A Circle and its 359 359 Sectors

Geometry Level 3

Jason was told to bring one circular paper with a radius of 1 1 dm and 359 359 pieces of paper. In those 359 359 papers, there must be one that is 1 360 \frac{1}{360} of the area of the aforementioned circular paper, there must be 2 360 \frac{2}{360} of the area, 3 360 \frac{3}{360} , 4 360 \frac{4}{360} , and so on until there must be a piece of paper that is 359 360 \frac{359}{360} of the area of the circular paper.

What is the exact area (in square dm) of paper Jason will be needing?

Note: The answer should be in the nearest thousandths.


The answer is 567.057.

Kaizen Cyrus by Kaizen Cyrus , 18, Philippines

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1 solution

Jordan Cahn
Nov 1, 2018

Note that the full circle Jason brings in is simply 360 360 \frac{360}{360} of a circle. Since the circle has radius 1 dm 1\text{ dm} , it has area π dm 2 \pi\text{ dm}^2 . Thus we need to compute ∑ n = 1 360 n 360 π = π 360 ∑ n = 1 360 n = π 360 ⋅ 360 ⋅ 361 2 = 361 π 2 ≈ 567.057 dm 2 \begin{aligned} \sum_{n=1}^{360}\frac{n}{360}\pi &= \frac{\pi}{360} \sum_{n=1}^{360} n \\ &= \frac{\pi}{360} \cdot \frac{360\cdot 361}{2} \\ &= \frac{361\pi}{2} \approx \boxed{567.057 \text{ dm}^2} \end{aligned}

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