Copyright © University of Cambridge. All rights reserved.
'Pentabuild' printed from http://nrich.maths.org/
Show menu
|
The film strip shows the steps in the construction of the
regular pentagon.
Copy this straight edge and compass construction for yourself
and explain why it produces a regular pentagon.
The description of the construction below, and the information
in the notes, should help you to explain the construction.
|
Here are the steps shown in the film sequence:
1. Draw a circle $C_1$ centre $O$ diameter $PQ$.
The circle $C_1$ has radius
1 unit what is its equation?
2. Draw the perpendicular bisector of $PQ$ cutting $PQ$ at $O$
and $C_1$ at $A$ and $Y$.
3. Draw perpendicular bisectors of $PO$ and $OQ$ cutting $PQ$
at $R$ and $S$.
Find the length
$YS$
4. Draw circles $C_2$ and $C_3$ centres $R$ and $S$ and radii
$RO$ and $SO$.
5. Join $R$ and $S$ to the point $Y$ cutting $C_2$ at $T$ and
$U$ and $C_3$ at $V$ and $W$.
6. Draw circle $C_4$ centre $Y$ radius $YW=YU$ cutting $C_1$
at $D$ and $C$.
What is the equation of
$C_4$? Find the value of $y$ at the intersection of $C_1$ and
$C_4$ .
7. Draw circle $C_5$ centre $Y$ radius $YT=YV$ cutting $C_1$
at $E$ and $B$.
What is the equation of
$C_5$ ?
Find the value of $y$ at the intersection of $C_1$ and
$C_5$.
At $B$ and $E$ $x^2 + y^2 +2y +1 = 2y + 2 = (3 + \sqrt 5)/2$
so
8. Join $AB$, $BC$, $CD$, $DE$, $EA$.
How would you adapt this construction to produce a regular
decagon?