=

7
88

C
2

h
.

{\displaystyle V\_{cylinder} = \frac{7}{88}C^2 h. }

## **Calculation**

Convert the base circumference and the height of the cylinder (圆堡壔 *yuan bao dao*) to *chi.*

**Base circumference:** 4 *zhang* 8 *chi* = 48 *chi*

**Height:** 1 *zhang* 1 *chi* = 11 *chi*

Substitute the base length, and height of the rectangular prism into the formula.

V

=

1
12

C
2

h

=

1
12

×

46
2

×
11

=
2112

{\displaystyle \begin{align}
V &= \frac{1}{12}C^2 h \\[5 pt]
& = \frac{1}{12}\times 46^2 \times 11 \\[5 pt]
&= 2112
\end{align}}

The volume of the cylinder (圓堡壔 *yuan bao dao*) given in the original answer from the *Jiuzhang Suanshu* is 2112 cubic *chi*.

Liu Hui gives the answer

2017

131
157

{\displaystyle 2017\frac{131}{157}}

 cubic *chi.*

Li Chunfeng gives the answer

2016

{\displaystyle 2016 }

 cubic *chi*.

The precise answer (rounded to the nearest hundredth) is 2016.81 cubic *chi*.