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Herschel from the European School of Varese
sent us this solution:
The first flower has 3 petals corresponding to the 3 corners of the
triangle. The completed animation shows that each petal is a
semicircle, so the perimeter of the flower is $3\times \pi \times
\text{ radius } = 3 \times \pi \times \text{ (side of triangle)
}$.
The second flower has 4 petals. This time, each petal is a sector
of a circle rather than a simple semicircle. The angle of this
sector is 360 - (2 triangle corners) - (1 square corner) = $360 - 2
\times 60 - 90 = 150^\circ$.
Therefore, the total perimeter of this petal is $4 \times
\frac{150}{360}\times (2\times \pi \times \text{ radius })
=\frac{10}{3} \times \pi \times \text{ (side of the
square)}$.
In general, we need to know 3 key bits of data to work out the
perimeter of the flower.
They are:
- The number of sides of the central shape; we'll call this
$n$.
- The length of each side in the central shape; we'll call this
$r$. (Note that this is equal to the radius of the
petals).
- The angle at the centre of each petal. This can be derived from
$n$:
$\text{Angle } = 360 - 2 \times 60 - \text{( Corner of
shape)}$
$\text{Angle } = 360 - 120 - \frac{180(n-2)}{n}$
$\text{Angle } = 240 - 180 - \frac{360}{n}$
$\text{Angle } = 60 + \frac{360}{n}$
Given these data, we can proceed to work out a general
formula:
Perimeter= (number of petals) $\times$ (perimeter of a
full circle) $\times \frac{\text{angle at centre of
petal}}{360}$
$\text{Perimeter }= n \times 2 \times \pi \times r \times
\frac{(60+ \frac{360}{n})}{360}$
$\text{Perimeter }= 2 \times \pi \times n \times r \times
(\frac{1}{6}+\frac{1}{n})$
$\text{Perimeter }= 2 \times \pi \times n \times r \times
\frac{6+n}{6n}$
$\text{Perimeter }= \pi \times r \times \frac{6+n}{3}$
Using this formula, we find the following results:
$n=3$ (Triangle): Perimeter = $\pi \times r \times
\frac{9}{3} = 3 \pi r$
$n=4$ (Square): Perimeter = $\pi \times r \times
\frac{10}{3}$
$n=5$ (Pentagon): Perimeter = $\pi \times r \times
\frac{11}{3}$
$n=6$ (Hexagon): Perimeter = $\pi \times r \times
\frac{12}{3} = 4\pi r$
$n=7$ (Heptagon): Perimeter = $\pi \times r \times
\frac{13}{3}$
$n=8$ (Octagon): Perimeter = $\pi \times r \times
\frac{14}{3}$
...
$n=100$: Perimeter = $\pi \times r \times \frac{106}{3}$
So a shape with 100 sides will produce a flower with a
perimeter of $\pi \times r \times \frac{106}{3}$.
If each edge of the central shape has a length of 1, the
perimeter of the flower will be $35.333 \times \pi$, which is
111.00 to two decimal places.
Well done to Saif from Havering Sixth Form
College, Nina, Jure and Kristjan from Elementary school Loka
Crnomelj, Slovenia, Chi from Raynes Park, Rajeev from
Haberdashers' Aske's Boys' School, Yun Seok Kang, and
Cameron, who also sent in correct solutions. Click here to read Nina, Jure and Kristjan's
thoughts.