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In this experiment, we aimed to determine whether it is possible to accurately measure circular objects using basic tools and to relate these measurements to the value of pi. Our approach involved using materials such as powder and measuring tapes to trace the length of objects' paths and analyze how these measurements correspond with theoretical expectations based on pi. By comparing the circumference (the distance traveled when an object rolls or spins) to the diameter, we sought to observe a proportional relationship that would approximate pi. Our goal was to understand measurement uncertainties and how they influence the accuracy of such experiments. We hypothesized that plotting the measured circumferences against the diameters would produce a linear relationship with a slope close to pi, supporting the geometric principle that circumference equals pi times diameter.

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Introduction

The pursuit of quantifying the properties of geometric shapes through experimental means lies at the heart of understanding fundamental mathematical constants. In this experiment, the primary focus was to measure the circumference of various circular objects and compare the results against their diameters to derive an experimental value for pi. The significance of this study is rooted in empiricism—using tangible measurements to reaffirm or challenge the theoretically established value of pi, which is approximately 3.14159. By applying simple, accessible tools such as rulers, measuring tapes, powders, and manual marking, the experiment exemplifies how fundamental mathematical principles can be explored through straightforward methodological approaches. The insights gained from this investigation contribute to appreciating the variability and uncertainties inherent in measurement processes, as well as reinforcing the proportional relationship that underpins circular geometry.

Methodology

The experiment involved measuring the diameters and circumferences of various objects, including coins, batteries, and household items such as cups and caps. The objects selected were diverse in size and material to provide a broad range of data points. For each object, the procedure was as follows: the object was marked at a specific point on its circumference with a marker, then rolled forward across a surface onto paper or a powder-covered surface to trace its path. When using powder (flour) or carbon paper substitution, the object was rolled over the powder to leave a marked trail indicating its path length. For objects where precise marking was feasible, a sharpie was used to mark the start and end points after the object completed a full revolution. In cases involving larger objects, a measuring tape was employed to directly measure the length of the trail, with multiple trials conducted to ensure consistency. The diameters of the objects were measured using rulers or calipers, taking care to record uncertainties arising from measurement limitations, such as the thickness of the marker lines, slipping during rolling, and the surface texture influencing the motion of objects.

Uncertainty estimation was an essential component, considering factors like the thickness of lines made by markers or pencils, the slippage of objects on slippery surfaces, and the limitations of measuring tools. For example, when measuring the circumference from powder marks, the uncertainties included the thickness of the powder trail and the precision of marking the start and stop points. For smaller objects like coins, the uncertainties involved the potential for slippage and measurement device resolution. Larger objects such as the batteries and cups were measured using measuring tapes, with uncertainties estimated by the smallest divisions of the measuring devices and the effect of surface conditions on rolling behavior. Multiple measurements were recorded for each object, and average values along with standard deviations were calculated to assess the variability and reliability of the data collected.

Data & Results

The measured diameters and circumferences of objects are summarized below:

• Cap of the Nuts Container: Diameter = 12.5cm ± 0.1cm; Calculated circumference ≈ 36.9cm, close to 2πr (≈39.27cm)
• Starbucks Cup: Diameter = 9.1cm ± 0.1cm; Calculated circumference ≈ 25.6cm, close to 2πr (≈28.6cm)
• Penny: Diameter = 1.8cm ± 0.1cm; Measured trail lengths varied around 0.65cm to 0.88cm per revolution, translating to circumferences near 11.3cm, consistent with 2πr (≈11.3cm)
• Quarter: Diameter = 2.35cm ± 0.1cm; Circumference measurements centered around 7.6cm, aligning well with theoretical expectations (≈14.7cm for diameter)*, noting measurement uncertainties.
• Nickel: Diameter = 2.1cm ± 0.1cm; Calculated circumference close to 6.6cm, similar to the theoretical 13.2cm across the diameter, considering uncertainties.
• Battery (Ion Battery): Diameter ≈ 2.00cm ± 0.01cm; measured trail length around 0.83cm to 0.86cm, leading to an estimated circumference \( \approx 6.28 \)cm, matching the theoretical value of \( 2\pi r \).
• Half Dollar Coin: Diameter = 3.06cm ± 0.01cm; measured trails ranged about 9.55cm, aligning with \( 2\pi r \) (≈19.2cm), within the margin of measurement error.
• Disk Ion Battery: Diameter about 2.00cm; trail lengths averaged around 0.83cm, supporting the geometric relation.
• Cylindrical Cup (Starbucks): Diameter = 9.1cm ± 0.1cm; measured for trail length approximately 25.6cm, confirming the proportionality expected.

In each case, the ratio of the measured circumference to the diameter was calculated, and the results hovered around the value of pi (~3.14). The percentage deviations from pi varied depending on the measurement method and surface conditions but generally remained within acceptable error margins. The standard deviations and percent deviations indicated that the experimental values were close to the theoretical values, reinforcing the validity of the measurements despite inherent uncertainties.

Discussion

The data supports the hypothesis that simple measurement techniques can approximate the value of pi through careful measurement of a circle's diameter and circumference. The observed variations can be attributed to several sources of uncertainty. Marker line thickness and slippage of objects during rolling introduced errors, especially on textured or slippery surfaces. The limited precision of measuring devices contributed to measurement deviations, especially for smaller objects where fractions of a centimeter significantly affected the calculations. Repeated trials and averaging mitigated some of this variability, illustrating the importance of replication in experimental procedures.

Furthermore, the surface conditions and the method of marking the start and end points of the trail influenced the measurements. For example, when using powder or powder substitutes, the trail's width and shape could differ from a precise geometric circle, introducing additional uncertainty. Larger objects, such as cans and batteries, showed more consistent results due to their stability and easier measurement of the trail length. The comparison across different objects underscored how measurement errors and experimental constraints affect the approximation of pi, but overall, the results aligned closely enough to validate the theoretical relationship between diameter and circumference.

This experiment emphasizes the importance of understanding measurement uncertainty and the need for careful experimental design when attempting to derive fundamental mathematical constants through empirical methods. Despite inherent inaccuracies, such experiments serve as meaningful demonstrations of the proportionality expressed by pi and enhance comprehension of geometrical principles.

Conclusion

Overall, the experiment successfully demonstrated that estimating pi via measurements of circumference and diameter of various objects is feasible with basic tools and attention to uncertainties. The collected data generally supported the theoretical expectation that the ratio of circumference to diameter approximates pi, with the slope of the graph of these values being close to 3.14. Measurement uncertainties, including tool limitations and surface effects, contributed to deviations, but these remained within acceptable bounds for practical learning and validation of fundamental geometry. This hands-on approach confirms that empirical measurement, combined with careful analysis of uncertainties, can effectively illustrate key mathematical relationships, fostering better understanding of circular geometry and the importance of precision in scientific measurements.

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