## Puaka-James 'Height'

#### Measuring UC's tallest building with a protractor

Published 2021/10/29 by Jordan Hay

After seeing a video by Mathematics Youtuber Matt Parker I was inspired to construct my own protractor and measure the height of a building. The University of Canterbury's Central Library, Puaka-James Hight (Figure 1) with a height of 53 metres (Rubix, n.d.) was selected to be measured.

Figure 2 - Illustration of important quantities, full explanations of these given below. |

$b$ denotes the distance from the building at the first observation site.

$\theta$ denotes the angle to the top of the building at the first observation site.

$a$ denotes the distance between the two observation sites.

$\phi$ denotes the angle to the top of the building at the second observation site.

$h$ is the height of the building.

From this we can derive a formula for $h$ without the use of $b$

$$\begin{align*}&&h&=b\tan\theta\\&&h&=(b+a)\tan\phi\\&&&=b\tan\phi + a\tan\phi\\&\Rightarrow&a\tan\phi&=b\tan\theta - b\tan\phi\\&&&= b(\tan\theta - \tan\phi)\\&\Rightarrow&b&= \frac{a\tan\phi}{\tan\theta - \tan\phi}\\&\Rightarrow&h&= \frac{a\tan\theta\tan\phi}{\tan\theta - \tan\phi}\end{align*}\tag{1}$$

In order to measure the angles involved it was necessary to construct a protractor with a plumb-line. This was done with the use of Fusion 360 and an Ender 3 3D Printer. An initial black layer was laid down to give the degree markings, and a blue body was printed on top of this (Figure 3). The protractor is accurate to a degree, this may be improved with a larger print.

Figure 3 - The 90 degree protractor without plumbline. Loop for plumbline can be seen at top. Large radial marks are given at 15 degree intervals. |

Observations (Figure 4) were taken over 5 different sites (Figure 5). Latitude and longitude coordinate data were noted for each site using GPS. Raw data can be found in the Github Repository (Footnote 1).

Figure 5 - Observation Sites, image rendered with Folium using data © OpenStreetMapcontributors, under ODbL. |

The observations were then processed in Python (Footnote 1) using (Equation 1) giving the following results (Figure 6)

Figure 6 - Results processed with Python (Footnote 1). |

The final result with uncertainty $(60 \pm 10) \text{ m}$ agrees with the expected value of $53\text{ m}$ (Rubix, n.d.). This is more accurate than anticipated.

**Update (30/10/2021):**

Excluding the height predicted from the observation A1-A2 yields a mean of $52.79\text{ m}$, accounting for uncertainty gives $(53.0 \pm 7.0)\text{ m}$. This agrees far more closely with the expected value. However exclusion of that data point is arbitrary choice based upon prior knowledge of the expected height, doing this is not a valid scientific choice, rather it is an interesting observation. It should also be noted that the observation A2-A3 gives almost exactly the expected height but selection of this point is also arbitrary.

**Acknowledgements**

Thank you to Tanya Smith for supplying and installing the protactor's plumbline, Finn Trass for supplying the plumb bob, and William Beauchamp for his assistance with measurements.

**Footnotes**

1. Github Repository with Data, Processing, and Mapping. https://github.com/JHay0112/HightHeight ** **

#### References

*University of Canterbury - Puaka / James Hight*. https://rubix.nz/projects/university-of-canterbury-puaka-james-hight