**Question Corner and Discussion Area**

I want to know how to construct a pentagon. I have done it before, but I have forgotten how. I remember it is similar to constructing a hexagon, but a bit more difficult. Thanks.There are several ways to do it. Unfortunately we are very short-staffed right now and cannot spare the resources to hunt down the easiest and most elegant construction. However, the following method will work:

Constructing a pentagon is equivalent to dividing a circle (a full 360 degrees) up into five equal parts (angle 72 degrees each). The cosine of 72 degrees is (this can be found by starting with the equation , using trigonometric identities to write as a polynomial in , factoring and solving the resulting polynomial equation for ).

Therefore, this angle of 72 degrees can be constructed by building a right-angled triangle whose hypotenuse is 4 and whose adjacent side is of length . This latter length can be constructed by taking hypotenuse of a right triangle whose other sides have lengths 1 and 2, and subtracting length 1 from it.

The following procedure uses this idea to construct a pentagon:

Start with a circle *C*, with centre point *O*. Let *P* be a point on *C*.
Draw the perpendicular bisector *L* to segment *OP* (bisecting it at point
*Q*). Construct the midpoint *R* of *OQ*. (*RQ* is going to be our unit
length).

With centre *Q* and radius *RQ*, draw an arc intersecting *L* at point
*S*. Draw segment *OS*.
(This is the hypotenuse of a right triangle *OQS* whose other sides have
length 1 and 2, so *OS* has length ).

With centre *S* and radius *RQ* (= *QS*), draw an arc intersecting *OS* at point
*T*. (Now *OT* has length ).

Construct the line passing through point *T* at right angles to *OT*.
Let it intersect the circle *C* at point *U*.

Now the triangle *OTU* has hypotenuse of length *OU* = radius of *C* = 4,
and side length . Therefore, angle *UOT* is 72 degrees.
Extend segment *OT* past *S* until it meets the circle *C* at point *V*;
you have now constructed two vertices (*U* and *V*) of the pentagon.

To construct the remaining vertices: with centre *V* and radius *UV* draw
an arc intersecting *C* at point *W*. With centre *W* and the same radius,
draw an arc intersecting *C* at point *X*. Finally, with centre *X*
and the same radius, draw an arc intersecting *C* at point *Y*.
*UVWXY* will be a pentagon.

There are probably much more efficient ways to do it, but the above procedure will certainly work, for the reasons described. The procedure is illustrated below:

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