Remember that you want someone following behind you to see where
you went. Can yo work out how these patterns were created and
Polygons drawn on square dotty paper have dots on their perimeter (p) and often internal (i) ones as well. Find a relationship between p, i and the area of the polygons.
Can you prove that the sum of the distances of any point inside a
square from its sides is always equal (half the perimeter)? Can you
prove it to be true for a rectangle or a hexagon?
Ned, who has just left Christ Church
Cathedral School in Oxford, sent a nice solution saying
that: ``The Star Gazing problem is easy. The 6-point star is
made up of 6 equilateral triangles, and so is the hexagon inside
it, so the ratio of the `points' to the `inside' is 1:1''.
Ned then found the area of the octagon
by dividing it up (a square, 4 rectangles and 4 triangles) and used
Pythagoras theorem to find the lengths needed and finally found the
ratio of the areas of the `points' to the `octagon inside' to be 1:
(1 + $\sqrt 2$).
The second part is most easily done by taking the shorter sides
of the triangles which make up the star to have length 1 unit, so
that the area of each triangle is 1/2 and the total area of the
`points' is 4 square units . The hypotenuse of each triangle has
length$\sqrt 2$ (by Pythagoras Theorem) which gives the
lengths of the edges of the octagon. To find the area of the
octagon take the square and chop off four triangles at the
The length of the side of the square is (2 + $\sqrt 2$)
the area of the octagon = (2 + $\sqrt 2)^2$ - 2
which simplifies to 4 + 4$\sqrt 2$.
Finally the ratio of areas is 4 : (4 + 4$\sqrt 2$) which
simplifies to 1 : (1 + $\sqrt 2$ ).