You may also like

problem icon

Pent

The diagram shows a regular pentagon with sides of unit length. Find all the angles in the diagram. Prove that the quadrilateral shown in red is a rhombus.

problem icon

Pentakite

ABCDE is a regular pentagon of side length one unit. BC produced meets ED produced at F. Show that triangle CDF is congruent to triangle EDB. Find the length of BE.

problem icon

Golden Mathematics

A voyage of discovery through a sequence of challenges exploring properties of the Golden Ratio and Fibonacci numbers.

Pentabuild

Stage: 5 Challenge Level: Challenge Level:2 Challenge Level:2

regular pentagon
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.
Full Screen Version
This text is usually replaced by the Flash movie.
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?