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```Perimeter and area -Pick’s Theroem
The Royal Institution of Great Britain Mathematics Masterclasses www.rigb.org
Further ideas
Overview
The relation between the perimeter of closed curves (in a 2d plane) and the area enclosed by
them is not straightforward. There are two classic results,
I. For planar regions of fixed area (say 1 square unit), the circle is the shape with minimum
perimeter.
II. On the other hand, one can find closed curves (e.g. non-convex - star-shaped or zig-zag ones)
with unit area but perimeter as large as desired.
But there is much to be explored between these two extremes. First off, one can distinguish
between two cases, one - where the curves are drawn/cut in rigid material (think of openings in
walls for doors and windows, metal buckles and washers), two – where the curve is flexible
(holes cut in paper or rubber, loops made of string).
The Rigid World
In the rigid category there are some interesting problems –
1. What is the longest pencil/stick that can be placed within a given rectangle, or 3d equivalents
– what is the largest framed picture, table, chest, ... that can be moved through a given door,
corridor or staircase? Even trained mathematicians can have a much poorer intuition than
experienced movers when it comes to the latter problems.
At a basic level, the above considerations lead to the concepts of the diagonal and diameter (for
general closed curves not just a circle – more on this below). Problems relating to moving a
cupboard around an L-shaped corridor need a higher understanding of geometry (there are
related optimisation problems).
2. What is the smallest area in which you can reverse a given car? What about reversing a
needle? This is an area of considerable research! Explore ‘the Kakeya Problem’
http://en.wikipedia.org/wiki/Kakeya_conjecture#Kakeya_conjecture
3a. Why is it recommended that manhole covers be circular?
3b. Why are the UK 20p and 50p coins shaped the way they are? Explore ‘curves of constant
width’ http://en.wikipedia.org/wiki/Curve_of_constant_width
4. Can you make a hole in a solid cube so that a cube the same size (or even slightly larger) can
pass through the hole?
Supple Shapes
1. There is a beautiful historic problem in this category – the founding of
the city of Carthage (now in Tunisia) by Queen Dido circa 800 BC. When
princess Elissa, refugee from Tyre, landed on the Libyan coast and asked
the local chieftains for land to settle in, she was told she could have as
Further questions? Contact [email protected]
Perimeter and area -Pick’s Theroem
The Royal Institution of Great Britain Mathematics Masterclasses www.rigb.org
much land as she could encompass using the hide of an ox. Elissa is supposed to have cut the
hide into thin strips, joined them and formed a semi-circle and, using the river bank as a side,
staked out a D-shaped region – to become the city of Carthage. Students can be given the
original open-ended problem or a more structured one, and their solutions can provide a
starting point for a discussion on area and perimeter.
2. Is it possible to cut a hole in a 1p sized circle so that a 2p coin can go through it? Is it possible
for you to cut a hole in an A4 size sheet of paper and walk right through that hole? If no, explain
why. If yes, cut such a hole and demonstrate your solution.
Piece-wise linear curves
There are some interesting relationships, and nice opportunities for learners to investigate, if
one restricts oneself to polygons, especially ones where the vertices have to lie on a fixed
square grid (lattice).
1. As a start consider the following problem - draw a variety of shapes formed by attaching 16
different unit squares. The region has to be connected and two squares that meet must share a
complete edge, not just a vertex. Further, the region cannot have any holes. Students need to
calculate the (external) perimeter for each of the regions they have. They should notice that a
square has minimal perimeter and there is some ‘order’ to the other perimeters, even those of
‘twisty’ shapes. Can they explain any observations or put forward some conjectures.
2. The above leads to Pick’s Theorem related investigations (see separate sheet). Note that in
this case one is allowed to have triangles and other polygons, as long as vertices lie on dots in a
given square grid. The goal is to find a simple formula linking the dots on the perimeter of a
polygon, the dots in the interior, and the area of the polygon.
One should note that this does not give a direct relation between perimeter and area, as using
certain triangles, one can have polygons with half-unit area and arbitrarily large perimeter,
though with only three dots involved.1
One of the interesting aspects of the Pick’s Theorem investigation is that there are three
variables (one can think of the ‘dots on the perimeter’ and ‘dots inside’ as independent variables
with the area depending on them) and one can use it as an example for communicating the
power of organised work. Students can find relationships using random collections of polygons
(it is quite educative to see/discuss the set of polygons a student or group of students use) or
use structured approaches. One option is to keep one of the variables constant (either ‘dots on
the perimeter’ or ‘dots inside’), increase the other in regular increments, and look for some
patterns/relations.
1
There is the separate yet related question of ‘what is the longest piece-wise straight curve that you can
draw, where you are only allowed to turn at dots that are on a given finite square grid?
Further questions? Contact [email protected]
Perimeter and area -Pick’s Theroem
The Royal Institution of Great Britain Mathematics Masterclasses www.rigb.org
An underlying idea that is worth getting across is that area increases as the square of ‘diameter’
for squares and rectangles, and if one keeps width constant, then area increases linearly with
length.
Fractal curves
Given the notion that one can have finite area with arbitrarily long perimeter, one can put a
further restriction, that the examples not just have thin long extensions, but be within a
bounded region (say rectangle). This can lead to the idea of fine zig-zagging boundaries, or
independently, one can ask the question ‘how long is the coast of Britain’? This beautiful
historical problem led to the study of fractal curves and there is much educational material here.
There are also worksheets and video-excerpts from the 2006 Ri Christmas Lectures.
http://en.wikipedia.org/wiki/How_Long_Is_the_Coast_of_Britain%3F_Statistical_SelfSimilarity_and_Fractional_Dimension
http://en.wikipedia.org/wiki/Fractal
http://www.rigb.org/christmaslectures06/20.html
Further questions? Contact [email protected]
```